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authorNorbert Preining <norbert@preining.info>2022-09-29 03:01:36 +0000
committerNorbert Preining <norbert@preining.info>2022-09-29 03:01:36 +0000
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treee14867dbe48dd37735c781960e975735d624fe0a /graphics/pgf/contrib/tzplot/tzplot-doc-D-v2.0.tex
parent47f9bcb9721388f6d98bf3b04ecd5fadc571b221 (diff)
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-%!TEX root = tzplot-doc.tex
-%\begin{document}
-
-%%%==========================
-\part{Plotting Graphs}
-%%%==========================
-\label{p:plotting}
-
-
-%%==================================
-\chapter{Axes}
-\label{c:axes}
-
-
-
-
-%%------------------------------------------------------------
-\section{Draw axes}
-\label{s:tzaxes}
-
-\subsection{\protect\cmd{\tzaxes}}
-\label{ss:tzaxes}
-
-Basically, \icmd{\tzaxes}|(<x1,y1>)(<x2,y2>)| draws the $x$ axis from |<x1>| to |<x2>| and the $y$ axis from |<y1>| to |<y2>|.
-The coordinate |(<x1,y1>)| represents the origin and |(<x2,y2>)| represents the opposite corner of the rectangle formed by the two coordinates.
-
-|\tzaxes| takes only one coordinate |(<x2,y2>)| as a mandatory argument, in which case the coordinate |(<x1,y1>)| is considered as |(0,0)|.
-
-\begin{tzdef}{}
-% syntax: minimal
-\tzaxes(<x2,y2>){<x-axis name>}{<y-axis name>}
-% syntax: full
-\tzaxes[<opt>]<x-shift,y-shift>(<x1,y1>)(<x2,y2>)
- {<x-axis name>}[<node opt>]{<y-axis name>}[<node opt>]
-% defaults
- [->]<0,0>(0,0)(<m>){}[right]{}[above]
-% arguments
- [#1]: line style, arrow type (for x-axis & y-axis)
- <#2>: axes shift coor %% axes intersect at (#2)
- (#3): (x1,y1) %% origin: if omitted, regarded as (0,0)
- (#4): (x2,y2) %% opposite corner: mandatory
- {#5}: x-axis name
- [#6]: x-axis name option %% node option
- {#7}: y-axis name
- [#8]: y-axis name option %% node option
-\end{tzdef}
-
-Here, |(|\ixxw{<m>}|)| stands for a mandatory argument.
-
-\begin{tzcode}{.3}
-% tzaxes
-\begin{tikzpicture}[scale=.7]
-\tzhelplines(3,3)
-\tzaxes(3,3){$x$}{$y$}
-\end{tikzpicture}
-\end{tzcode}
-
-\paragraph{Shift}
-By default, the $x$ and $y$ axes intersect at $(0,0)$.
-Specifying the option |<x-shift,y-shift>| moves the axes to intersect at $(\text{\ttfamily<x-shift,y-shift>})$.
-
-
-\begin{tzcode}{.3}
-% \tzaxes: shift
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(-1,-1)(7,7)
-\tzshoworigin*
-\tzaxes[thick,blue]<1,2>(7,7){$x$}{$y$} %%
-\tzaxes[-,dashed](7,7){$X$}[below]{$Y$}[left]
-\tzaxes<6,1>(6,1)(3,6){$a$}[left]{$b$} %%
-\end{tikzpicture}
-\end{tzcode}
-
-
-\subsection{\protect\cmd{\tzaxes*}}
-\label{ss:tzaxes*}
-
-The starred version \icmd{\tzaxes*} sets the current state to a \iisw{bounding box} when the macro |\tzaxes| execution is complete. It is recommended for you to use |\tzaxes*| as the first graphics command in |tikzpicture| environment or before any larger graphics.
-
-\begin{tzcode}{.3}
-\begin{tikzpicture}[scale=.5]
-\tzaxes*(8,5){$x$}{$f(x)$} % bounding box
-\tzhelplines(-2,-1)(10,8)
-\tzto[out=90,in=-135,dashed](-2,8)(12,-2)
-\tzbezier[blue](-1,-1)(3,-2)(7,12)(10,10)
-\end{tikzpicture}
-\end{tzcode}
-
-
-%%------------------------------------------------------------
-\section{\protect\cmd{\tzaxisx} and \protect\cmd{\tzaxisy}}
-\label{s:tzaxisx}
-
-\icmd{\tzaxisx} draws only the $x$ axis.
-
-\begin{tzdef}{}
-% syntax
-\tzaxisx[<opt>]<y-shift>{<from>}{<to>}{<x-axis name>}[<node opt>]
-% defaults
- [->]<0>{<m>}{<m>}{}[right]
-% arguments:
- [#1]: line style, arrow type (for x-axis)
- <#2>: y-shift of x-axis
- {#3}: x-axis starts from %% mandatory
- {#4}: x-axis runs to %% mandatory
- {#5}: x-axis name
- [#6]: x-axis name option %% node option
-\end{tzdef}
-
-\icmd{\tzaxisy} draws only the $y$ axis.
