% Copyright 2010 by Till Tantau
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\section{Axes}
\label{section-dv-axes}


\subsection{Overview}

When a data point is visualized, the most obvious way of creating a
visual representation of its many attributes is to
vary \emph{where} the data point is shown. The data visualization
system uses \emph{axes} to turn data point attributes into positions
on a page. The simplest -- and most common -- use of axes is to vary
the horizontal position of data points according to one attribute and
to vary the vertical position according to another attribute. In
contrast, in a polar plot one attribute dictates the distance of the
data point from the origin and another attribute describes the
angle. From the data visualization engine's point of view, in both
cases two \emph{axes} are involved.

In addition to specifying how the value of a certain attribute is
converted into a displacement on the page, an axis is also typically
(but not always) visualized (``drawn'') somewhere on the page. In this case, it is
also customary to add a visual representation on this axis of which
attribute values correspond to which positions on the page --
something commonly known as \emph{ticks}. Similar to ticks, \emph{grid
  lines} also indicate positions where a certain attribute has a
certain value, but instead of just indicating a single position on an
axis, a grid line goes through all points that share an attribute
value.

In the following, in Section~\ref{section-dv-axes-main} we first have
a look at how axes can be defined and configured. As you will see, a
lot of powerful configurations are available, but you will rarely
define and configure an axis from scratch. Rather, it is more common
to use a preconfigured axis
instead. Section~\ref{section-dv-axis-systems} introduces \emph{axis
  systems}, which are predefined bundles of axes. You can define your
own axis systems, but, again, in most cases it will suffice to just
use one of the many preconfigured axis systems and use a few options
to configure it so that it fits your
need. Section~\ref{section-dv-ticks-and-grids} explains how ticks and
grid lines can be configured. Again, several layers of
options allow you to configure the way ticks look and where they are
placed in great detail.

This section documents the standard axis systems that are always
available. For polar axis systems, a special library needs to be
loaded, which is documented in Section~\ref{section-dv-polar}.



\subsection{Basic Configuration of Axes}
\label{section-dv-axes-main}

Inside the data visualization system, an \emph{axis} is roughly a
``systematic, named way of mapping an attribute to a 
position on a page''. For instance, the classical ``$x$-axis'' is the
``systematic way of mapping the value of the |x| attribute of data
points to a horizontal position on the page''. An axis is \emph{not}
its visual representation (such as the horizontal line with the ticks
drawn to represent the $x$-axis), but a visual representation can be
created once an axis has been defined.

The transformation of an attribute value (such as the value |1000000000|
for the |x| attribute) to a specific displacement of the corresponding
data point on the page involves two steps:
\begin{enumerate}
\item First, the range of possible values such as $[-5.6\cdot
  10^{12},7.8\cdot 10^{12}]$ must be mapped to a ``reasonable''
  interval such as $[0\mathrm{cm},5\mathrm{cm}]$ or
  $[0^\circ,180^\circ]$. \tikzname's drawing routines will only be
  able to cope with values from such a ``reasonable'' interval.
\item Second, the values from the reasonable interval must be mapped
  to a transformation.
\end{enumerate}
The first step is always the same for all axes, while the second
requires different strategies. For this reason, the command
|new axis base| is used to create a ``basic'' axis that has a
``scaling mapper'', whose job it is to map the range of values of a
specific attribute to a reasonable interval, but such a basic axis
does not define an actual transformation object. For this second step,
additional objects such as a |linear transformer| need to be created
separately. 


\subsubsection{Usage}

To create an axis, the key |new axis base| is used first. Since this key
does not create a transformation object, users typically do not use
this key directly. Rather, it is used internally by other keys
that create ``real'' axes. These keys are listed in
Section~\ref{section-dv-reference-axis-types}. 

\begin{key}{/tikz/data visualization/new axis base=\meta{axis name}}
  This key defines a new axis for the current data visualization
  called \meta{name}. This has two effects:
  \begin{enumerate}
  \item A so called \emph{scaling mapper} is created that will monitor
    a certain attribute, rescale it, and map it to another
    attribute. (This will be explained in detail in a moment.)
  \item The \meta{axis name} is made available as a key that can be
    used to configure the axis:
    \begin{key}{/tikz/data visualization/\meta{axis name}=\meta{options}}
      This key becomes available once |new axis base=|meta{axis name} has
      been called. It will execute the \meta{options} with the path
      prefix |/tikz/data visualization/axis options|.
\begin{codeexample}[code only]
[new axis base=my axis,
 my axis={attribute=some attribute}]
\end{codeexample}
    \end{key}
  \item The \meta{axis name} becomes part of the current set of
    axes. This set can be accessed through the following key:
    \begin{key}{/tikz/data visualization/all axes=\meta{options}}
      This key passes the \meta{options} to all axes inside the
      current scope, just as if you had written \meta{some axis
        name}|=|\meta{options} for each \meta{some axis name} in the
      current scope, including the just-created name \meta{axis name}.
    \end{key}
  \end{enumerate}
  There are many \meta{options} that can be passed to a newly created
  axis. They are explained in the rest of this section.
\end{key}

Note the |new axis base| does \emph{not} cause attributes to be mapped to
positions on a page. Rather, special keys like |new Cartesian axis|
first use |new axis base| to create an axis and then create an internal
object that performs a linear mapping of the attribute to 
positions along a vectors.


\subsubsection{The Axis Attribute}
\label{section-dv-axis-attribute}

The first main job of an axis is to map the different values of some
attribute to a reasonable interval. To achieve this, the following
options are important (recall that these options are passed to the key
whose name is the name of the axis):

\begin{key}{/tikz/data visualization/axis options/attribute=\meta{attribute}}
  Specifies that the axis is used to transform the data points
  according the different values of the key
  |/data point/|\meta{attribute}. For instance, when we create a
  classical two-dimensional Cartesian coordinate system, then there
  are two axes called |x axis| and |y axis| that monitor the values of
  the attributes |/data point/x| and |/data point/y|, respectively:
\begin{codeexample}[code only]
  [new axis base=x axis,
   new axis base=y axis,
   x axis={attribute=x},
   y axis={attribute=y}]
\end{codeexample}
  In another example, we also create an |x axis| and a
  |y axis|. However, this time, we want to plot the values of the
  |/data point/time| attribute on the $x$-axis and, say, the value of
  the |height| attribute on the $y$-axis:
\begin{codeexample}[code only]
  [new axis base=x axis,
   new axis base=y axis,
   x axis={attribute=time},
   y axis={attribute=height}]
\end{codeexample}
  During the data visualization, the \meta{attribute} will be
  ``monitored'' during the survey phase. This means that for each data
  point, the current value of |/data point/|\meta{attribute} is
  examined and the minimum value of all of these values as well as the
  maximum value is recorded internally. Note that this works even when
  very large numbers like |100000000000| are involved.

  Here is a real-life example. The |scientific axes| create two axes,
  called |x axis| and |y axis|, respectively.
\begin{codeexample}[]
\tikz \datavisualization [scientific axes,
                          x axis={attribute=people, length=2.5cm, ticks=few},
                          y axis={attribute=year},
                          visualize as scatter]
  data {
    year, people
    1900, 100
    1910, 200
    1950, 200
    1960, 250
    2000, 150
  };
\end{codeexample}
\end{key}


\subsubsection{The Axis Attribute Range Interval}

Once an attribute has been specified for an axis, the data
visualization engine will start monitoring this value. This means that
before anything actual visualization is done, a ``survey phase'' is
used to determine the range of values encountered for the attribute
for all data points. This range of values results in what is called
the \emph{attribute range interval}. Its minimum is the smallest value
encountered in the data and its maximum is the largest value.

Even though the attribute range interval is computed automatically and
even though you typically do not need to worry about it, there are
some situations where you may wish to set or enlarge the attribute
range interval:
\begin{itemize}
\item You may wish to start the interval with $0$, even though the
  range of values contains only positive values.
\item You may wish to slightly enlarge the interval so that, say, the
  maximum is some ``nice'' value like |100| or |60|.
\end{itemize}

The following keys can be used to influence the size of the attribute
range interval:
\begin{key}{/tikz/data visualization/axis options/include
    value=\meta{list of value}}
  This key ``fakes'' data points for which the attribute's values are
  in the comma-separated \meta{list of values}. For instance, when you
  write |include value=0|, then the attribute range interval is
  guaranteed to contain |0| -- even if the actual data points are all
  positive or all negative.
\begin{codeexample}[]
\tikz \datavisualization [scientific axes, all axes={length=3cm},
                          visualize as line]
  data [format=function] {
    var x : interval [5:10];
    func y = \value x * \value x;
  };
\end{codeexample}  
\begin{codeexample}[]
\tikz \datavisualization [scientific axes, all axes={length=3cm},
                          visualize as line,
                          x axis={include value=20},
                          y axis={include value=0}]
  data [format=function] {
    var x : interval [5:10];
    func y = \value x * \value x;
  };
\end{codeexample}  
\end{key}

\begin{key}{/tikz/data visualization/axis options/min value=\meta{value}}
  This key allows you to simply set the minimum value, regardless of
  which values are present in the actual data. This key should be used
  with care: If there are data points for which the attribute's value
  is less than \meta{value}, they will still be depicted, but
  typically outside the normal visualization area. Usually, saying
  |include value=|\meta{value} will achieve the same as saying
  |min value=|\meta{value}, but with less danger of creating
  ill-formed visualizations.
\end{key}

\begin{key}{/tikz/data visualization/axis options/max value=\meta{value}}
  Works like |min value|.
\end{key}


\subsubsection{Scaling: The General Mechanism}

The above key allows us specify which attribute should be
``monitored''. The next key is used to specify what should happen with
the observed values.

\begin{key}{/tikz/data visualization/axis options/scaling=\meta{scaling spec}}
  The \meta{scaling spec} must have the following form:
  \begin{quote}
    \meta{$s_1$}| at |\meta{$t_1$}| and |\meta{$s_2$}| at |\meta{$t_2$}
  \end{quote}
  This means that monitored values in the interval $[s_1,s_2]$ should
  be mapped to values the ``reasonable'' interval $[t_1,t_2]$,
  instead. For instance, we might write
\begin{codeexample}[code only]
[y axis = {scaling = 1900 at 0cm and 2000 at 5cm}]
\end{codeexample}
  in order to map dates between 1900 and 2000 to the dimension
  interval $[0\mathrm{cm},5\mathrm{cm}]$.
\begin{codeexample}[]
\tikz \datavisualization
   [scientific axes,
    x axis={attribute=people, length=2.5cm, ticks=few},
    y axis={attribute=year, scaling=1900 at 0cm and 2000 at 5cm},
    visualize as scatter]
  data {
    year, people
    1900, 100
    1910, 200
    1950, 200
    1960, 250
    2000, 150
  };
\end{codeexample}
  So much for the basic idea. Let us now have a detailed look at what
  happens.

  \medskip
  \textbf{Number format and the min and max keywords.}
  The source values $s_1$ and $s_2$ are typically just numbers like
  |3.14| or |10000000000|. However, as described in
  Section~\ref{section-dv-expressions}, you can also specify
  expressions like |(pi/2)|, provided that (currently) you put them in
  parentheses.

  Instead of a number, you may alternatively also use the two key
  words |min| and |max| for $s_1$ and/or $s_2$. In this case, |min|
  evaluates to the smallest value observed for the attribute in the
  data, symmetrically |max| evaluates to the largest values. For
  instance, in the above example with the |year| attribute ranging
  from |1900| to |2000|, the keyword |min| would stand for |1900| and
  |max| for |2000|. Similarly, for the |people| attribute |min| stands
  for |100| and |max| for |250|. Note that |min| and |max| can only be
  used for $s_1$ and $s_2$, not for $t_1$ and $t_2$.

  A typical use of the |min| and |max| keywords is to say
\begin{codeexample}[code only]
scaling = min at 0cm and max at 5cm    
\end{codeexample}
  to map the complete range of values into an interval of length of
  5cm. 

  The interval  $[s_1,s_2]$ need not contain all values that the
  \meta{attribute} may attain. It is permissible that values are less
  than $s_1$ or more than $s_2$.
  
  \medskip
  \textbf{Linear transformation of the attribute.}
  As indicated earlier, the main job of an axis is to map values from
  a ``large'' interval $[s_1,s_2]$ to a more reasonable interval
  $[t_1,t_2]$. Suppose
  that for the current data point the value of the key
  |/data point/|\meta{attribute} is the number $v$. In the simplest 
  case, the following happens: A new value $v'$ is computed so that
  $v' = t_1$ when $v=s_1$ and $v'=t_2$ when $v=s_2$ and $v'$ is some
  value in between $t_1$ and $t_2$ then $v$ is some value in between
  $s_1$ and $s_2$. (Formally, in this basic case $v' = t_1 +
  (v-s_1)\frac{t_2-t_1}{s_2-s_1}$.)

  Once $v'$ has been computed, it is stored in the key
  |/data point/|\meta{attribute}|/scaled|. Thus, the ``reasonable''
  value $v'$ does not replace the value of the attribute, but it is
  placed in a different key. This means that both the original value
  and the more ``scaled'' values are available when the data point is
  visualized.

  As an example, suppose you have written
\begin{codeexample}[code only]
[x axis = {attribute = x, scaling=1000 at 20 and 2000 at 30}]
\end{codeexample}
  Now suppose that |/data point/x| equals |1200| for a data
  point. Then the key |/data point/x/scaled| will be set to |22| when
  the data point is being visualized.
  
  \medskip
  \textbf{Nonlinear transformations of the attribute.}
  By default, the transformation of $[s_1,s_2]$ to $[t_1,t_2]$ is the
  linear transformation described above. However, in some case you may
  be interested in a different kind of transformation: For example, in
  a logarithmic plot, values of an attribute may range between, say,
  |1| and |1000| and we want an axis of length |3cm|. So, we would
  write
  \begin{codeexample}[code only]
[x axis = {attribute = x, scaling=1 at 0cm and 1000 at 3cm}]
  \end{codeexample}
  Indeed, |1| will now be mapped to position |0cm| and |1000| will be
  mapped to position |3cm|. Now, the value |10| will be mapped to
  approximately |0.03cm| because it is (almost) at one percent between
  |1| and |1000|. However, in a logarithmic plot we actually want |10|
  to be mapped to the position |1cm| rather than |0.03cm| and we want
  |100| to be mapped to the position |2cm|. Such a mapping a
  \emph{nonlinear} mapping between the intervals.

  In order to achieve such a nonlinear mapping, the |function| key can
  be used, whose syntax is described in a moment. The effect of this key is
  to specify a function $f \colon \mathbb R \to \mathbb R$ like, say,
  the logarithm function. When such a function is specified, the
  mapping of $v$ to $v'$ is computed as follows:
  \begin{align*}
    v' = t_1 + (f(s_2) - f(v))\frac{t_2 - t_1}{f(s_2)-f(s_1)}.
  \end{align*}
  
  The syntax of the |function| key is described next, but you
  typically will not call this key directly. Rather, you will use a
  key like |logarithmic| that installs appropriate code for the
  |function| key for you.
  \begin{key}{/tikz/data visualization/axis options/function=\meta{code}}
    The \meta{code} should specify a function $f$ that is applied
    during the transformation of the interval $[s_1,s_2]$ to the
    interval $[t_1,t_2]$ in the following way: When the \meta{code} is
    called, the macro |\pgfvalue| will have been set to an internal
    representation of the to-be-transformed value~$v$. You can then
    call the commands of the math-micro-kernel of the data
    visualization system, see Section~\ref{section-dv-math-kernel}, to
    compute a new value. This new value must once more be stored in
    |\pgfvalue|.

    The most common use of this key is to say
\begin{codeexample}[code only]
some axis={function=\pgfdvmathln{\pgfvalue}{\pgfvalue}} 
\end{codeexample}
    This specifies that the function $f$ is the logarithm
    function.
\begin{codeexample}[]
\tikz \datavisualization
   [scientific axes,
    x axis={ticks={major={at={1,10,100,1000}}},
             scaling=1 at 0cm and 1000 at 3cm,
             function=\pgfdvmathln{\pgfvalue}{\pgfvalue}},
    visualize as scatter]
  data [format=named] {
    x={1,100,...,1000}, y={1,2,3}
  };
\end{codeexample}
    Another possibility might be to use the square-root function, instead:
\begin{codeexample}[]
\tikz \datavisualization
   [scientific axes,
    x axis={ticks=few,
            scaling=1 at 0cm and 1000 at 3cm,
            function=\pgfdvmathunaryop{\pgfvalue}{sqrt}{\pgfvalue}},
    visualize as scatter]
  data [format=named] {
    x={0,100,...,1000}, y={1,2,3}
  };
\end{codeexample}
  \end{key}
  
  \medskip
  \textbf{Default scaling.}
  When no scaling is specified, it may seem natural to use $[0,1]$
  both as the source and the target interval. However, this would not
  work when the logarithm function is used as transformations: In this
  case the logarithm of zero would be computed, leading to an
  error. Indeed, for a logarithmic axis it is far more natural to use
  $[1,10]$ as the source interval and $[0,1]$ as the target interval.