-
-\begin{tzdef}{}
-% syntax
-\tzaxisy[<opt>]<x-shift>{<from>}{<to>}{<y-axis name>}[<node opt>]
-% defaults
- [->]<0>{<m>}{<m>}{}[above]
-% arguments:
- [#1]: line style, arrow type (for y-axis)
- <#2>: x-shift of y axis
- {#3}: y-axis starts from %% mandatory
- {#4}: y-axis runs to %% mandatory
- {#5}: y-axis name
- [#6]: y-axis name option %% node option
-\end{tzdef}
-
-
-
-\begin{tzcode}{.3}
-% \tzaxisx, \tzaxisy
-\begin{tikzpicture}[scale=.5]
-\tzshoworigin
-\tzhelplines(5,5)
-\tzaxisx[blue]{-1}{5}{$xx$}
-\tzaxisy[red] {-1}{5}{$yy$}[green]
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-\begin{tzcode}{.3}
-% \tzaxisx, \tzaxisy: shift
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(7,7)
-\tzshoworigin
-\tzaxisx{-1}{7}{$xx$}[near end,below]
-\tzaxisy{-1}{7}{$yy$}[midway,sloped,auto]
-\tzaxisx[-,dashed,red]<2>{-1}{7}{$xx'$}[blue]
-\tzaxisy[-,dashed,blue]<3>{-1}{7}{$yy'$}
-\end{tikzpicture}
-\end{tzcode}
-
-
-%%------------------------------------------------------------
-\section{Display the origin}
-\label{s:tzshoworigin}
-
-\subsection{\protect\cmd{\tzshoworigin}}
-\label{ss:tzshoworigin}
-
-
-\icmd{\tzshoworigin} prints `0' (approximately) at the bottom left of the origin |(0,0)|, by default.
-
-\begin{tzdef}{}
-% syntax
-\tzshoworigin<shift coor>(<origin>){<text>}[<node opt>]
-% default
- <>(0,0){0}[below left,text height=1.25ex,text depth=.25ex]
-\end{tzdef}
-
-All arguments of |\tzshoworigin| are optional.
-
-\begin{tzcode}{.3}
-% \tzshoworigin
-\begin{tikzpicture}[scale=.45]
-\tzhelplines(8,5)
-\tzshoworigin
-\tzaxes(8,5)
-\end{tikzpicture}
-\end{tzcode}
-
-You can change the text by specifying the curly brace option |{<text>}|, like, for example, |\tzshoworigin{$O$}|.
-You can also change the coordinate of origin by the option |(<origin>)|.
-Specifying the option |<shift coor>| also moves the origin.
-
-
-\begin{tzcode}{.3}
-% \tzshoworigin: shift
-\begin{tikzpicture}[scale=.45]
-\tzhelplines(-2,-1)(8,6)
-\tzshoworigin{$O_1$} %%
-\tzaxes(8,6){$x$}{$y$}
-\tzshoworigin<2,1>{$O_2$}[red] %%
-\tzaxes[dashed]<2,1>(8,6)
-\end{tikzpicture}
-\end{tzcode}
-
-
-\subsection{\protect\cmd{\tzshoworigin*}}
-\label{ss:tzshoworigin*}
-
-
-\icmd{\tzshoworigin*} prints a node dot at the origin with no text by default.
-Internally the dot is processed by |\tzdot*|. All arguments are optional.
-
-\begin{tzdef}{}
-% syntax
-\tzshoworigin*[<dot opt>]<shift coor>(<origin>){<text>}[<node opt>](<dot size>)
-% default
- *[]<>(0,0){}[below left,text height=1.25ex,text depth=.25ex](2.4pt)
-\end{tzdef}
-
-\begin{tzcode}{.3}
-% \tzshoworigin*
-\begin{tikzpicture}[scale=.45]
-\tzhelplines(-2,-1)(6,4)
-\tzshoworigin*
-\tzaxes(-2,-1)(6,4)
-\end{tikzpicture}
-\end{tzcode}
-
-You can add text with the option |{<text>}|.
-The default size of the dot is |2.4pt|, and it can be changed with the last option |(<dot size>)|. You can change the dot style using the first optional argument |[<dot opt>]|.
-You can also move the dot by specifying the option |<shift coor>|.
-
-\begin{tzcode}{.3}
-% \tzshoworigin*
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(-1,-1)(7,7)
-\tzshoworigin*[blue]{$o_1$}[red](5pt)
-\tzaxes(-1,-1)(7,7){$x$}{$y$}
-\tzaxes<6,4>(6,4)(1,1){$x'$}[left]{$y'$}[below]
-\tzshoworigin*[fill=none]<6,4>{$O'$}[blue,ar](3pt)
-\end{tikzpicture}
-\end{tzcode}
-
-\remark
-For |\tzshoworigin*|, text for the origin and the dot are placed independently. In other words, the position of node text does not depend on the size of a node dot.