  For these reasons, the default value for the |scaling| that is used
  when no value is specified explicitly can be set using a special
  key:   
  \begin{key}{/tikz/data visualization/axis options/scaling/default=\meta{text}}
    The \meta{text} is used as |scaling| whenever no other scaling is
    specified. This key is mainly used when a transformation function
    is set using |function|; normally, you will not use this key
    directly. 
  \end{key}
\end{key}

Most of the time, you will not use neither the |scaling| nor the
|function| key directly, but rather you will use one of the following
predefined styles documented in the following. 


\subsubsection{Scaling: Logarithmic Axes}

\begin{key}{/tikz/data visualization/axis options/logarithmic}
  When this key is used with an axis, three things happen:
  \begin{enumerate}
  \item The transformation |function| of the axis is setup to the
    logarithm.
  \item The strategy for automatically generating ticks and grid lines
    is set to the |exponential strategy|, see
    Section~\ref{section-dv-exponential-strategy} for details.
  \item The default scaling is setup sensibly.
  \end{enumerate}
  All told, to turn an axis into a logarithmic axis, you just need to
  add this option to the axis.
\begin{codeexample}[width=8cm]
\tikz \datavisualization [scientific axes, 
                          x axis={logarithmic},
                          y axis={logarithmic},
                          visualize as line]
 data [format=function] {
   var x : interval [0.01:100];
   func y = \value x * \value x;
 };
\end{codeexample}
  Note that this will work with any axis,
  including, say, the degrees on a polar axis:
\begin{codeexample}[]
\tikz \datavisualization
    [new polar axes, 
     angle axis={logarithmic, scaling=1 at 0 and 90 at 90},
     radius axis={scaling=0 at 0cm and 100 at 3cm},
     visualize as scatter]
  data [format=named] {
    angle={1,10,...,90}, radius={1,10,...,100}
  };
\end{codeexample}  
\begin{codeexample}[]
\tikz \datavisualization
    [new polar axes, 
     angle axis={degrees},
     radius axis={logarithmic, scaling=1 at 0cm and 100 at 3cm},
     visualize as scatter]
  data [format=named] {
    angle={1,10,...,90}, radius={1,10,...,100}
  };
\end{codeexample}  
\end{key}



\subsubsection{Scaling: Setting the Length or Unit Length}

\begin{key}{/tikz/data visualization/axis options/length=\meta{dimension}}
  Sets |scaling| to |min at 0cm and max at |\meta{dimension}. The
  effect is that the range of all values of the axis's attribute will
  be mapped to an interval of exact length \meta{dimension}.
  
\begin{codeexample}[]
\tikz \datavisualization [scientific axes,
                          x axis={length=3cm},
                          y axis={length=2cm},
                          all axes={ticks=few},
                          visualize as line]
    data {
      x, y
      10, 10
      20, 20
      15, 30
      13, 20
    };
\end{codeexample}
\begin{codeexample}[]
\tikz \datavisualization [scientific axes,
                          x axis={length=3cm},
                          y axis={length=4cm},
                          all axes={ticks=few},
                          visualize as line]
    data {
      x, y
      10, 10
      20, 20
      15, 30
      13, 20
    };
\end{codeexample}
  
\end{key}

\begin{key}{/tikz/data visualization/axis
    options/unit length=\meta{dimension}\opt{| per |\meta{number}| units|}}
  Sets |scaling| to |0 at 0cm and 1 at |\meta{dimension}. In other
  words, this key allows you to specify how long a single unit should
  be. This key is particularly useful when you wish to ensure that the
  same scaling is used across multiple axes or pictures.  
\begin{codeexample}[]
\tikz \datavisualization [scientific axes,
                          all axes={ticks=few, unit length=1mm},
                          visualize as line]
    data {
      x, y
      10, 10
      40, 20
      15, 30
      13, 20
    };
\end{codeexample}
  The optional |per |\meta{number}| units| allows you to apply more
  drastic scaling. Suppose that you want to plot a graph where one
  billion corresponds to one centimeter. Then the unit length would be
  need to be set to a hundredth of a nanometer -- much too small for
  \TeX\ to handle as a dimension. In this case, you can write
  |unit length=1cm per 1000000000 units|:
\begin{codeexample}[]
\tikz \datavisualization
  [scientific axes,
   x axis={unit length=1mm per 1000000000 units, ticks=few},
   visualize as line]
 data {
   x, y
   10000000000, 10
   40000000000, 20
   15000000000, 30
   13000000000, 20
 };
\end{codeexample}
\end{key}
\begin{key}{/tikz/data visualization/axis
    options/power unit length=\meta{dimension}}
  This key is used in conjunction with the |logarithmic| setting. It
  cases the |scaling| to be set to
  |1 at 0cm and 10 at |\meta{dimension}. This causes a ``power unit,''
  that is, one power of ten in a logarithmic plot, to get a length of
  \meta{dimension}. Again, this key is useful for ensuring that the
  same scaling is used across multiple axes or pictures.  
\begin{codeexample}[width=8cm]
\tikz \datavisualization
  [scientific axes, 
   y axis={logarithmic, power unit length=1mm, grid},
   visualize as line]
 data {
   x, y
   0, 0.0000000001
   1, 1
   2, 100000
   3, 100000000000
   4, 10000000000000000000000000000000
   5, 500000000
   6, 5000000000000000000
 };
\end{codeexample}
\end{key}



\subsubsection{Axis Label}

An axis can have a \emph{label}, which is a textual representation of
the attribute according to which the axis varies the position of the
page. You can set the attribute using the following key:

\begin{key}{/tikz/data visualization/axis
    options/label=\opt{|\char`\{[|\meta{options}|]|}\meta{text}\opt{|\char`\}|}
  (default \normalfont axis's label in math mode)}
  This key sets the label of an axis to \meta{text}. This text will
  typically be placed inside a |node| and the \meta{options} can be
  used to further configure the way this node is rendered. The
  \meta{options} will be executed with the path prefix
  |/tikz/data visualization/|, so you need to say |node style| to
  configure the styling of a node, see Section~\ref{section-dv-style}.
\begin{codeexample}[]
\tikz \datavisualization [
    scientific axes,
    x axis = {label, length=2.5cm},
    y axis = {label={[node style={fill=blue!20}]{$x^2$}}},
    visualize as smooth line]
 data [format=function] {
   var x : interval [-3:5];
   func y = \value x * \value x;
 };
\end{codeexample}  
\end{key}

Note that using the |label| key does not actually cause a node to be
created, because it is somewhat unclear where the label should be
placed. Instead, the |visualize label| key is used (typically
internally by an axis system) to show the label at some sensible
position. This key is documented in
Section~\ref{section-dv-visualize-label}. 



\subsubsection{Reference: Axis Types}

\label{section-dv-reference-axis-types}

As explained earlier, when you use |new axis base| to create a new axis,
a powerful scaling and attribute mapping mechanism is installed, but
no mapping of values to positions on the page is performed. For this,
a \emph{transformation object} must be installed. The following keys
take care of this for you. Note, however, that even these keys do not
cause a visual representation of the axis to be added to the
visualization -- this is the job of an axis system, see
Section~\ref{section-dv-axis-systems}. 

\begin{key}{/tikz/data visualization/new Cartesian axis=\meta{name}}
  This key creates a new ``Cartesian'' axis, named \meta{name}. For
  such an axis, the (scaled) values of the axis's attribute are
  transformed into a displacement on the page along a straight
  line. The following key is used to configure in which ``direction''
  the axis points:
  \begin{key}{/tikz/data visualization/axis options/unit
      vector=\meta{coordinate} (initially {(1pt,0pt)})}
    Recall that an axis takes the values of an attribute and rescales
    them so that they fit into a ``reasonable'' interval
    $[t_1,t_2]$. Suppose that $v'$ is the rescaled dimension in (\TeX)
    points. Then when the data point is visualized, the coordinate
    system will be shifted by $v'$ times the \meta{coordinate}.

    As an example, suppose that you have said
    |scaling=0 and 10pt and 50 and 20pt|. Then when the underlying
    attribute has the value |25|, it will be mapped to a $v'$ of $15$
    (because |25| lies in the middle of |0| and |50| and |15pt| lies
    in the middle of |10pt| and |20pt|). This, in turn, causes the
    data point to be displaced by $15$ times the \meta{coordinate}.

    The bottom line is that the \meta{coordinate} should usually
    denote a point that is at distance |1pt| from the origin and that
    points into the direction of the axis.
\begin{codeexample}[]
\begin{tikzpicture}
  \draw [help lines] (0,0) grid (3,2);

  \datavisualization
    [new Cartesian axis=x axis, x axis={attribute=x},
     new Cartesian axis=y axis, y axis={attribute=y},
     x axis={unit vector=(0:1pt)},
     y axis={unit vector=(60:1pt)},
     visualize as scatter]
  data {
    x, y
    0, 0
    1, 0
    2, 0
    1, 1
    2, 1
    1, 1.5
    2, 1.5
  };
\end{tikzpicture}
\end{codeexample}
\end{key}
\end{key}



\subsection{Axis Systems}
\label{section-dv-axis-systems}


An \emph{axis system} is, as the name suggests, a whole family of axes
that act in concert. For example, in the ``standard'' axis system there
is a horizontal axis called the $x$-axis that monitors the |x|
attribute (by default, you can change this easily) and a vertical axis
called the $y$-axis. Furthermore, a certain number of ticks are added
and labels are placed at sensible positions.

\subsubsection{Usage}

Using an axis system is usually pretty easy: You just specify a key
like |scientific axes| and the necessary axes get initialized with
sensible default values. You can then start to modify these default
values, if necessary.

First, you can (and should) set the attributes to which the difference 
axes refer. For instance, if the |time| attribute is plotted along the
$x$-axis, you would write
\begin{codeexample}[code only]
x axis = {attribute = time}
\end{codeexample}

Second, you may wish to modify the lengths of the axes. For this, you
can use keys like |length| or further keys as described in the
references later on.

Third, you may often wish to modify how many ticks and grid lines are
shown. By default, no grid lines are shown, but you can say the
following in order to cause grid lines to be shown:
\begin{codeexample}[code only]
all axes={grid}
\end{codeexample}
Naturally, instead of |all axes| you can also specify a single axis,
causing only grid lines to be shown for this axis.
In order to change the number of ticks that are shown, you can say
\begin{codeexample}[code only]
all axes={ticks=few}  
\end{codeexample}
or also |many| instead of |few| or even |none|. Far more fine-grained
control over the tick placement and rendering is possible, 
see Section~\ref{section-dv-ticks-and-grids} for details.

Fourth, consider adding units (like ``cm'' for centimeters or
``$\mathrm m/\mathrm s^2$'' for acceleration) to your ticks:
\begin{codeexample}[code only]
x axis={ticks={tick unit=cm}}, y axis={ticks={tick unit=m/s^2}}  
\end{codeexample}


Finally, consider adding labels to your axes. For this, use the label
option:
\begin{codeexample}[code only]
x axes={time $t$ (ms)}, y axis={distance $d$ (mm)}  
\end{codeexample}

Here is an example that employs most of the above features:

\begin{codeexample}[width=8.5cm]
\tikz \datavisualization [
  scientific axes=clean,
  x axis={attribute=time, ticks={tick unit=ms},
    label={elapsed time}},
  y axis={attribute=v, ticks={tick unit=m/s},
    label={speed of disc}},
  all axes=grid,
  visualize as line]
data {
  time, v
  0, 0
  1, 0.001
  2, 0.002
  3, 0.004
  4, 0.0035
  5, 0.0085
  6, 0.0135
};
\end{codeexample}



\subsubsection{Reference: Scientific Axis Systems}

\begin{key}{/tikz/data visualization/scientific axes=\opt{\meta{options}}}
  This key installs a two-dimensional coordinate system based on the
  attributes |/data point/x| and |/data point/y|.

\begin{codeexample}[width=7cm]
\begin{tikzpicture}
  \datavisualization [scientific axes,
                      visualize as smooth line]
    data [format=function] {
      var x : interval [0:100];
      func y = sqrt(\value x);
    };
\end{tikzpicture}
\end{codeexample}

  This axis system is usually a good choice to depict ``arbitrary two
  dimensional data.'' Because the axes are automatically scaled, you
  do not need to worry about how large or small the values will
  be. The name |scientific axes| is intended to indicate that this
  axis system is often used in scientific publications.

  You can use the \meta{options} to fine tune the axis system. The
  \meta{options} will be executed with the following path prefix:
\begin{codeexample}[code only]
/tikz/data visualization/scientific axes    
\end{codeexample}
  All keys with this prefix can thus be passed as \meta{options}.

  This axis system will always distort the
  relative magnitudes of the units on the two axis. If you wish the
  units on both axes to be equal, consider directly specifying the
  unit length ``by hand'':

\begin{codeexample}[]
\begin{tikzpicture}
  \datavisualization [visualize as smooth line,
                      scientific axes,
                      all axes={unit length=1cm per 10 units, ticks={few}}]
    data [format=function] {
      var x : interval [0:100];
      func y = sqrt(\value x);
    };
\end{tikzpicture}
\end{codeexample}

  The |scientific axes| have the following properties:
  \begin{itemize}
  \item The |x|-values are surveyed and the $x$-axis is then scaled
    and shifted so  that it has the length specified by the following key.
    \begin{key}{/tikz/data visualization/scientific
        axes/width=\meta{dimension} (initially 5cm)}
    \end{key}
    The minimum value is at the left end of the axis and at the canvas
    origin. The maximum value is at the right end of the axis.
  \item The |y|-values are surveyed and the $y$-axis is then scaled so
    that is has the length specified by the following key.
    \begin{key}{/tikz/data visualization/scientific
        axes/height=\meta{dimension}}
      By default, the |height| is the golden ratio times the |width|.
    \end{key}
    The minimum value is at the bottom of the axis and at the canvas
    origin. The maximum value is at the top of the axis.
  \item Lines (forming a frame) are depicted at the minimum and
    maximum values of the axes in 50\% black.
  \end{itemize}

  The following keys are executed by default as options: |outer ticks|
  and |standard labels|.

  You can use the following style to overrule the defaults:

  \begin{stylekey}{/tikz/data visualization/every scientific axes}
  \end{stylekey}
\end{key}

The  keys described in the following can be used to fine-tune the way
the scientific axis system is rendered.

\begin{key}{/tikz/data visualization/scientific axes/outer ticks}
  This causes the ticks to be drawn `` on the outside'' of the frame
  so that they interfere as little as possible with the data. It is
  the default. 

\begin{codeexample}[width=7cm]
\begin{tikzpicture}
  \datavisualization [scientific axes=outer ticks,
                      visualize as smooth line]
    data [format=function] {
      var x : interval [-12:12];
      func y = \value x*\value x*\value x;
    };
\end{tikzpicture}
\end{codeexample}
\end{key}

\begin{key}{/tikz/data visualization/scientific axes/inner ticks}
  This axis system works like |scientific axes|, only the ticks are on
  the ``inside'' of the frame.

\begin{codeexample}[width=7cm]
\begin{tikzpicture}
  \datavisualization [scientific axes=inner ticks,
                      visualize as smooth line]
    data [format=function] {
      var x : interval [-12:12];
      func y = \value x*\value x*\value x;
    };
\end{tikzpicture}
\end{codeexample}

  This axis system is also common in publications, but the ticks tend
  to interfere with marks if they are near to the border as can be
  seen in the following example:
\begin{codeexample}[]
\begin{tikzpicture}
  \datavisualization [scientific axes={inner ticks, width=3.2cm},
                      style sheet=cross marks,
                      visualize as scatter/.list={a,b}]
    data [set=a] {
      x, y
      0, 0
      1, 1
      0.5, 0.5
      2, 1
    }
    data [set=b] {
      x, y
      0.05, 0
      1.5, 1
      0.5, 0.75
      2, 0.5
    };
\end{tikzpicture}
\end{codeexample}

\end{key}

\begin{key}{/tikz/data visualization/scientific axes/clean}
  The axes and the ticks are completely removed from the actual data,
  making this axis system especially useful for scatter plots, but
  also for most other scientific plots.

\begin{codeexample}[width=7.5cm]
\tikz \datavisualization [
  scientific axes=clean,
  visualize as smooth line]
data [format=function] {
  var x : interval [-12:12];
  func y = \value x*\value x*\value x;
};
\end{codeexample}

  The distance of the axes from the actual plot is given by the
  padding of the axes.
\end{key}


For all scientific axis systems, different label placement strategies
can be specified. They are discussed in the following.

\begin{key}{/tikz/data visualization/scientific axes/standard labels}
  As the name suggests, this is the standard placement strategy. The
  label of the $x$-axis is placed below the center of the $x$-axis,
  the label of the $y$-axis is rotated by $90^\circ$ and placed left
  of the center of the $y$-axis.
\begin{codeexample}[width=8cm]
\tikz \datavisualization
 [scientific axes={clean, standard labels},
  visualize as smooth line,
  x axis={label=degree $d$,
    ticks={tick unit={}^\circ}},
  y axis={label=$\sin d$}]
data [format=function] {
  var x : interval [-10:10] samples 10;
  func y = sin(\value x);
};
\end{codeexample}
\end{key}

\begin{key}{/tikz/data visualization/scientific axes/upright labels}
  Works like |scientific axes standard labels|, only the label of the
  $y$-axis is not rotated.
\begin{codeexample}[width=8cm]
\tikz \datavisualization [
  scientific axes={clean, upright labels},
  visualize as smooth line,
  x axis={label=degree $d$,
    ticks={tick unit={}^\circ}},
  y axis={label=$\cos d$, include value=1,
    ticks={style={
        /pgf/number format/precision=4,
        /pgf/number format/fixed zerofill}}}]
data [format=function] {
  var x : interval [-10:10] samples 10;
  func y = cos(\value x);
};
\end{codeexample}
\end{key}


\begin{key}{/tikz/data visualization/scientific axes/end labels}
  Places the labels at the end of the $x$- and the $y$-axis, similar
  to the axis labels of a school book axis system.
\begin{codeexample}[width=8cm]
\tikz \datavisualization [
  scientific axes={clean, end labels},
  visualize as smooth line,
  x axis={label=degree $d$,
    ticks={tick unit={}^\circ}},
  y axis={label=$\tan d$}]
data [format=function] {
  var x : interval [-80:80];
  func y = tan(\value x);
};
\end{codeexample}
\end{key}



\subsubsection{Reference: School Book Axis Systems}

\begin{key}{/tikz/data visualization/school book axes=\meta{options}}
  This axis system is intended to ``look like'' the coordinate systems
  often used in school books: The axes are drawn in such a way that
  they intersect to origin. Furthermore, no automatic
  scaling is done to ensure that the lengths of units are the same in
  all directions.

  This axis system must be used with care -- it is nearly always
  necessary to specify the desired unit length by hand using the
  option |unit length|. If the magnitudes of the units on the two axes
  differ, different unit lengths typically need to be specified for
  the different axes.

  Finally, if the data is ``far removed'' from the origin, this
  axis system will also ``look bad.''

\begin{codeexample}[]
\begin{tikzpicture}
  \datavisualization [school book axes, visualize as smooth line]
    data [format=function] {
      var x : interval [-1.3:1.3];
      func y = \value x*\value x*\value x;
    };
\end{tikzpicture}
\end{codeexample}

  The stepping of the ticks is one unit by default. Using keys like
  |ticks=some| may help to give better steppings.
  
  The \meta{options} are executed with the key itself as path
  prefix. Thus, the following subkeys are permissible options:
  \begin{key}{/tikz/data visualization/school book axes/unit=\meta{value}}
    Sets the scaling so that 1\,cm corresponds to \meta{value}
    units. At the same time, the stepping of the ticks will also be
    set to \meta{value}.
\begin{codeexample}[]
\begin{tikzpicture}
  \datavisualization [school book axes={unit=10},
                      visualize as smooth line,
                      clean ticks,
                      x axis={label=$x$},
                      y axis={label=$f(x)$}]
    data [format=function] {
      var x : interval [-20:20];
      func y = \value x*\value x/10;
    };
\end{tikzpicture}
\end{codeexample}
  \end{key}

  \begin{key}{/tikz/data visualization/school book axes/standard labels}
    This key makes the label of the $x$-axis appear at the right end of
    this axis and it makes the label of the $y$-axis appear at the top
    of the $y$-axis.

    Currently, this is the only supported placement strategy for the
    school book axis system.
\begin{codeexample}[]
\begin{tikzpicture}
  \datavisualization [school book axes={standard labels},
                      visualize as smooth line,
                      clean ticks,
                      x axis={label=$x$},
                      y axis={label=$f(x)$}]
    data [format=function] {
      var x : interval [-1:1];
      func y = \value x*\value x + 1;
    };
\end{tikzpicture}
\end{codeexample}
  \end{key}
\end{key}



\subsubsection{Advanced Reference: Underlying Cartesian Axis Systems}

The axis systems described in the following are typically not used
directly by the user. The systems setup \emph{directions} for several
axes in some sensible way, but they do not actually draw anything on
these axes. For instance, the |xy Cartesian| creates two axes called
|x axis| and |y axis| and makes the $x$-axis point right and the
$y$-axis point up. In contrast, an axis system like |scientific axes|
uses the axis system |xy Cartesian| internally and then proceeds to
setup a lot of keys so that the axis lines are drawn,
ticks and grid lines are drawn, and labels are placed at the correct
positions.

\begin{key}{/tikz/data visualization/xy Cartesian}
  This axis system creates two axes called |x axis| and |y axis| that
  point right and up, respectively. By default, one unit is mapped to
  one cm.

\begin{codeexample}[]
\begin{tikzpicture}
  \datavisualization [xy Cartesian, visualize as smooth line]
    data [format=function] {
      var x : interval [-1.25:1.25];
      func y = \value x*\value x*\value x;
    };
\end{tikzpicture}
\end{codeexample}


  \begin{key}{/tikz/data visualization/xy axes=\meta{options}}
    This key applies the \meta{options} both to the |x axis| and the
    |y axis|.
  \end{key}

\end{key}


\begin{key}{/tikz/data visualization/xyz Cartesian cabinet}
  This axis system works like |xy Cartesian|, only it
  \emph{additionally} creates an axis called |z axis| that points left
  and down. For this axis, one unit corresponds to $\frac{1}{2}\sin
  45^\circ\mathrm{cm}$. This is also known as a cabinet projection.

  \begin{key}{/tikz/data visualization/xyz axes=\meta{options}}
    This key applies the \meta{options} both to the |x axis| and the
    |y axis|.
  \end{key}

\end{key}


\begin{key}{/tikz/data visualization/uv Cartesian}
  This axis system works like |xy Cartesian|, but it introduces two
  axes called |u axis| and |v axis| rather than the |x axis| and the
  |y axis|. The idea is that in addition to a ``major''
  $xy$-coordinate system this is also a ``smaller'' or ``minor''
  coordinate system in use for depicting, say, small vectors with
  respect to this second coordinate system.

  \begin{key}{/tikz/data visualization/uv axes=\meta{options}}
    Applies the \meta{options} to both the |u axis| and the |y axis|.
  \end{key}

\end{key}

\begin{key}{/tikz/data visualization/uvw Cartesian cabinet}
  Like |xyz Cartesian cabinet|, but for the $uvw$-system.

  \begin{key}{/tikz/data visualization/uvw axes=\meta{options}}
    Like |xyz axes|.
  \end{key}
\end{key}



\subsection{Ticks and Grids}
\label{section-dv-ticks-and-grids}


\subsubsection{Concepts}

A \emph{tick} is a small visual indication on an axis of the value of
the axis's attribute at the position where the tick is shown. A tick
may be accompanied additionally by a textual representation, but it
need not. A \emph{grid line} is similar to a tick, but it is not an
indication on the axis, but rather a whole line that indicates all
positions where the attribute has a certain value. Unlike ticks,
grid lines (currently) are not accompanied by a textual
representation.

Just as for axes, the data visualization system decouples the
specification of which ticks are present \emph{in principle} from
where they are visualized. In the following, I describe how you
specify which ticks and grid lines you would like to be drawn and how
they should look like (their styling). The axis system of your choice
will then visualize the ticks at a sensible position for the chosen
system. For details on how to change where whole axis is shown along
with its ticks, see Section~\ref{section-dv-visualize-ticks}. 

Specifying which ticks you are interested in is done as follows:
First, you use |ticks| key (or, for specifying which grid lines  should
be present, the |grid| key). This key takes several possible options,
described in detail in the following, which have different effects:
\begin{enumerate}
\item Keys like |step=10| or |minor steps between steps| cause a
  ``semi-automatic'' computation of possible steps. Here, you
  explicitly specify the stepping of steps, but the first stepping and
  their number are computed automatically according to the range of
  possible values for the attribute.
\item Keys like |few|, |some|, or |many| can be passed to |ticks| in
  order to have \tikzname\ compute good tick positions
  automatically. This is usually what you want to happen, which is why
  most axis system will implicitly say |ticks={some}|.
\item Keys like |at| or |also at| provide ``absolute control'' over
  which ticks or grid lines are shown. For these keys, you can not only
  specify at what value a tick should be shown, but also its styling
  and also whether it is a major, minor, or subminor tick or grid line.
\end{enumerate}

In the following, the main keys |ticks| and |grids| are documented
first. Then the different kinds of ways of specifying where ticks or
grid lines should be shown are explained.  


\subsubsection{The Main Options: Tick and Grid}

\begin{key}{/tikz/data visualization/axis options/ticks=\meta{options}
  (default some)}
  This key can be passed to an axis in order to configure which ticks
  are present for the axis. The possible \meta{options} include, for
  instance, keys like |step|, which is used to specify a stepping for
  the ticks, but also keys like |major| or |minor| for specifying the
  positions of major and minor ticks in detail. The list of possible
  options is described in the rest of this section.

  Note that the |ticks| option will only configure which ticks should
  be shown in principle. The actual rendering is done only when the
  |visualize ticks| key is used, documented in
  Section~\ref{section-dv-visualize-ticks}, which is typically done
  only internally by an axis system.

  The \meta{options} will be executed with the path prefix
  |/tikz/data visualization/|. When the |ticks| key is used multiple
  times for an axis, the \meta{options} accumulate.
\begin{codeexample}[width=6cm]
\tikz \datavisualization [
  scientific axes, visualize as line,
  x axis={ticks={step=24, minor steps between steps=3},
          label=hours}]
  data {
    x, y
    0, 0
    10, 0
    20, 0.5
    30, 0.75
    40, 0.7
    50, 0.6
    60, 0.5
    70, 0.45
    80, 0.47
  };
\end{codeexample}
\end{key}

\begin{key}{/tikz/data visualization/axis options/grid=\meta{options}
    (default at default ticks)}
  This key is similar to |ticks|, only it is used to configure where
  grid lines should be shown rather than ticks. In particular, the
  options that can be passed to the |ticks| key can also be passed to
  the |grid| key. Just like |ticks|, the \meta{options} only specify
  which grid lines should be drawn in principle; it is the job of the
  |visualize grid| key to actually cause any grid lines to be shown.

  If you do not specify any \meta{options}, the default text
  |at default ticks| is used. This option causes grid lines to be
  drawn at all positions where ticks are shown by default. Since this
  usually exactly what you would like to happen, most of the time you
  just need to |all axes=grid| to cause a grid to be shown.
\end{key}

\begin{key}{/tikz/data visualization/axis options/ticks and grid=\meta{options}}
  This key passes the \meta{options} to both the |ticks| key and also
  to the |grid| key. This is useful when you want to specify some
  special points explicitly where you wish a tick to be shown and also
  a grid line.
\begin{codeexample}[]
\tikz \datavisualization
  [scientific axes,
   visualize as smooth line,
   all axes= {grid, unit length=1.25cm},
   y axis={ ticks=few },
   x axis={ ticks=many, ticks and grid={ major also at={(pi/2) as $\frac{\pi}{2}$}}}]
  data [format=function] {
    var x : interval [-pi/2:3*pi] samples 50;
    func y = sin(\value x r);
  };
\end{codeexample}
\end{key}


\subsubsection{Semi-Automatic Computation of Tick and Grid Line Positions}
\label{section-dv-concept-tick-placement-strategies}

Consider the following problem: The data visualization engine
determines that in a plot the $x$-values vary between $17.4$ and
$34.5$. In this case, we 
certainly do not want, say, ten ticks at exactly ten evenly spaced
positions starting with $17.4$ and ending with $34.5$, because this
would yield ticks at positions like $32.6$. Ticks should be placed at
``nice'' positions like $20$, $25$, and $30$.

Determining which positions are ``nice'' is somewhat difficult. In the
above example, the positions $20$, $25$, and $30$ are certainly nice,
but only three ticks may be a bit few of them. Better might be the
tick positions $17.5$, $20$, $22.5$, through to $32.5$. However, users
might prefer even numbers over fractions like $2.5$ as the stepping.

A \emph{tick placement strategy} is a method of automatically deciding
which positions are \emph{good} for placing ticks. The data
visualization engine comes with a number of predefined strategies, but
you can also define new ones yourself. When the data visualization is
requested to automatically determine
``good'' positions for the placement of ticks on an axis, it uses one
of several possible \emph{basic strategies}. These strategies differ
dramatically in which tick positions they will choose: For a range of
values between $5$ and $1000$, a |linear steps| strategy might place
ticks at positions $100$, $200$, through to $1000$, while an
|exponential steps| strategy would prefer the tick positions $10$,
$100$ and $1000$. The exact number and values of the tick positions
chosen by either strategy can be fine-tuned using additional options
like |step| or |about|.

Here is an example of the different stepping chosen when one varies
the tick placement strategy:

\begin{codeexample}[]
\begin{tikzpicture}
  \datavisualization [scientific axes, visualize as smooth line]
    data [format=function] {
      var x : interval [1:11];
      func y = \value x*\value x;
    };
\end{tikzpicture}
\qquad
\begin{tikzpicture}
  \datavisualization [scientific axes, visualize as smooth line,
    y axis={exponential steps},
    x axis={ticks={quarter about strategy}},
  ]
    data [format=function] {
      var x : interval [1:11];
      func y = \value x*\value x;
    };
\end{tikzpicture}
\end{codeexample}

Two strategies are always available: |linear steps|, which yields
(semi)automatic ticks are evenly spaced positions, and
|exponential steps|, which yields (semi)automatic steps at positions
at exponentially increasing positions -- which is exactly what is
needed for logarithmic plots. These strategies are details in
Section~\ref{section-dv-strategies}.

The following options are used to configure tick placement strategies
like |linear steps|. Unlike the basic choice of a placement strategy,
which is an axis option, the following should be passed to the option
|ticks| or |grid| only. So, you would write
things like |x axis={ticks={step=2}}|, but |x axis={linear steps}|.

\begin{key}{/tikz/data visualization/step=\meta{value} (initially 1)}
  The value of this key is used to determine the spacing of the major
  ticks. The key is used by the |linear steps| and |exponential steps|
  strategies, see the explanations in
  Section~\ref{section-dv-strategies} for details. Basically, all
  ticks are placed at all multiples of \meta{value} that lie in the
  attribute range interval.
\begin{codeexample}[]
\tikz \datavisualization [
    school book axes, visualize as smooth line,
    y axis={ticks={step=1.25}},
  ]
    data [format=function] {
      var x : interval [0:3];
      func y = \value x*\value x/2;
    };
\end{codeexample}
\end{key}

\begin{key}{/tikz/data visualization/minor steps between
    steps=\meta{number} (default 9)}
  Specifies that between any two major steps (whose positions are
  specified by the |step| key), there should be \meta{number} many
  minor steps. Note that the default of |9| is exactly the right
  number so that each interval between two minor steps is exactly a
  tenth of the size of a major step. See also
  Section~\ref{section-dv-strategies} for further details.
\begin{codeexample}[]
\begin{tikzpicture}
  \datavisualization [school book axes, visualize as smooth line,
    x axis={ticks={minor steps between steps=3}},
    y axis={ticks={minor steps between steps}},
  ]
    data [format=function] {
      var x : interval [-1.5:1.5];
      func y = \value x*\value x;
    };
\end{tikzpicture}
\end{codeexample}
\end{key}

\begin{key}{/tikz/data visualization/phase=\meta{value} (initially 0)}
  See Section~\ref{section-dv-strategies} for details on how the phase
  of steps influences the tick placement.
\end{key}



\subsubsection{Automatic Computation of Tick and Grid Line Positions}

The |step| option gives you ``total control'' over the stepping of
ticks on an axis, but you often do not know the correct stepping in
advance. In this case, you may prefer to have a good value for |step|
being computed for you automatically.

Like the |step| key, these options are passed to the |ticks|
option. So, for instance, you would write |x axis={ticks={about=4}}|
to request about four ticks to be placed on the $x$-axis.


\begin{key}{/tikz/data visualization/about=\meta{number}}
  This key asks the data visualization to place \emph{about}
  \meta{number} many ticks on an axis. It is not guaranteed that
  \emph{exactly} \meta{number} many ticks will be used, rather the
  actual number will be the closest number of ticks to \meta{number}
  so that their stepping is still ``good''. For instance, when you say
  |about=10|, it may happen that exactly |10|, but perhaps even |13| ticks are
  actually selected, provided that these numbers of ticks lead to good
  stepping values like |5| or |2.5| rather than numbers like |3.4| or
  |7|. The method that is used to determine which steppings a deemed to
  be ``good'' depends on the current tick placement strategy.

  \medskip
  \textbf{Linear steps.}
  Let us start with |linear steps|: First, the difference between the
  maximum value $v_{\max}$ and the minimum value $v_{\min}$ on the
  axis is computed; let us call it $r$ for ``range.'' Then, $r$ is
  divided by \meta{number},
  yielding a target stepping~$s$. If $s$ is a number like $1$ or $5$
  or $10$, then this number could be used directly as the new value of
  |step|. However, $s$ will typically something strange like $0.02345$
  or $345223.76$, so $s$ must be replaced by a better value like $0.02$
  in the first case and perhaps $250000$ in the second case.

  In order to determine which number is to be used, $s$ is rewritten
  in the form $m \cdot 10^k$ with $1 \le m < 10$ and $k \in \mathbb
  Z$. For instance, $0.02345$ would be rewritten as $2.345 \cdot
  10^{-2}$ and $345223.76$ as $3.4522376 \cdot 10^5$. The next step
  is to replace the still not-so-good number $m$ like $2.345$ or
  $3.452237$ by a ``good'' value $m'$. For this, the current value of
  the |about strategy| is used:
  \begin{key}{/tikz/data visualization/about strategy=\meta{list}}
    The \meta{list} is a comma-separated sequence of pairs
    \meta{threshold}/\meta{value} like for instance |1.5/1.0| or
    |2.3/2.0|. When a good value $m'$ is sought for a given $m$, we
    iterate over the list and find the first pair
    \meta{threshold}/\meta{value} where \meta{threshold}
    exceeds~$m$. Then $m'$ is set to \meta{value}. For instance, if
    \meta{list} is |1.5/1.0,2.3/2.0,4/2.5,7/5,11/10|, which is the
    default, then for $m=3.141$ we would get $m'=2.5$ since $4 >
    3.141$, but $2.3 \le 3.141$. For $m=6.3$ we would get $m'=5$.
  \end{key}
  Once $m'$ has been determined, the stepping is set to $s' = m'
  \cdot 10^k$.

  % Define an axis type
  \tikzdatavisualizationset{
    one dimensional axis/.style={
      new Cartesian axis=axis,
      axis={
        attribute=main,
        unit vector={(0pt,1pt)},
        visualize axis={style=->},
        visualize ticks={major={tick text at low},direction axis=perpendicular},
        length=3cm
      },
      new Cartesian axis=perpendicular,
      perpendicular={
        attribute=perp,
        unit vector={(1pt,0pt)},
        include value=0,
        include value=1
      }
    }
  }

  \def\showstrategy#1{

  % Show the effect for the different strategies
    \medskip
    \begin{tikzpicture}
      \foreach \max/\about [count=\c] in {10/5,20/5,30/5,40/5,50/5,60/5,70/5,80/5,90/5,100/5,100/3,100/10}
      {
        \begin{scope}[xshift=\c pt*30]
          \datavisualization [#1,
          one dimensional axis,
          axis={
            ticks={about=\about},
            include value=0,
            include value=\max
          }
          ];

          \node at (0,-5mm) [anchor=mid] {\texttt{\about}};
        \end{scope}
      }

      \node at (30pt,-5mm) [anchor=mid east] {\texttt{about=\ \ }};
  \end{tikzpicture}
}

  The net effect of all this is that for the default strategy the
  only valid stepping are the values $1$, $2$, $2.5$ and $5$ and every
  value obtainable by multiplying one of these values by a power of
  ten. The following example shows the effects of, first, setting
  |about=5| (corresponding to the |some| option) and then having axes
  where the minimum value is always |0| and where the maximum value
  ranges from |10| to |100| and, second, setting |about| to the values
  from |3| (corresponding to the |few| option) and to |10|
  (corresponding to the |many| option) while having the
  minimum at |0| and the maximum at |100|:

  \showstrategy{standard about strategy}

  \medskip
  \textbf{Exponential steps.}
  For |exponential steps| the strategy for determining a good stepping
  value is similar to |linear steps|, but with the following
  differences:
  \begin{itemize}
  \item Naturally, since the stepping value refers to the exponent,
    the whole computation of a good stepping value needs to be done
    ``in the exponent.'' Mathematically spoken, instead of considering
    the difference $r = v_{\max} - v_{\min}$, we consider the difference $r =
    \log v_{\max} - \log v_{\min}$. With this difference, we still
    compute $s = r / \meta{number}$ and let $s = m \cdot 10^k$ with $1
    \le m < 10$.
  \item It makes no longer sense to use values like $2.5$ for $m'$
    since this would yield a fractional exponent. Indeed, the only
    sensible values for $m'$ seem to be $1$, $3$, $6$, and
    $10$. Because of this, the |about strategy| is ignored and one of
    these values or a multiple of one of them by a power of ten is
    used.
  \end{itemize}

  The following example shows the chosen steppings for a maximum
  varying from $10^1$ to $10^5$ and from $10^{10}$ to $10^{50}$ as
  well as for $10^{100}$ for |about=3|:

  \medskip
  \begin{tikzpicture}
    \foreach \max [count=\c] in {1,...,5,10,20,...,50,100}
      {
        \begin{scope}[xshift=\c pt*40]
          \datavisualization [
          one dimensional axis,
          axis={
            logarithmic,
            ticks={about=3},
            include value=1,
            include value=1e\max
          }
          ];
        \end{scope}
      }
  \end{tikzpicture}


  \medskip
  \textbf{Alternative strategies.}

  In addition to the standard |about strategy|, there are some
  additional strategies that you might wish to use instead:

  \begin{key}{/tikz/data visualization/standard about
      strategy}
    Permissible values for $m'$ are: $1$, $2$, $2.5$, and~$5$. This
    strategy is the default strategy.
  \end{key}

  \begin{key}{/tikz/data visualization/euro about strategy}
    Permissible values for $m'$ are: $1$, $2$, and~$5$. These are the
    same values as for the Euro coins, hence the
    name.

    \showstrategy{euro about strategy}
  \end{key}

  \begin{key}{/tikz/data visualization/half about strategy}
    Permissible values for $m'$: $1$ and $5$. Use this
    strategy if only powers of $10$ or halves thereof seem logical.

    \showstrategy{half about strategy}
  \end{key}

  \begin{key}{/tikz/data visualization/decimal about strategy}
    The only permissible value for $m'$ is $1$. This is an even more
    radical version of the previous strategy.

    \showstrategy{decimal about strategy}
  \end{key}

  \begin{key}{/tikz/data visualization/quarter about strategy}
    Permissible values for $m'$ are: $1$, $2.5$, and $5$.

    \showstrategy{quarter about strategy}
  \end{key}

  \begin{key}{/tikz/data visualization/int about strategy}
    Permissible values for $m'$ are: $1$, $2$, $3$, $4$, and $5$.

    \showstrategy{int about strategy}
  \end{key}
\end{key}

\begin{key}{/tikz/data visualization/many}
  This is an abbreviation for |about=10|.
\end{key}

\begin{key}{/tikz/data visualization/some}
  This is an abbreviation for |about=5|.
\end{key}

\begin{key}{/tikz/data visualization/few}
  This is an abbreviation for |about=3|.
\end{key}

\begin{key}{/tikz/data visualization/none}
  Switches off the automatic step computation. Unless you use |step=|
  explicitly to set a stepping, no ticks will be (automatically)
  added.
\end{key}



\subsubsection{Manual Specification of Tick and Grid Line Positions}

The automatic computation of ticks and grid lines will usually do a
good job, but not always. For instance, you might wish to have ticks
exactly at, say, prime numbers or at Fibonacci numbers or you might
wish to have an additional tick at $\pi$. In these cases you need more
direct control over the specification of tick positions.

First, it is important to understand that the data visualization
system differentiates between three kinds of 
ticks and grid lines: major, minor, and subminor. The major ticks are
the most prominent ticks where, typically, a textual representation of
the tick is shown; and the major grid lines are the thickest. The
minor ticks are smaller, more numerous, and lie between major
ticks. They are used, for instance, to indicate positions in the
middle between major ticks or at all integer positions between major
ticks. Finally, subminor ticks are even smaller than minor ticks and
they lie between minor ticks. 

Four keys are used to configure the different kinds:

\begin{key}{/tikz/data visualization/major=\meta{options}}
  The key can be passed as an option to the |ticks| key and also to
  the |grid| key, which in turn is passed as an option to an axis. The
  \meta{options} passed to |major| specify at which positions
  major ticks/grid lines should be shown (using the |at| option and
  |also at| option) and also any special styling. The different
  possible options are described later in this section.
\begin{codeexample}[]
\tikz \datavisualization
  [ school book axes, visualize as smooth line,
    x axis={ticks={major={at={1, 1.5, 2}}}}]
  data [format=function] {
    var x : interval [-1.25:2];
    func y = \value x * \value x / 2;
  };
\end{codeexample}
\end{key}

\begin{key}{/tikz/data visualization/minor=\meta{options}}
  Like |major|, only for minor ticks/grid lines.
\begin{codeexample}[]
\tikz \datavisualization
  [ school book axes, visualize as smooth line,
    x axis={grid={minor={at={1, 1.5, 2}}}}]
  data [format=function] {
    var x : interval [-1.25:2];
    func y = \value x * \value x / 2;
  };
\end{codeexample}
\end{key}

\begin{key}{/tikz/data visualization/subminor=\meta{options}}
  Like |major|, only for subminor ticks/grid lines.
\end{key}


\begin{key}{/tikz/data visualization/common=\meta{options}}
  This key allows you to specify \meta{options} that apply to |major|,
  |minor| and |subminor| alike. It does not make sense to use |common|
  to specify positions (since you typically do not want both a major
  and a minor tick at the same position), but it can be useful to
  configure, say, the size of all kinds of ticks:
\begin{codeexample}[]
\tikz \datavisualization
  [ school book axes, visualize as smooth line,
    x axis={ticks={minor steps between steps, common={low=0}}} ]
  data [format=function] {
    var x : interval [-1.25:2];
    func y = \value x * \value x / 2;
  };
\end{codeexample}
\end{key}


The following keys can now be passed to the |major|, |minor|, and
|subminor| keys to specify where ticks or grid lines should be shown: 

\begin{key}{/tikz/data visualization/at=\meta{list}}
  Basically, the \meta{list} must be a list of values that is
  processed with the |\foreach| macro (thus, it can contain ellipses
  to specify ranges of value). Empty values are skipped.

  The effect of passing |at| to a |major|, |minor|, or |subminor| key
  is that ticks or grid lines on the axis will be placed exactly at
  the values in \meta{list}. Here is an example:
\begin{codeexample}[]
\tikz \datavisualization
  [ school book axes, visualize as smooth line,
    x axis={ticks={major={at={-1,0.5,(pi/2)}}}}]
  data [format=function] {
    var x : interval [-1.25:2];
    func y = \value x * \value x / 2;
  };
\end{codeexample}  
  When this option is used, any previously specified tick positions
  are overwritten by the values in \meta{list}. Automatically computed
  ticks are also overwritten. Thus, this option gives you complete
  control over where ticks should be placed.

  Normally, the individual values inside the \meta{list} are just
  numbers that are specified in the same way as an attribute
  value. However, such a value may also contain the keyword |as|,
  which allows you so specify the styling of the tick in
  detail. Section~\ref{section-dv-ticks-styling} details how this
  works.

  It is often a bit cumbersome that one has to write things like
\begin{codeexample}[code only]
some axis = {ticks = {major = {at = {...}}}}    
\end{codeexample}
  A slight simplification is given by the following keys, which can be
  passed directly to |ticks| and |grid|:
  \begin{key}{/tikz/data visualization/major at=\meta{list}}
    A shorthand for |major={at={|\meta{list}|}}|.
  \end{key}
  \begin{key}{/tikz/data visualization/minor at=\meta{list}}
    A shorthand for |major={at={|\meta{list}|}}|.
  \end{key}
  \begin{key}{/tikz/data visualization/subminor at=\meta{list}}
    A shorthand for |major={at={|\meta{list}|}}|.
  \end{key}
\end{key}

\begin{key}{/tikz/data visualization/also at=\meta{list}}
  This key is similar to |at|, but it causes ticks or grid lines to be
  placed at the positions in the \meta{list} \emph{in addition} to the
  ticks that have already been specified either directly using |at| or
  indirectly using keys like |step| or |some|. The effect of multiple
  calls of this key accumulate. However, when |at| is used after an
  |also at| key, the |at| key completely resets the positions where
  ticks or grid lines are shown.
\begin{codeexample}[]
\tikz \datavisualization
  [ school book axes, visualize as smooth line,
    x axis={grid, ticks and grid={major={also at={0.5}}}}]
  data [format=function] {
    var x : interval [-1.25:2];
    func y = \value x * \value x / 2;
  };
\end{codeexample}  
  As for |at|, there are some shorthands available:
  \begin{key}{/tikz/data visualization/major also at=\meta{list}}
    A shorthand for |major={also at={|\meta{list}|}}|.
  \end{key}
  \begin{key}{/tikz/data visualization/minor also at=\meta{list}}
    A shorthand for |major={also at={|\meta{list}|}}|.
  \end{key}
  \begin{key}{/tikz/data visualization/subminor also at=\meta{list}}
    A shorthand for |major={also at={|\meta{list}|}}|.
  \end{key}
\end{key}


\subsubsection{Styling Ticks and Grid Lines: Introduction}

\label{section-dv-ticks-styling}

When a tick, a tick label, or a grid line is visualized on the page, a
whole regiment of styles influences the appearance. The reason for
this large number of interdependent styles is the fact that we often
wish to influence only a very certain part of how a tick is rendered
while leaving the other aspects untouched: Sometimes we need to modify
just the font of the tick label; sometimes we wish to change the
length of the tick label and the tick label position at the same time;
sometimes we wish to change the color of grid line, tick, and tick
label; and sometimes we wish to generally change the thickness of all
ticks.

Let us go over the different kinds of things that can be styled (grid
lines, ticks, and tick labels) one by one and let us have a look at
which styles are involved. We will start with the grid lines, since they
turn out to be the most simple, but first let us have a look at the
general |style| and |styling| mechanism that is used in many placed in
the following:

\subsubsection{Styling Ticks and Grid Lines: The Style and Node Style Keys}

\label{section-dv-style}

All keys of the data visualization system have the path prefix
|/tikz/data visualization|. This is not only true for the main keys
like |scientific axes| or  |visualize as line|, but also for keys that
govern how ticks are visualized. In particular, a style like
|every major grid| has the path prefix |/tikz/data visualization| and
all keys stored in this style are also executed with this path
prefix.

Normally, this does not cause any trouble since most of the keys and
even styles used in a data visualization are intended to configure
what is shown in the visualization. However, at some point, we may
also with to specify options that no longer configure the
visualization in general, but specify the appearance of a line or a
node on the \tikzname\ layer.

Two keys are used to ``communicate'' with the \tikzname\ layer:

\begin{key}{/tikz/data visualization/style=\meta{\tikzname\ options}}
  This key takes options whose path prefix is |/tikz|, not
  |/tikz/data visualization|. These options will be \emph{appended} to
  a current list of such options (thus, multiple calls of this key
  accumulate). The resulting list of keys is not executed
  immediately, but it will be executed whenever the data visualization
  engine calls the \tikzname\ layer to draw something (this placed
  will be indicated in the following).
\begin{codeexample}[]
\tikz \datavisualization
  [scientific axes,
   all axes={ticks={style=blue}, length=3cm},
   y axis={grid, grid={minor steps between steps, major={style=red}}}, 
   visualize as line]
  data [format=function] {
    var x : interval [5:10];
    func y = \value x * \value x;
  };
\end{codeexample}
\end{key}

\begin{key}{/tikz/data visualization/styling}
  Executing this key will cause all ``accumulated'' \tikzname\ options
  from previous calls to the key |/tikz/data visualization/style| to be
  executed. Thus, you use |style| to set \tikzname\ options, but you
  use |styling| to actually apply these options. Usually, you do not
  call this option directly since this application is only done deep
  inside the data visualization engine.
\end{key}

Similar to |style| (and |styling|) there also exist the 
|node style| (and |node styling|) key that takes \tikzname\ options that
apply to nodes only -- in addition to the usual |style|.

\begin{key}{/tikz/data visualization/node style=\meta{\tikzname\ options}}
  This key works like |style|, only it has an effect only on nodes
  that are created during a data visualization. This includes tick
  labels and axis labels: 
\begin{codeexample}[]
\tikz \datavisualization
  [scientific axes,
   all axes={ticks={node style=red}, length=3cm},
   visualize as line]
  data [format=function] {
    var x : interval [5:10];
    func y = \value x * \value x;
  };
\end{codeexample}
  Note that in the example the ticks themselves (the little thicker
  lines) are not red.
\end{key}

\begin{key}{/tikz/data visualization/node styling}
  Executing this key will cause all ``accumulated'' node stylings to
  be executed. 
\end{key}


\subsubsection{Styling Ticks and Grid Lines: Styling Grid Lines}
\label{section-dv-styling-grid-lines}

When a grid line is visualized, see
Section~\ref{section-dv-visualize-gridlines} for details on when this
happens, the following styles are executed in the specified order. 

\begin{enumerate}
\item
  |grid layer|.
\item
  |every grid|.
\item |every major grid| or |every minor grid| or
  |every subminor grid|, depending on the kind of grid line.
\item
  locally specified options for the individual grid line, see
  Section~\ref{section-dv-local-styles}. 
\item
  |styling|, see Section~\ref{section-dv-style}.
\end{enumerate}

All of these keys have the path prefix
|/tikz/data visualization|. However, the options stored in the first
style (|grid layer|) and also in the last (|styling|) are executed
with the path prefix |/tikz| (see Section~\ref{section-dv-style}).

Let us now have a look at these keys in detail:

\begin{stylekey}{/tikz/data visualization/grid layer (initially on background layer)}
  \label{section-dv-grid-layer}%
  This key is used to specified the \emph{layer} on which grid lines
  should be drawn (layers are explained in
  Section~\ref{section-tikz-backgrounds}). By default, all grid lines
  are placed on the |background| layer and thus behind the data
  visualization. This is a sensible strategy since it avoids obscuring
  the more important data with the far less important grid
  lines. However, you can change this style to ``get the grid lines to
  the front'':
\begin{codeexample}[]
\tikz \datavisualization
  [scientific axes,
   all axes={
     length=3cm,
     grid,
     grid={minor steps between steps}
   },
   grid layer/.style=, % none, so on top of data (bad idea)
   visualize as line]
  data [format=function] {
    var x : interval [5:10];
    func y = \value x * \value x;
  };
\end{codeexample}   
  When this style is executed, the keys stored in the style will be
  executed with the prefix |/tikz|. Normally, you should only set this
  style to be empty or to |on background layer|.
\end{stylekey}

\begin{stylekey}{/tikz/data visualization/every grid}
  This style provides overall configuration options for grid lines. By
  default, it is set to the following:
\begin{codeexample}[code only]
low=min, high=max
\end{codeexample}
  This causes grid lines to span all possible values when they are
  visualized, which is usually the desired behaviour (the |low| and
  |high| keys are explained in
  Section~\ref{section-dv-visualize-ticks}. You can append 
  the |style| key to this style to configure the overall appearance of
  grid lines. In the following example we cause all grid lines to be
  dashed (which is not a good idea in general since it creates a
  distracting background pattern).
\begin{codeexample}[]
\tikz \datavisualization
  [scientific axes,
   all axes={length=3cm, grid},
   every grid/.append style={style=densely dashed},
   visualize as line]
  data [format=function] {
    var x : interval [5:10];
    func y = \value x * \value x;
  };
\end{codeexample}   
\end{stylekey}

\begin{stylekey}{/tikz/data visualization/every major grid}
  This style configures the appearance of major grid lines. It does so
  by calling the |style| key to setup appropriate \tikzname\ options
  for visualizing major grid lines. The default definition of this
  style is:
\begin{codeexample}[code only]
style = {help lines, thin, black!25}
\end{codeexample}
  In the following example, we use thin major blue grid lines:
\begin{codeexample}[]
\tikz \datavisualization
  [scientific axes,
   all axes={
     length=3cm,
     grid,
     grid={minor steps between steps}
   },
   every major grid/.style = {style={blue, thin}},
   visualize as line]
  data [format=function] {
    var x : interval [5:10];
    func y = \value x * \value x;
  };
\end{codeexample}   
  As can be seen, this is not exactly visually pleasing. The default
  settings for the grid lines should work in most situations; you may
  wish to increase the blackness level, however, when you experience
  trouble during printing or projecting graphics.
\end{stylekey}

\begin{stylekey}{/tikz/data visualization/every minor grid}
  Works like |every major grid|. The default is
\begin{codeexample}[code only]
style = {help lines, black!25}
\end{codeexample}
\end{stylekey}

\begin{stylekey}{/tikz/data visualization/every subminor grid}
  Works like |every major grid|. The default is
\begin{codeexample}[code only]
style = {help lines, black!10}
\end{codeexample}
\end{stylekey}



\subsubsection{Styling Ticks and Grid Lines: Styling Ticks and Tick Labels}

\label{section-dv-styling-ticks}

Styling ticks and tick labels is somewhat similar to styling grid
lines. Let us start with the tick \emph{mark}, that is, the small line
that represents the tick. When this mark is drawn, the following styles are applied:

\begin{enumerate}
\item
  |every ticks|.
\item |every major ticks| or |every minor ticks| or
  |every subminor ticks|, depending on the kind of ticks to be
  visualized. 
\item
  locally specified options for the individual tick, see
  Section~\ref{section-dv-local-styles}. 
\item
  |tick layer|
\item
  |every odd tick| or |every even tick|, see Section~\ref{section-dv-stacking}.
\item
  |draw|
\item
  |styling|, see Section~\ref{section-dv-style}.
\end{enumerate}

For the tick label node (the node containing the textual
representation of the attribute's value at the tick position), the
following styles are applied:

\begin{enumerate}
\item
  |every ticks|.
\item |every major ticks| or |every minor ticks| or
  |every subminor ticks|, depending on the kind of ticks to be
  visualized. 
\item
  locally specified options for the individual tick, see
  Section~\ref{section-dv-local-styles}. 
\item
  |tick node layer| 
\item
  |every odd tick| or |every even tick|, see Section~\ref{section-dv-stacking}.
\item
  |styling|, see Section~\ref{section-dv-style}.
\item
  |node styling|, see Section~\ref{section-dv-style}.
\end{enumerate}

\begin{stylekey}{/tikz/data visualization/every ticks}
  This style allows you to configure the appearance of ticks using the
  |style| and |node style| key. Here is (roughly) the default
  definition of this style:
\begin{codeexample}[code only]
node style={
  font=\footnotesize,
  inner sep=1pt,
  outer sep=.1666em,
  rounded corners=1.5pt
}    
\end{codeexample}
\end{stylekey}

\begin{stylekey}{/tikz/data visualization/every major ticks}
  The default is
\begin{codeexample}[code only]
  style={line cap=round}, tick length=2pt
\end{codeexample}
\end{stylekey}

\begin{stylekey}{/tikz/data visualization/every minor ticks}
  The default is
\begin{codeexample}[code only]
  style={help lines,thin, line cap=round}, tick length=1.4pt
\end{codeexample}
\end{stylekey}

\begin{stylekey}{/tikz/data visualization/every subminor ticks}
  The default is
\begin{codeexample}[code only]
  style={help lines, line cap=round}, tick length=0.8pt
\end{codeexample}
\end{stylekey}


\begin{stylekey}{/tikz/data visualization/tick layer (initially on background layer)}
  Like |grid layer|, this key specifies on which layer the ticks
  should be placed.
\end{stylekey}

\begin{stylekey}{/tikz/data visualization/tick node layer (initially
    \normalfont empty)}
  Like |tick layer|, but now for the nodes. By default, tick nodes are
  placed on the main layer and thus on top of the data in case that
  the tick nodes are inside the data.
\end{stylekey}


\subsubsection{Styling Ticks and Grid Lines: Exceptional Ticks}

You may sometimes wish to style a few ticks differently from the other
ticks. For instance, in the axis system |school book axes| there
should be a tick label at the |0| position only on one axis and then
this label should be offset a bit. In many cases this is easy to
achieve: When you add a tick ``by hand'' using the |at| or |also at|
option, you can add any special options in square brackets.

However, in some situations the special tick position has been
computed automatically for you, for instance by the |step| key or by
saying |tick=some|. In this case, adding a tick mark with the desired
options using |also at| would cause the tick mark with the correct
options to be shown in addition to the tick mark with the wrong
options. In cases like this one, the following option may be helpful:

\begin{key}{/tikz/data visualization/options at=\meta{value} |as [|\meta{options}|]|}
  This key causes the \meta{options} to be executed for any tick
  mark(s) at \meta{value} in addition to any options given already for
  this position:
\begin{codeexample}[width=7cm]
\tikz \datavisualization [
  scientific axes,
  visualize as smooth line,
  x axis={ticks={major={
    options at = 3    as [no tick text],
    also at    = (pi) as
      [{tick text padding=1ex}] $\pi$}}}]
data [format=function] {
  var x : interval[0:2*pi];
  func y = sin(\value x r);
};
\end{codeexample}
\end{key}

\begin{key}{/tikz/data visualization/no tick text at=\meta{value}}
  Shorthand for |options at=|\meta{value}| as [no tick text]|.  
\end{key}



\subsubsection{Styling Ticks and Grid Lines: Styling and Typesetting a Value}

\label{section-dv-local-styles}
\label{section-dv-tick-labels}


The |at| and |also at| key allow you to provide a comma-separated
\meta{list} of \meta{value}s where ticks or grid lines should be
placed. In the simplest case, the \meta{value} is simply a
number. However, the general syntax allows three different kinds of
\meta{value}s: 

\begin{enumerate}
\item
  \meta{value}
\item
  \meta{value} |as| |[|\meta{local options}|]|
\item
  \meta{value} |as| \opt{|[|\meta{local options}|]|} \meta{text}
\end{enumerate}

In the first case, the \meta{value} is just a number that is
interpreted like any other attribute value.

In the second case, where the keyword |as| is present, followed by
some option in square brackets, but nothing following the closing
square bracket, when the tick or grid line at position \meta{value} is
shown, the \meta{local options} are executed first. These can use the
|style| key or the |node style| key to configure the appearance of
this single tick or grid line. You can also use keys like |low| or
|high| to influence how large the grid lines or the ticks are or keys
like |tick text at low| to explicitly hide or show a tick label.

In the third case, which is only important for |ticks| and not for
|grid|, the same happens as in the second case, but the 
text that is shown as tick label is \meta{text} rather than the 
automatically generated tick label. This automatic generation of tick
labels is explained in the following.

\begin{codeexample}[]
\tikz \datavisualization
  [scientific axes=clean,
   x axis={length=2.5cm, ticks={major at={
         5,
         6 as [style=red],
         7 as [{style=blue, low=-1em}],
         8 as [style=green] $2^3$,
         10 as ten
       }}},
   visualize as line]
  data [format=function] {
    var x : interval [5:10];
    func y = \value x * \value x;
  };
\end{codeexample}


A value like ``2'' or ``17'' could just be used as \meta{text} to be
displayed in the node of a tick label. However, things are more
difficult when the to-be-shown value is $0.0000000015$, because we
then would typically (but not always) prefer something like $1.5 \cdot
10^{-9}$ to be shown. Also, we might wish a unit to be added like
$23\mathrm{m}/\mathrm{s}$. Finally, we might wish a number like
$3.141$ to be replaced by $\pi$. For these reasons, the data
visualization system does not simply put the to-be-shown value in a
node as plain text. Instead, the number is passed to a
\emph{typesetter} whose job it is to typeset this number nicely using
\TeX's typesetting capabilities. The only exception is, as indicated
above, the third syntax version of the |at| and |also at| keys, where
\meta{text} is placed in the tick label's node, regardless of what the
typesetting would usually do.

The text produced by the automatic typesetting is computed as follows:
\begin{enumerate}
\item The current contents of the key |tick prefix| is put into the node.
\item This is followed by a call of the key |tick typesetting| which
  gets the \meta{value} of the tick as its argument in scientific
  notation.
\item This is followed by the contents of the key |tick suffix|.
\end{enumerate}

Let us have a look at these keys in detail:

\begin{key}{/tikz/data visualization/tick prefix=\meta{text}
    (initially \normalfont empty)}
  The \meta{text} will be put in front of every typeset tick:
\begin{codeexample}[]
\tikz \datavisualization
  [scientific axes, all axes={ticks=few, length=2.5cm},
   x axis={ticks={tick prefix=$\langle$, tick suffix=$]$}},
   visualize as line]
  data [format=function] {
    var x : interval [5:10];
    func y = \value x * \value x;
  };
\end{codeexample}
\end{key}

\begin{key}{/tikz/data visualization/tick suffix=\meta{text}
    (initially \normalfont empty)}
  Works like |tick prefix|. This key is especially useful for adding
  units like ``cm'' or ``$\mathrm m/\mathrm s$'' to every tick
  label. For this reason, there is a (near) alias that is easier to memorize:
  \begin{key}{/tikz/data visualization/tick unit=\meta{roman math text}}
    A shorthand for |tick suffix={$\,\rm|\meta{roman math text}|$}|:
\begin{codeexample}[]
\tikz \datavisualization
  [scientific axes, all axes={length=3cm},
   x axis={ticks={tick unit=s}},
   y axis={ticks={tick unit=m/s^2}},
   visualize as line]
  data [format=function] {
    var x : interval [5:10];
    func y = \value x * \value x;
  };
\end{codeexample}
  \end{key}
\end{key}

\begin{key}{/tikz/data visualization/tick typesetting=\meta{value}}
  The key gets called for each number that should be typeset. The
  argument \meta{value} will be in scientific notation (like |1.0e1|
  for $10$). By default, this key applies |\pgfmathprintnumber| to its
  argument. This command is a powerful number printer whose
  configuration is documented in
  Section~\ref{pgfmath-numberprinting}. 

  You are invited to code underlying this key so that a different
  typesetting mechanism is used. Here is a (not quite finished)
  example that shows how, say, numbers could be printed in terms of
  multiples of $\pi$:
\begin{codeexample}[]
\def\mytypesetter#1{%
  \pgfmathparse{#1/pi}%
  \pgfmathprintnumber{\pgfmathresult}$\pi$%
}
\tikz \datavisualization
  [school book axes, all axes={unit length=1.25cm},
   x axis={ticks={step=(0.5*pi), tick typesetter/.code=\mytypesetter{##1}}},
   y axis={include value={-1,1}},
   visualize as smooth line]
  data [format=function] {
    var x : interval [0.5:7];
    func y = sin(\value x r);
  };
\end{codeexample}
\end{key}




\subsubsection{Stacked Ticks}

\label{section-dv-stacking}



Sometimes, the text of tick labels are so long or so numerous that the
text of adjacent tick labels overlap (or have too little padding):
\begin{codeexample}[]
\tikz \datavisualization [scientific axes,
                          all axes={length=2.5cm},
                          visualize as smooth line]
  data [format=function] {
    var y : interval[-100:100];
    func x = \value y*\value y;
  };
\end{codeexample}
There are two ways to address this problem:
\begin{itemize}
\item One can rotate the labels on horizontal axes:
\begin{codeexample}[]
\tikz \datavisualization [scientific axes,
                          all axes={length=2.5cm},
                          x axis={ticks={node style={rotate=90, anchor=east}}},
                          visualize as smooth line]
  data [format=function] {
    var y : interval[-100:100];
    func x = \value y*\value y;
  };
\end{codeexample}
  This is often a good solution, but may be hard to read. Also
  consider rotating labels only by $45^\circ$ or $30^\circ$.
\item One can specify different shifts of the nodes for the different
  ticks, whereby the ticks text no longer overlap.
\begin{codeexample}[]
\tikz \datavisualization [scientific axes,
                          all axes={length=2.5cm},
                          x axis={ticks={major at={0,4000,8000,
                                2000 as [node style={yshift=-1em}],
                                6000 as [node style={yshift=-1em}],
                                10000 as [node style={yshift=-1em}]}}},
                          visualize as smooth line]
  data [format=function] {
    var y : interval[-100:100];
    func x = \value y*\value y;
  };
\end{codeexample}
  However, specifying shifts ``by hand'' in the above way is not
  always an option, especially when the tick positions should be
  computed automatically. Instead, the |stack| option can be used,
  which is much easier to use and gives better results:
\begin{codeexample}[]
\tikz \datavisualization [scientific axes,
                          all axes={length=2.5cm}, x axis={ticks=stack},
                          visualize as smooth line]
  data [format=function] {
    var y : interval[-100:100];
    func x = \value y*\value y;
  };
\end{codeexample}
\end{itemize}


The |stack| option is actually just a style that gives you access to
the general even/odd mechanism for ticks with labels. Whenever a tick
mark is created where a tick label is also to be drawn, two special
things happen:

\begin{enumerate}
\item For every odd tick mark, the |every odd tick| style is executed,
  for every even tick mark the |every even tick|. Here, ``odd'' and
  ``even'' are with respect to the order in which the ticks have been
  added to the list of |at| positions for each major, minor, or
  subminor tick list, not with respect to the order in which they will
  appear on the axis. Thus, when you write 
\begin{codeexample}[code only]
ticks={major at={1,2,3,4}, major at={0,-1,-2}, minor at={9,8,7}}
\end{codeexample}
  then for |1|, |3|, |0|, and |-2| as well as |9| and |7| the key
  |every odd tick| will be executed, while |every even tick| will be
  executed for positions |2|, |4|, |-1|, and also |8|.
\item When a tick node label is shown at the |low| position of the
  tick mark, the dimension stored in the key
  |tick text low even padding| is added to the |low| value. Provided
  that this padding is not zero (which is the default), the length of
  the even tick marks will be increased and the tick label node will
  be placed at a greater distance from the axis.

  Similar keys exist for padding ticks with labels at high positions
  and also at even positions.
\end{enumerate}


\begin{key}{/tikz/data visualization/tick text low even
    padding=\meta{dimension} (initially 0pt)}
  When a tick label is shown at the low position of an even tick, the
  \meta{distance} is added to the |low| value, see also
  Section~\ref{section-dv-visualize-ticks}.
\begin{codeexample}[]
\tikz \datavisualization [scientific axes,
                          all axes={length=2.5cm},
                          x axis={ticks={tick text low even padding=-1em}},
                          visualize as smooth line]
  data [format=function] {
    var y : interval[-100:100];
    func x = \value y*\value y;
  };
\end{codeexample}
  Note that \meta{dimension} should usually be non-positive.
\end{key}

The following keys work similarly:
\begin{key}{/tikz/data visualization/tick text low odd padding=\meta{dimension} (initially 0pt)}
\end{key}
\begin{key}{/tikz/data visualization/tick text high even padding=\meta{dimension} (initially 0pt)}
\end{key}
\begin{key}{/tikz/data visualization/tick text high odd padding=\meta{dimension} (initially 0pt)}
\end{key}

\begin{key}{/tikz/data visualization/tick text odd
    padding=\meta{dimension}}
  A shorthand for setting |tick text odd low padding| and
  |tick text odd high padding| at the same time.
\end{key}

\begin{key}{/tikz/data visualization/tick text even
    padding=\meta{dimension}}
  A shorthand for setting |tick text even low padding| and
  |tick text even high padding| at the same time.
\end{key}

\begin{key}{/tikz/data visualization/tick text 
    padding=\meta{dimension}}
  Sets all text paddings to \meta{dimension}.
\end{key}

\begin{key}{/tikz/data visualization/stack=\meta{dimension} (default 1em)}
  Shorthand for |tick text even padding=|\meta{dimension}.  
\begin{codeexample}[]
\tikz \datavisualization [scientific axes,
                          all axes={length=2.5cm},
                          x axis={ticks={stack=1.5em}},
                          visualize as smooth line]
  data [format=function] {
    var y : interval[-100:100];
    func x = \value y*\value y;
  };
\end{codeexample}
\end{key}

\begin{key}{/tikz/data visualization/stack'=\meta{dimension}}
  Shorthand for |tick text odd padding=|\meta{dimension}. The
  difference to |stack| is that the set of value that are
  ``lowered'' is exactly exchanged with the set of value ``lowered''
  by |stack|.
\begin{codeexample}[]
\tikz \datavisualization [scientific axes,
                          all axes={length=2.5cm},
                          x axis={ticks=stack'},
                          visualize as smooth line]
  data [format=function] {
    var y : interval[-100:100];
    func x = \value y*\value y;
  };
\end{codeexample}
\end{key}

Note that the above keys have an effect on all tick labels of an axis,
also on special ticks that you may have added using the |also at|
key. When using the |stack| key, you should specify a |tick text padding|
explicitly for such keys:

\begin{codeexample}[width=7cm]
\tikz \datavisualization
  [scientific axes,
   x axis={ticks={stack, many, major also at=
     {(pi) as [{tick text padding=2.5em}] $\pi$}}},
   visualize as smooth line]
  data [format=function] {
    var x : interval[0:(2*pi)];
    func y = sin(\value x r);
  };
\end{codeexample}


\subsubsection{Reference: Basic Strategies}

\label{section-dv-strategies}

\begin{key}{/tikz/data visualization/axis options/linear steps}
  This strategy places ticks at positions that are evenly spaced by
  the current value of |step|.

  In detail, the following happens: Let $a$ be the minimum value of the
  data values along the axis and let $b$ be the maximum. Let the
  current \emph{stepping} be $s$ (the stepping is set using the |step|
  option, see below) and let the current \emph{phasing} be $p$ (set
  using the |phase|) option. Then ticks are placed all positions
  $i\cdot s + p$ that lie in the interval $[a,b]$, where $i$ ranges
  over all integers.

  The tick positions computed in the way described above are
  \emph{mayor} step positions. In addition to these, if the key
  |minor steps between steps| is set to some number $n$, then $n$ many
  minor ticks are introduced between each two mayor ticks (and also
  before and after the last mayor tick, provided the values still lie
  in the interval $[a,b]$). Note that is $n$ is $1$, then one minor tick
  will be added in the middle between any two mayor ticks. Use a value
  of $9$ (not $10$) to partition the interval between two mayor ticks
  into ten equally sized minor intervals.

\begin{codeexample}[]
\begin{tikzpicture}
  \datavisualization
    [scientific axes={inner ticks, width=3cm},
     x axis={ticks={step=3, minor steps between steps=2}},
     y axis={ticks={step=.36}},
     visualize as scatter]
    data {
      x, y
      17, 30
      34, 32
    };
\end{tikzpicture}
\end{codeexample}
\end{key}

\label{section-dv-exponential-strategy}

\begin{key}{/tikz/data visualization/axis options/exponential steps}
  This strategy produces ticks at positions that are appropriate for
  logarithmic plots. It is automatically selected when you use the
  |logarithmic| option with an axis.

  In detail, the following happens: As for |linear steps| let numbers
  $a$, $b$, $s$, and $p$ be given. Then, mayor ticks are placed at all
  positions $10^{i\cdot s+p}$ that lie in the interval $[a,b]$ for $i
  \in \mathbb Z$.

  The minor steps are added in the same way as for |linear steps|. In
  particular, they interpolate \emph{linearly} between mayor steps.

\begin{codeexample}[]
\begin{tikzpicture}
  \datavisualization
    [scientific axes,
     x axis={logarithmic, length=2cm, ticks={step=1.5}},
     y axis={logarithmic, ticks={step=1, minor steps between steps=9}},
     visualize as scatter]
    data {
      x, y
      1, 10
      1000, 1000000
    };
\end{tikzpicture}
\end{codeexample}
\end{key}



\subsubsection{Advanced: Defining New Placement Strategies}

\begin{key}{/tikz/data visualization/axis options/tick placement
    strategy=\meta{macro}} 
  This key can be used to install a so-called \emph{tick placement
    strategy}. Whenever |visualize ticks| is used to request some
  ticks to be visualized, it is checked whether some automatic ticks
  should be created. This is the case when the following key is set:
  \begin{key}{/tikz/data visualization/compute step=\meta{code}}
    The \meta{code} should compute a suitable value for the stepping
    to be used by the \meta{macro} in the tick placement strategy.

    For instance, the |step| key sets |compute step| to
    |\def\tikz@lib@dv@step{#1}|. Thus, when you say |step=5|, then the
    desired stepping of |5| is communicated to the \meta{macro} via the
    macro |\tikz@lib@dv@step|.
  \end{key}

  Provided |compute step| is set to some nonempty value, upon
  visualization of ticks the \meta{macro} is executed. Typically,
  \meta{macro} will first call the \meta{code} stored in the key
  |compute step|. Then, it should implement some strategy then uses
  the value of the computed or desired stepping to create appropriate
  |at| commands. To be precise, it should set the keys |major|,
  |minor|, and/or |subminor| with some appropriate |at| values.

  Inside the call of \meta{macro}, the macro |\tikzdvaxis| will have
  been set to the name of the axis for which default ticks need to be
  computed. This allows you to access the minimum and the maximum
  value stored in the |scaling mapper| of that axis.
\begin{codeexample}[width=7cm]
\def\silly{
  \tikzdatavisualizationset{major={at={
        2,3,5,7,11,13}}}
}
\begin{tikzpicture}
  \datavisualization [
    scientific axes, visualize as scatter,
    x axis={tick placement strategy=\silly}
    ]
    data {
      x, y
      0, 0
      15, 15
    };
\end{tikzpicture}
\end{codeexample}
\end{key}





\subsection{Advanced: Creating New Axis Systems}

The data visualization library comes with a number of predefined axis
systems, like |scientific axes=clean|, but it is also possible and
to define new axis systems. Doing so involves the following steps:

\begin{enumerate}
\item Creating a number of axes.
\item Configuring attributes of these axes like their length or
  default scaling.
\item Creating visual representations of the axes.
\item Creating visual representations of the ticks and grid lines.
\end{enumerate}

The first step uses |new ... axis| keys to create new axes, the
last steps use |visualize ...| keys to create the visual
representations of the axes.

Note that the axis system has no control over the actual attribute
value ranges and neither over which ticks need to be drawn. The axis
system can only provide good defaults and then specify \emph{how} the
ticks or labels should be drawn and \emph{where} on the page -- but
not at which values.

In the following, as a running example let us develop an axis system
|our system| that does the following: For the $x$-axis is looks like a
normal scientific axis system, but there are actually two $y$-axes:
One at the left and one at the right, each using a different
attribute, but both coexisting in the same picture.

\subsubsection{Creating the Axes}

A new axis system is created as a style key with the prefix
|/tikz/data visualization|. Thus, we would write:
\begin{codeexample}[code only]
\tikzset{
  data visualization/our system/.style={
    ...
  }
}
\end{codeexample}

In our system we need three axis: The $x$-axis, the left axis and the
right axis. Since all of these axes are Cartesian axes, we write the
following:
\begin{codeexample}[code only]
\tikzset{
  data visualization/our system/.style={
    new Cartesian axis=x axis,
    new Cartesian axis=left axis,
    new Cartesian axis=right axis,
    x axis={attribute=x},
    left axis={unit vector={(0cm,1pt)}},
    right axis={unit vector={(0cm,1pt)}},
  }
}
\end{codeexample}
As can be seen, we also configure things so that the $x$-axis will use
the |x| attribute by default (users can later change this by saying
|x axis={attribute=|\meta{some other attribute}|}|), but we do not
configure the attributes of the |left axis| nor the |right axis|. We
also make the left and right axis point upward (the |x axis| needs no
configuration here since a Cartesian axis points right by default). The
reason is the |left| would not be a particularly good attribute name
and this way we ensure that users have to pick names themselves
(hopefully good ones).

The next step is to define a standard scaling for the axes. Here, we
can use the same as for |scientific axes|, so we would add the
following keys to the definition of |our system|:
\begin{codeexample}[code only]
x axis    ={length=\pgfkeysvalueof{/tikz/data visualization/scientific axes/width}},
left axis ={length=\pgfkeysvalueof{/tikz/data visualization/scientific axes/height}},
right axis={length=\pgfkeysvalueof{/tikz/data visualization/scientific axes/height}}
\end{codeexample}

We now already have enough to try our system, although we will not yet
see any axes or ticks, but we will see the correct scaling of the
attributes. Let us first define a data group:
\begin{codeexample}[]
\tikz \datavisualization data group {people and money} = {
    data [set=people 1] {
      time, people
      1900, 1000000000
      1920, 1500000000
      1930, 2000000000
      1980, 3000000000
    }
    data [set=people 2] {
      time, people
      1900, 2000000000
      1920, 2500000000
      1940, 4000000000
      2000, 5700000000
    }
    data [set=money 1] {
      time, money
      1910, 1.1
      1920, 2
      1930, 5
      1980, 2
    }
    data [set=money 2] {
      time, money
      1950, 3
      1960, 3
      1970, 4
      1990, 3.5
    } 
  };
\end{codeexample}

\tikzdatavisualizationset{
  our system/.style={
    new Cartesian axis=x axis,
    new Cartesian axis=left axis,
    new Cartesian axis=right axis,
    x axis={attribute=x},
    left axis={unit vector={(0cm,1pt)}},
    right axis={unit vector={(0cm,1pt)}},
    x axis    ={length=\pgfkeysvalueof{/tikz/data visualization/scientific axes/width}},
    left axis ={length=\pgfkeysvalueof{/tikz/data visualization/scientific axes/height}},
    right axis={length=\pgfkeysvalueof{/tikz/data visualization/scientific axes/height}}
  }
}
\begin{codeexample}[]
\tikz \datavisualization [
    our system,
    x axis={attribute=time, length=4cm},
    left axis ={attribute=money},
    right axis={attribute=people},
    visualize as line/.list={people 1, people 2, money 1, money 2},
    people 1={style={visualizer color=blue}},
    people 2={style={visualizer color=blue!50}},
    money 1={style={visualizer color=red}},
    money 2={style={visualizer color=red!50}}]
  data group {people and money};  
\end{codeexample}



\subsubsection{Visualizing the Axes}

\label{section-dv-visualize-axis}

We must now show the axes themselves. For this we can use the
|visualize axis| key:

\begin{key}{/tikz/data visualization/axis options/visualize axis=\meta{options}}
  This key is passed to an axis as an option. It causes a visual
  representation of the axis to be created during the data visualization. The
  \meta{options} are used to determine where the axis should be drawn
  and how long it should be. We can specify, for instance, that an
  axis should be drawn at the minimum value of another axis or where
  another axis has the value |0|.
  
  \medskip
  \textbf{The goto, high, and low Keys.}
  In our example, the |left axis| should be shown at the left hand
  side. This is the position where the |x axis| has its minimum
  value. To specify this, we would use the following code:
\begin{codeexample}[code only]
left axis={ visualize axis={ x axis={ goto=min } }
\end{codeexample}
  As can be seen, we can pass another axis as an \meta{option} to
  |visualize axis|, where we pass the following key to the axis in
  turn: 
  \begin{key}{/tikz/data visualization/axis options/goto=\meta{value}}
    The key can be passed to an axis. It will set the attribute
    monitored by the axis to the given \meta{value}, which is usually
    some number. However, \meta{value} may also be one of the
    following, which causes a special behaviour:
    \begin{itemize}
    \item
      |min|: The attribute is set to the minimal value that the
      attribute has attained along this axis.
    \item
      |max|: Like |min|.
    \item
      |padded min|: This will also set the \meta{attribute} monitored
      by the axis to the same value  as |min|. Additionally, however,
      the subkey |/data point/|\meta{attribute}|/offset| is set to the
      current padding for the minimum, see the description of
      |padding min| later on. The effect of this is that the actual
      point ``meant'' by the attribute is offset by this padding along
      the attribute's axis.
    \item
      |padded max|: Like |padded min|.
    \end{itemize}
  \end{key}

  The |right axis| would be visualized the same way, only at
  |goto=max|. The $x$-axis actually need to be visualized
  \emph{twice}: Once at the bottom and once at the top. Thus, we need
  to call |visualize axis| twice for this axis:
  
\tikzdatavisualizationset{
  our system/.style={
    new Cartesian axis=x axis,
    new Cartesian axis=left axis,
    new Cartesian axis=right axis,
    x axis={attribute=x},
    left axis={unit vector={(0cm,1pt)}},
    right axis={unit vector={(0cm,1pt)}},
    x axis    ={length=\pgfkeysvalueof{/tikz/data visualization/scientific axes/width}},
    left axis ={length=\pgfkeysvalueof{/tikz/data visualization/scientific axes/height}},
    right axis={length=\pgfkeysvalueof{/tikz/data visualization/scientific axes/height}}
  }
}
\begin{codeexample}[]
\tikzset{
  data visualization/our system/.append style={
    left axis= {visualize axis={x axis=   {goto=min}}},
    right axis={visualize axis={x axis=   {goto=max}}},
    x axis=    {visualize axis={left axis={goto=min}},
                visualize axis={left axis={goto=max}}},
 }
}
\tikz \datavisualization [
    our system,
    x axis={attribute=time, length=4cm},
    left axis ={attribute=money},
    right axis={attribute=people},
    visualize as line/.list={people 1, people 2, money 1, money 2}]
  data group {people and money};  
\end{codeexample}

  There is another key that is similar to |goto|, but has a slightly
  different semantics:
  \begin{key}{/tikz/data visualization/axis options/goto pos=\meta{fraction}}
    The key works like |goto|, only the \meta{fraction} is not
    interpreted as a value but as a fraction of the way between the
    minimum and the maximum value for this axis.

    Suppose that for an axis the attribute range interval is
    $[500,1000]$ and the reasonable interval is $[1,3]$. Then for a
    \meta{fraction} of |0|, the mapping process would choose value
    $1$ from the reasonable interval, for a \meta{fraction} of |1| the
    position $3$ from the reasonable interval, and for a
    \meta{fraction} or |0.25| the position $1.5$ since it is one
    quarter at the distance from $1$ to $3$.

    Note that neither the attribute range interval nor the
    transformation function for the attribute are important for the
    |goto pos| option -- the \meta{fraction} is computed with respect
    to the reasonable interval. Also note that the values of the
    actual attribute corresponding to the fractional positions in the
    reasonable interval are not computed.
\begin{codeexample}[]
\tikzset{
  data visualization/our system/.append style={
    x axis=    {visualize axis={left axis={goto pos=0.25}},
                visualize axis={left axis={goto pos=0.5}}},
 }
}
\tikz \datavisualization [
    our system,
    x axis={attribute=time, length=4cm},
    left axis ={attribute=money},
    right axis={attribute=people},
    visualize as line/.list={people 1, people 2, money 1, money 2}]
  data group {people and money};  
\end{codeexample}
  \end{key}

  By default, when an axis is visualized, it spans the set of all
  possible values for the monitored attribute, that is, from |min| to
  |max|. However, there are actually two keys that allow you to adjust
  this:
  \begin{key}{/tikz/data visualization/low=\meta{value}}
    This is the attribute value where the axis visualization
    starts. The same special values as for |goto| are permissible
    (like |min| or |padded min|, but also |0| or |1|).
  \end{key}
  \begin{key}{/tikz/data visualization/high=\meta{value}}
    Like |low|, only for where the axis ends.
  \end{key}

  By default, |low=min| and |high=max| are set for an axis
  visualization. Another sensible setting is |low=padded min| and
  |high=padded max|. The following key provides a shorthand for this:
  \begin{key}{/tikz/data visualization/padded}
    Shorthand for |low=padded min, high=padded max|.
  \end{key}
  As an example, consider the |scientific axes=clean|. Here, each axis
  is actually drawn three times: Once at the minimum, once at the
  maximum and then once more at the padded minimum. 

    
  \medskip
  \textbf{The axis line.}
  When an axis is drawn, \tikzname\ does not simply draw a straight
  line from the |low| position to the |high| position. In reality, the
  data visualization system uses the two commands |\pgfpathdvmoveto|
  and |\pgfpathdvlineto| internally. These will replace the straight
  line by a curve in certain situations. For instance, in a polar
  coordinate system, if an axis should be drawn along an angle axis
  for a fixed radius, an arc will be used instead of a straight line.
    
  \medskip
  \textbf{Styling the axis.}
  As can be seen, we now get the axis we want (but without the ticks,
  visualizing them will be explained later). The axis is, however,
  simply a black line. We can \emph{style} the axis in a manner
  similar to styling ticks and grid lines, see
  Section~\ref{section-dv-style}. In detail, the following styles get
  executed:
  \begin{enumerate}
  \item |axis layer|
  \item |every axis|
  \item |styling|
  \end{enumerate}
  Additionally, even before |every axis| is executed, |low=min| and
  |high=max| are executed.
  
  \begin{stylekey}{/tikz/data visualization/axis layer (initially on
      background layer)}
    The layer on which the axis is drawn. See the description of
    |grid layer| on page~\ref{section-dv-grid-layer} for details.
  \end{stylekey}
  
  \begin{stylekey}{/tikz/data visualization/every axis}
    Put styling of the axis here. It is usually a good idea to set
    this style to |style={black!50}|.
  \end{stylekey}

  Recall that the |styling| key is set using the |style| key, see
  Section~\ref{section-dv-style}. 
\tikzset{
}
\begin{codeexample}[]
\tikzset{
  data visualization/our system/.append style={
    every axis/.style={style=black!50}, % make this the default
    left axis= {visualize axis={x axis=   {goto=min}, style=red!75}},
    right axis={visualize axis={x axis=   {goto=max}, style=blue!75}},
    x axis=    {visualize axis={left axis={goto=min}},
                visualize axis={left axis={goto=max}}},
 }
}
\tikz \datavisualization [
    our system,
    x axis={attribute=time, length=4cm},
    left axis ={attribute=money},
    right axis={attribute=people},
    visualize as line/.list={people 1, people 2, money 1, money 2}]
  data group {people and money};  
\end{codeexample}
\tikzset{
  data visualization/our system/.append style={
    every axis/.style={style=black!50}, % make this the default
    left axis= {visualize axis={x axis=   {goto=min}, style=red!75}},
    right axis={visualize axis={x axis=   {goto=max}, style=blue!75}},
    x axis=    {visualize axis={left axis={goto=min}},
                visualize axis={left axis={goto=max}}},
 }
}
  
  \medskip
  \textbf{Padding the Axis.}
  When an axis is visualized, it is often a good idea to make it ``a
  little bit longer'' or to ``remove it a bit from the border,'' because
  the visualization of an axis should not interfere with the actual
  data. For this reason, a \emph{padding} can be specified for axes:
  
  \begin{key}{/tikz/data visualization/axis options/padding min=\meta{dimension}}
    This is the dimension that is used whenever |goto=padded min| is
    used. The \meta{dimension} is then put into the |offset| subkey of
    the attribute monitored by the axis. When a data point is
    transformed by a linear transformer and when this subkey is
    nonzero, this offset is added. (For an angle axis of a polar
    transformer, the \meta{dimension} is interpreted as an additional
    angle rather than as an additional distance). Note that
    \meta{dimension} should typically be negative since 
    ``adding the \meta{dimension}'' will then make the axis longer
    (because it starts at a smaller value). The standard
    axis systems set the padding to some default and take its value
    into account: 
\begin{codeexample}[width=8cm]
\begin{tikzpicture}
  \datavisualization [scientific axes=clean,
                      x axis={padding min=-1cm},
                      visualize as smooth line]
    data [format=function] {
      var x : interval [-3:5];
      func y = \value x * \value x;
    };
\end{tikzpicture}
\end{codeexample}

  Using padded and using the |padded| key, we can visualize our axis
  ``a little removed from the actual data'':
\begin{codeexample}[]
\tikzset{
  data visualization/our system/.append style={
    all axes=  {padding=.5em},
    left axis= {visualize axis={x axis=   {goto=padded min}, padded}},
    right axis={visualize axis={x axis=   {goto=padded max}, padded}},
    x axis=    {visualize axis={left axis={goto=padded min}, padded},
                visualize axis={left axis={goto=padded max}, padded}},
 }
}
\tikz \datavisualization [
    our system,
    x axis={attribute=time, length=3cm},
    left axis ={attribute=money},
    right axis={attribute=people},
    visualize as line/.list={people 1, people 2, money 1, money 2}]
  data group {people and money};  
\end{codeexample}  
  \end{key}

  \begin{key}{/tikz/data visualization/axis options/padding max=\meta{dimension}}
    Works like |padding min|, but \meta{dimension} should typically be positive.
  \end{key}
  
  \begin{key}{/tikz/data visualization/axis options/padding=\meta{dimension}}
    Sets both |padding min| to the negated value of \meta{dimension} and
    |padding max| to \meta{dimension}.
  \end{key}
\end{key}



\subsubsection{Visualizing Grid Lines}

\label{section-dv-visualize-gridlines}

As explained earlier, the |grid| key is used to specify at which
positions grid lines should be drawn in principle. However, this key
does not actually cause any grid lines to be drawn. Instead, the
|visualize grid| key is used by the axis system to specify how grid
lines are drawn.

\begin{key}{/tikz/data visualization/axis options/visualize grid=\meta{options}}
  This key is passed to an axis. It causes grid lines to be drawn at
  the positions specified by the |grid| key for this axis. The
  \meta{options} govern where and how the grid lines will be drawn.
  
  \medskip
  \textbf{The direction axis.}
  At first sight, one might expect that the grid lines for an axis
  should simply be drawn perpendicular to the axis between the minimum
  and maximum value of the axis. However, things are somewhat more
  difficult in reality:
  \begin{enumerate}
  \item A grid line is supposed to indicate all positions where a
    certain attribute attains a fixed value. But, then, a grid line
    does not really need to be a grid \emph{line}. Consider for
    instance a three dimensional axis system. A ``grid line'' for the
    $x$-coordinate |3| would actually     be a ``grid plane.''
  \item For a polar coordinate  system and a fixed radius, this set of
    positions at a certain radius is not a straight line, but an
    arc. For more complicated coordinate systems such as the one
    arising from three-dimensional spherical projections, a grid line
    may well be a fairly involved curve. 
  \end{enumerate}
  The |visualize grid| command addresses these complications as
  follows:
  \begin{enumerate}
  \item A grid line is always a line, not a plane or a volume. This
    means that in the example of a three dimensional axis system and
    the $x$-attribute being |3|, one would have to choose whether the
    grid line should go ``along'' the $y$-axis or ``along'' the
    $z$-axis for this position. One can, however, call the
    |visualize grid| command twice, once for each direction, to cause
    grid lines to be shown for both directions.
  \item A grid line is created by moving to a start position and then
    doing a lineto to the target position. However, the ``moveto''
    and ``lineto'' are done by calling special commands of the data
    visualization system. These special commands allow coordinate
    system to ``notice'' that the line is along an axis and will allow
    them to replace the straight line by an appropriate curve. The
    polar axes systems employ this strategy, for instance.
  \end{enumerate}
  
  By the above discussion, in order to create a grid line for
  attribute $a$ having value $v$, we need to specify an axis ``along''
  which the line should be drawn. When there  are only two axes, this
  is usually ``the other axis''. This ``other axis'' is specified using the following key:
  \begin{key}{/tikz/data visualization/direction axis=\meta{axis name}}
    You must pass this key as an \meta{option} each time you use
    |visualize axis|. When the grid line is drawn, the attribute $a$
    is set to $v$ and the axis \meta{axis name}'s attribute is set
    once to the current value of |low| and once to |high|. Then a line
    is drawn between these two positions using |\pgfpathdvlineto|. 
  \end{key}
  The |low| and |high| keys are the same as the ones used in the
  |visualize axis| key.
  
\begin{codeexample}[]
\tikz \datavisualization [
    xyz Cartesian cabinet,
    all axes={visualize axis={low=0, style=->}},
    x axis={visualize grid={direction axis=y axis}, grid=many},
    visualize as scatter]
  data {
    x, y, z
    0, 0, 1
    0, 1, 0
    2, 2, 2
  };  
\end{codeexample}  
\begin{codeexample}[]
\tikz \datavisualization [
    xyz Cartesian cabinet,
    all axes={visualize axis={low=0, style=->}, grid=many},
    x axis={visualize grid={direction axis=z axis}}, 
    z axis={visualize grid={direction axis=x axis},
            visualize grid={direction axis=y axis},}, 
    visualize as scatter]
  data {
    x, y, z
    0, 0, 1
    0, 1, 0
    2, 2, 2
  };  
\end{codeexample}  

  \medskip
  \textbf{Styling the grid lines.}
  When a grid line is draw, styles are applied as described in
  Section~\ref{section-dv-styling-grid-lines}.
  
  \medskip
  \textbf{The major, minor, and subminor grid lines.}
  The |grid| option allows you to specify for each kind of grid line
  (major, minor, or subminor) a set of different values for which
  these grid lines should be drawn. Correspondingly, it is also
  possible to configure for each kind of grid line how it should be
  drawn. For this, the |major|, |minor|, |subminor|, and also the
  |common| keys can be used inside the \meta{options} of
  |visualize grid|. While as option to |grid| these keys are used to
  specify |at| values, as options of |visualize grid| they are used to
  configure the different kinds of grid lines.

  Most of the time, no special configuration is necessary since all
  styling is best done by configuring keys like
  |every major grid|. You need to use a key like |major| only if you
  wish to configure for instance the |low| or |high| values of a
  |major| grid line differently from those of |minor| grid lines --
  are rather unlikely setting -- or when the styling should deviate
  from the usual settings.
\begin{codeexample}[]
\tikz \datavisualization [
    xy Cartesian,
    all axes={visualize axis={low=0, style=->},
              grid={some, minor steps between steps}},
    x axis=  {visualize grid={
                direction axis=y axis,
                minor={low=0.25, high=1.75, style=red!50}}},
    visualize as scatter]
  data {
    x, y
    0, 0
    3, 3
  };  
\end{codeexample}  
\end{key}

Returning to the example of |our system| with the two axis systems, it
is straight-forward to configure the grid lines of the $x$-axis: The
direction axis is either of the other two axis (they point in the same
direction and they have the same range). For the other two axes, we
visualize one grid independently of the other, using different colors.


\begin{codeexample}[]
\tikzset{
  data visualization/our system/.append style={
    x axis=    {visualize grid={direction axis=left axis}},
    left axis= {visualize grid={direction axis=x axis,
                                common={style=red!50}}},
    right axis={visualize grid={direction axis=x axis,
                                common={style=blue!50}}},
  }
}
\tikz \datavisualization [
    our system,
    x axis={attribute=time, length=3cm, grid=many},
    left axis ={attribute=money, grid=some},
    right axis={attribute=people, grid=few},
    visualize as line/.list={people 1, people 2, money 1, money 2}]
  data group {people and money};  
\end{codeexample}



\subsubsection{Visualizing the Ticks and Tick Labels}

\label{section-dv-visualize-ticks}

\begin{key}{/tikz/data visualization/axis options/visualize ticks=\meta{options}}
  Visualizing a tick involves (possibly) drawing a tick mark and
  adding (possibly) the tick node. The process is similar to
  |visualize grid|: Users use the |ticks| key to configure how many
  ticks they would like for an axis and at which positions. The axis
  system uses the |visualize ticks| key to specify where these ticks
  should actually be shown.

  Unlike grid lines, which are typically only visualized once for each
  combination of an axis and a direction axis, tick marks might be
  visualized at different places for the same axis. Consider for
  instance the |scientific axes|:
\begin{codeexample}[]
\tikz \datavisualization [scientific axes, all axes={length=3cm},
                          x axis={ticks={stack}},
                          visualize as smooth line]
  data [format=function] {
    var x : interval [0:2];
    func y = \value  x*\value x;
  };
\end{codeexample}
  Have a look at the ticks on the $y$-axis: There are ticks at values |0|,
  |1|, |2|, |3|, and~|4|. These are visualized both at the left
  side (where the tick nodes are also shown) and additionally also at
  the right side, but only as small marks. Similarly, the ticks on the
  $x$-axis appear at the bottom, but also (in much simpler versions)
  at the top. Both for the $x$-axis and for the $y$-axis the
  |visualize ticks| key was called twice.

  \medskip
  \textbf{The tick marks.}
  Drawing a tick mark is quite similar to visualizing a grid line;
  indeed a tick mark can be thought of as a ``mini grid line'': Just
  like a grid line it ``points a long an axis.'' However, a tick will
  always be a short straight line -- even when the coordinate system
  is actually twisted (experimentation has shown that ticks that
  follow the curvature of the coordinate system like grid lines are
  hard to recognize). For this reason, the |low| and |high| keys have
  a different meaning from the one used with the |visualize grid|
  key. In detail to configure the size and position of a tick mark for
  the value $v$ of attribute $a$, proceed as follows:
  \begin{itemize}
  \item The |visualize ticks| key will have setup attribute $a$ to be
    equal to $v$.
  \item You should now use the |goto| or |goto pos| key together with all
    \emph{other} axes to configure at which position with respect to
    these other options the tick mark should be shown. For instance,
    suppose we want tick marks in |our system| for the $x$-axis at the
    bottom and at the top. This corresponds to once setting the
    |left axis| to its minimal value and once to its maximal value:

\begin{codeexample}[]
\tikzset{
  data visualization/our system/.append style={
    x axis={visualize ticks={direction axis=left axis, left axis={goto=min}},
            visualize ticks={direction axis=left axis, left axis={goto=max}},
    }
  }
}
\tikz \datavisualization [
    our system,
    x axis={attribute=time, length=3cm, ticks=many},
    left axis ={attribute=money},
    right axis={attribute=people},
    visualize as line/.list={people 1, people 2, money 1, money 2}]
  data group {people and money};  
\end{codeexample}    
  \item In the above example, we may wish to shorten the ticks a bit
    at the bottom and at the top. For this, we use the |low| and
    |high| key:
    \begin{key}{/tikz/data visualization/low=\meta{dimension}}
      When used with the |visualize ticks| option, the |low| key
      contains a dimension that specifies the extend of the tick going
      ``toward the minimum'' of the direction axis. More precisely, when a tick
      mark is visualized, a unit tangent vector at the current data point
      in the direction of the |direction axis| is computed and this
      vector is multiplied by \meta{dimension} to compute the start
      position of the tick line. The end position is given by this
      vector times the |high| value.

      Note that the \meta{dimension} should usually be negative for
      the |low| key and positive for the |high| key.

      For tick marks where a tick label node is shown, the
      \meta{dimension} is increased by the current values of keys like
      |tick text even low padding|, see
      Section~\ref{section-dv-stacking} for details.
    \end{key}
    \begin{key}{/tikz/data visualization/high=\meta{dimension}}
      Like |low|.
    \end{key}
    \begin{key}{/tikz/data visualization/tick length=\meta{dimension}}
      Shorthand for |low=-|\meta{dimension}|, high=|\meta{dimension}.
    \end{key}
    
    What we want to happen is that in the upper visualization of the
    ticks the |low| value is |0pt|, while in the lower one the |high|
    value is |0pt|:    
\begin{codeexample}[]
\tikzset{
  data visualization/our system/.append style={
    x axis={
      visualize ticks={direction axis=left axis,high=0pt,left axis={goto=min}},
      visualize ticks={direction axis=left axis,low=0pt,left axis={goto=max}},
    }
  }
}
\tikz \datavisualization [
    our system,
    x axis={attribute=time, length=3cm, ticks=many},
    left axis ={attribute=money},
    right axis={attribute=people},
    visualize as line/.list={people 1, people 2, money 1, money 2}]
  data group {people and money};  
\end{codeexample}
\end{itemize}
  In order to style the tick mark, use the styling mechanism that is
  detailed in Section~\ref{section-dv-styling-ticks}.

  \medskip
  \textbf{The tick label node.}
  At certain tick positions, we may wish to add a node indicating the
  value of the attribute at the given position. The |visualize ticks|
  command has no influence over which text should be shown at a node
  -- the text is specified and typeset as explained in
  Section~\ref{section-dv-tick-labels}.

  Each time |visualize ticks|, for each tick position up to two tick
  label nodes will be created: One at the |low| position and one at
  the |high| position. The following keys are used to configure which
  of these cases happen:
  \begin{key}{/tikz/data visualization/tick text at
      low=\opt{\meta{true or false}} (default true)}
    Pass this option to |visualize ticks| when you want tick label
    nodes to be placed at the |low| position of each tick mark.
\begin{codeexample}[]
\tikzset{
  data visualization/our system/.append style={
    x axis={
      visualize ticks={direction axis=left axis, left axis={goto=min},
                       high=0pt, tick text at low, stack},
      visualize ticks={direction axis=left axis, left axis={goto=max},
                       low=0pt, tick text at high, stack}
    }
  }
}
\tikz \datavisualization [
    our system,
    x axis={attribute=time, length=3cm, ticks=some},
    left axis ={attribute=money},
    right axis={attribute=people},
    visualize as line/.list={people 1, people 2, money 1, money 2}]
  data group {people and money};  
\end{codeexample}
  \end{key}
  \begin{key}{/tikz/data visualization/tick text at
      high=\opt{\meta{true or false}} (default true)}
    Like |tick text at low|. 
  \end{key}
  
  \begin{key}{/tikz/data visualization/no tick text}
    Shorthand for |tick text at low=false, tick text at high=false|. 
\begin{codeexample}[]
\tikz \datavisualization [scientific axes, all axes={length=3cm},
                          x axis={ticks={
                              major also at={6.5 as [no tick text]}}},
                          visualize as smooth line]
  data [format=function] {
    var x : interval [5:10];
    func y = \value x * \value x;
  };
\end{codeexample}  
  \end{key}
  
  When a tick label node is to be placed at the low or the high
  position, the next step is to determine the exact position and the
  correct anchor of the node. This is done as follows:
  \begin{itemize}
  \item In order to compute an appropriate |anchor|, the tick mark is
    considered: This is a short line pointing in a certain
    direction. For a tick label node at the |low| position, the
    |anchor| attribute is setup in such a way that the node label will
    be below the |low| position when the tick mark direction points
    up, it will be to the right when the direction points left, above
    when it points down, and so on also for diagonal
    directions. Similarly, for the |high| position, when the direction
    points up, the node will be placed above the tick mark and so on.

    This computation is done automatically.
  \item The tick label node is styled. The styles that are applied are
    described in Section~\ref{section-dv-styling-ticks}.
  \item A tick label node for the |low| position is usually anchored
    at this |low| position, but an additional padding will be
    added as described in Section~\ref{section-dv-stacking}.
  \end{itemize}
\end{key}




\subsubsection{Visualizing the Axis Labels}

\label{section-dv-visualize-label}

The |label| option can be used with an axis to specify a text should
be shown next to the axis to indicates which attribute this axis
refers to. Like |ticks| or |grid|, the |label| option does not
actually draw the label, this is the job of the |visualize label| key,
which is configured by the axis system.

\begin{key}{/tikz/data visualization/axis options/visualize label=\meta{options}}
  The \meta{options} should be used to configure a ``good place'' for
  the axis label. Usually, you will use the |goto| or the |goto pos|
  key.

  For the example of |our system|, we would like the label of the
  |x axis| to be placed below at the middle of the axis, so we use
  |goto pos=.5| to determine this position. Concerning the other axes,
  we want it to be placed at the minimum position of the |left axis|
  with a lot of padding.
\begin{codeexample}[width=7cm]
\tikzdatavisualizationset{
  our system/.append style={
    x axis={visualize label={
        x axis={goto pos=.5},
        left axis={padding=1.5em, goto=padded min}}}
  }
}
\tikz \datavisualization [
    our system,     
    x axis={attribute=time, ticks=some, label},
    left axis ={attribute=money},
    right axis={attribute=people},
    visualize as line/.list={
      people 1, people 2, money 1, money 2}]
  data group {people and money};  
\end{codeexample}

  In the above example, the |padding| of |1.5em| was rather arbitrary
  and ``suboptimal''. It would be outright wrong if the labels on the |x axis| were larger or
  if they were missing. It would be better if the vertical position of
  the |x axis| label were always ``below'' all other options. For such
  cases a slightly strange approach is useful: You position the node
  using |node style={at=...}| where |at| 
  is now the normal \tikzname\ option that is used to specify the
  position of a node. Inside the |...|, you specify that the
  horizontal position should be the bottom of up-to-now-constructed
  data visualization and the vertical position should be at the
  ``origin,'' which is, however, the position computed by the |goto|
  keys for the axes:
\begin{codeexample}[width=7cm]
\tikzdatavisualizationset{
  our system/.append style={
    x axis={visualize label={
      x axis={goto pos=.5},
      node style={
        at={(0,0 |- data visualization bounding box.south)},
        below
} } } } }
\tikz \datavisualization [
    our system,     
    x axis={attribute=time, ticks=some, label=Year},
    left axis ={attribute=money},
    right axis={attribute=people},
    visualize as line/.list={
      people 1, people 2, money 1, money 2}]
  data group {people and money};  
\end{codeexample}

  Two additional keys are useful for positioning axis labels:
  \begin{key}{/tikz/data visualization/axis option/anchor at min}
    When passed to an axis, this key sets the |anchor| so that a node
    positioned at either the |min| or the |padded min| value of the
    axis will be placed ``nicely'' with respect to the axis. For
    instance, if the axis points upwards from the |min| value to the
    |max| value, the |anchor| would be set to |north| since this gives
    a label below the axis's start. Similarly, if the axis points
    right, the anchor would be set to |east|, and so on.
  \end{key}
  \begin{key}{/tikz/data visualization/axis option/anchor at max}
    Like |anchor at min|.
  \end{key}
\end{key}


\subsubsection{The Complete Axis System}

Here is the code for the complete axis system developed above and an
example of how it is used:

\begin{codeexample}[code only]
\tikzdatavisualizationset{ our system/.style={
  % The axes
  new Cartesian axis=x axis,     new Cartesian axis=left axis,         new Cartesian axis=right axis,
  % The directions of the axes
  all axes={padding=.5em},       left axis={unit vector={(0cm,1pt)}},  right axis={unit vector={(0cm,1pt)}},
  % The default attributes, other attributes must be configured
  x axis={attribute=x},
  % The lengths of the axes
  x axis    ={length=\pgfkeysvalueof{/tikz/data visualization/scientific axes/width}},
  left axis ={length=\pgfkeysvalueof{/tikz/data visualization/scientific axes/height}},
  right axis={length=\pgfkeysvalueof{/tikz/data visualization/scientific axes/height}},
  % The styling of the axes
  every axis/.style={style=black!50}, % make this the default
  % Visualizing the axes themselves
  left axis= {visualize axis={x axis=   {goto=padded min}, style=red!75, padded}},
  right axis={visualize axis={x axis=   {goto=padded max}, style=blue!75,padded}},
  x axis=    {visualize axis={left axis={goto=padded min}, padded},
              visualize axis={left axis={goto=padded max}, padded}},
  % Visualizing the grid, when requested
  x axis=    {visualize grid={direction axis=left axis}},
  left axis= {visualize grid={direction axis=x axis, common={style=red!50}}},
  right axis={visualize grid={direction axis=x axis, common={style=blue!50}}},
  % Visualizing the ticks, when requested
  left axis={visualize ticks={style={red!50!black}, direction axis=x axis, 
                              x axis={goto=padded min}, high=0pt, tick text at low}},
  right axis={visualize ticks={style={blue!80!black}, direction axis=x axis,
                              x axis={goto=padded max}, low=0pt, tick text at high}},
  x axis={visualize ticks={direction axis=left axis, left axis={goto=padded min}, high=0pt,
                           tick text at low},
          visualize ticks={direction axis=left axis, left axis={goto=padded max}, low=0pt}},
  % By default, there are ticks on all axes
  all axes={ticks},
  % Visualizing the axis labels, when requested
  x axis={visualize label={x axis={goto pos=.5}, node style={
        at={(0,0 |- data visualization bounding box.south)}, below}}},
  left axis={visualize label={left axis={goto pos=.5}, node style={
        at={(0,0 -| data visualization bounding box.west)}, rotate=90, anchor=south, red!50!black}}},
  right axis={visualize label={right axis={goto pos=.5}, node style={
        at={(0,0 -| data visualization bounding box.east)}, rotate=-90, anchor=south,  blue!80!black}}},
}}
\end{codeexample}


\tikzdatavisualizationset{
  our system/.style={
    % The axes
    new Cartesian axis=x axis,
    new Cartesian axis=left axis,
    new Cartesian axis=right axis,
    % The default attributes, other attributes must be configured
    x axis={attribute=x},
    % The directions of the axes
    all axes={padding=.5em},
    left axis={unit vector={(0cm,1pt)}},
    right axis={unit vector={(0cm,1pt)}},
    % The lengths of the axes
    x axis    ={length=\pgfkeysvalueof{/tikz/data visualization/scientific axes/width}},
    left axis ={length=\pgfkeysvalueof{/tikz/data visualization/scientific axes/height}},
    right axis={length=\pgfkeysvalueof{/tikz/data visualization/scientific axes/height}},
    % The styling of the axes
    every axis/.style={style=black!50}, % make this the default
    % Visualizing the axes themselves
    left axis= {visualize axis={x axis=   {goto=padded min}, style=red!75, padded}},
    right axis={visualize axis={x axis=   {goto=padded max}, style=blue!75,padded}},
    x axis=    {visualize axis={left axis={goto=padded min}, padded},
                visualize axis={left axis={goto=padded max}, padded}},
    % Visualizing the grid, when requested
    x axis=    {visualize grid={direction axis=left axis, padded}},
    left axis= {visualize grid={direction axis=x axis, padded, common={style=red!50}}},
    right axis={visualize grid={direction axis=x axis, padded, common={style=blue!50}}},
    % Visualizing the ticks, when requested
    left axis={
      visualize ticks={style={red!50!black}, direction axis=x axis, x axis={goto=padded min}, high=0pt, tick text at low}},
    right axis={
      visualize ticks={style={blue!80!black}, direction axis=x axis, x axis={goto=padded max}, low=0pt, tick text at high}},
    x axis={
      visualize ticks={direction axis=left axis, left axis={goto=padded min}, high=0pt, tick text at low},
      visualize ticks={direction axis=left axis, left axis={goto=padded max}, low=0pt}
    },
    % By default, there are ticks on all axes
    all axes={ticks},
    % Visualizing the axis labels, when requested
    x axis={visualize label={
      x axis={goto pos=.5}, node style={at={(0,0 |- data visualization bounding box.south)}, below}}},
    left axis={visualize label={
      left axis={goto pos=.5}, node style={
        at={(0,0 -| data visualization bounding box.west)}, rotate=90, anchor=south, red!50!black}}},
    right axis={visualize label={
      right axis={goto pos=.5}, node style={
        at={(0,0 -| data visualization bounding box.east)}, rotate=-90, anchor=south,  blue!80!black}}},
  }
}

\begin{codeexample}[]
\tikz \datavisualization [
  our system,     
  x axis={attribute=time, label=Year,
    ticks={tick text padding=2pt,  style={/pgf/number format/set thousands separator=}}},
  left axis={attribute=money, label=Spending,
    padding min=0, include value=0, grid,
    ticks={tick prefix=\$, style={/pgf/number format/fixed,
        /pgf/number format/fixed zerofill, /pgf/number format/precision=2}}},
  right axis={attribute=people,
    label=Population,
    padding min=0, include value=0,
    ticks={style=/pgf/number format/fixed}},
  visualize as line/.list={
    people 1, people 2, money 1, money 2},
  people 1={style={visualizer color=blue}},
  people 2={style={visualizer color=blue!50}},
  money 1={style={visualizer color=red}},
  money 2={style={visualizer color=red!50}} ]
data group {people and money};    
\end{codeexample}



\subsubsection{Using the New Axis System Key}

The axis system |our system| that we developed in the course of the
previous section is not yet ``configurable.'' The only configuration
that was possible was to ``misuse'' the |width| and |height| keys of
the |scientific axes|.

In order to make |our system| configurable so that we can say
|our system=|\meta{options}, where \meta{options} are executed with
the path prefix
\begin{codeexample}[code only]
/tikz/data visualization/our system  
\end{codeexample}
we can use the following key:

\begin{key}{/tikz/data visualization/new axis system=\marg{axis system
      name}\marg{axis setup}\marg{default options}\\ \marg{application
      options}} 
  The |new axis system| key takes four parameters. The first one,
  \meta{system name}, is the name of the to-be-created axis system,
  |our system| in our case. The |new axis system| will create the
  following new key:
  \begin{key}{/tikz/data visualization/\meta{axis system
        name}=\opt{\meta{options}}}
    When the key \meta{axis system name} is used, the following keys
    will be executed in the following order:
    \begin{enumerate}
    \item
      The \meta{axis setup} with the path prefix |/tikz/data visualization/|.
    \item
      The \meta{default options} with the same path prefix.
    \item
      The following style:
      \begin{stylekey}{/tikz/data visualization/every \meta{axis system name}}
        Even though this style has the path prefix
        |/tikz/data visualization| itself, the keys stored in this
        style will be executed with the path prefix
        |/tikz/data visualization/|\meta{axis system name}.
      \end{stylekey}
    \item
      The \meta{options} with the path prefix
      |/tikz/data visualization/|\meta{axis system name}.
    \item
      The \meta{application options} with the path prefix |/tikz/data visualization/|
    \end{enumerate}
  \end{key}

  Let us now have a look at what all of this means. First, the
  \meta{axis setup} will contain all options that setup the axis
  system in all ways that need not be configured. For instance, the
  \meta{axis setup} for the |scientific axes| will create an |x axis|
  and also a |y axis| (because these are always present), but will not
  setup the label visualization (because this can be configured in
  different ways). For |our system|, which cannot be configured at
  all, we would place all of our configuration in the \meta{axis
    setup}.

  The \meta{default options} can be used to pick default values that
  would usually be passed to the \meta{options} of the newly created
  axis system. For instance, for |scientific axis|, the
  \meta{default options} are set to |outer ticks,standard labels|,
  because these are the defaults.

  Finally, the \meta{application options} can be used to actually
  apply the configuration that has been chosen by the
  \meta{options}. The idea is that \meta{default options},
  \meta{options}, and also |every| \meta{axis system name} all have a
  chance of changing, re-changing and re-setting all sorts of styles
  and keys. Then, with the last change ``winning,'' the resulting
  setting of a style can be executed, which may then cause a label
  visualization to be installed.
\end{key}