-(In fact, the node text for the origin should look good with the `ticks labels', so it was not designed as a |label| for the node dot. This also means that the origin text cannot be positioned by an |<angle>|.)
-
-
-\begin{tzcode}{.3}
-% \tzshoworigin* (with tick labels)
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(-2,1)(6,5)
-\tzshoworigin*{0}(5pt)
-\tzaxes(-2,-1)(6,5){$x$}{$y$}
-\tzticks{1,...,5}{1,...,4}
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-%%------------------------------------------------------------
-\section{\protect\cmd{\tzaxesL(')}: L-type axes}
-\label{s:tzaxesL}
-
-\icmd{\tzaxesL} is similar to |\tzaxes|, but it draws only the `L' type axes with |(<x1,y1>)| as the origin and |(<x2,y2>)| as the opposite corner of the rectangle.
-Those two coordinates are mandatory.
-
-\begin{tzdef}{}
-% syntax
-\tzaxesL[<opt>]<shift coor>(<x1,y1>)(<x2,y2>)
- {<x-axis name>}[<node opt>]{<y-axis name>}[<node opt>]
-% defaults
- []<>(<m>)(<m>){}[right]{}[above]
-% arguments:
- [#1]: line style, arrow type
- <#2>: shift coordinate
- (#3): (x1,y1) %% mandatory
- (#4): (x2,y2) %% mandatory
- {#5}: x-axis name
- [#6]: x-axis name option %% node option
- {#7}: y-axis name
- [#8]: y-axis name option %% node option
-\end{tzdef}
-
-The \iisw{swap version} \icmd{\tzaxesL'} swaps |(<x1,y1>)| and |(<x2,y2>)|.
-That is, |\tzaxesL'(A)(B)| is equivalent to |\tzaxesL(B)(A)|.
-
-\begin{tzcode}{.3}
-% \tzaxesL, \tzaxesL'
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,7)
-\tzshoworigin
-\tzaxes(8,7){$x$}{$y$}
-\tzaxesL[red,thick](2,2)(6,5){$a$}{$b$}
-\tzaxesL'[blue,dashed,->](2,2)(6,5){$c$}{$d$}
-\tzaxesL'[->](7,1)(4,7){m}[draw,r]{n}[draw,circle]
-\end{tikzpicture}
-\end{tzcode}
-
-The option |<shift coor>| moves the whole L-type axes.
-The empty option |<>| is not allowed.
-
-\begin{tzcode}{.3}
-% \tzaxesL, \tzaxesL': shift
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,7)
-\tzshoworigin
-\tzaxes(8,7){$x$}{$y$}
-\tzaxesL[red,thick](2,2)(6,5){$a$}{$b$}
-\tzaxesL[red,thick]<.5,.5>(2,2)(6,5){$a'$}{$b'$}
-\tzaxesL'[blue,dashed,->](2,2)(6,5){$c$}{$d$}
-\tzaxesL'[blue,dashed,->]<-1,-1>(2,2)(6,5){$c'$}{$d'$}
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-%%==================================
-\chapter{Ticks}
-\label{c:ticks}
-
-
-%%------------------------------------------------------------
-\section{\protect\cmd{\tzticks}: Tick labels}
-\label{s:tzticks}
-
-By default, \icmd{\tzticks} prints tick labels and draws zero length tick marks, i.e. from |(0pt)| to |(0pt)|.
-
-\begin{tzdef}{}
-% syntax: minimal
-\tzticks{<x-ticks pos>}{<y-ticks pos>}
-% syntax: medium
-\tzticks(<x-from:x-to>){<x-ticks pos/labels>}[<node opt>]
- (<y-from:y-to>){<y-ticks pos/labels>}[<node opt>]
-% syntax: full
-\tzticks[<opt>]<x-shift,y-shift>
- (<x-from:x-to>){<x-ticks pos/labels>}[<node opt>]
- (<y-from:y-to>){<y-ticks pos/labels>}[<node opt>]
-% defaults
- []<0,0>(0pt:0pt){<m>}[text height=1.25ex,text depth=.25ex,below](0pt:0pt){}[left]
-\end{tzdef}
-
-\paragraph{Tick labels}
-Internally, |\tzticks| uses \Tikz's |foreach| operation.
-So you need to provide comma separated lists to print tick labels.
-If only one comma separated list is specified, it is for $x$ tick labels.
-
-\begin{tzcode}{.3}
-% \tzticks
-\begin{tikzpicture}[scale=.4,font=\scriptsize]
-\tzhelplines(10,10)
-\tzaxes(-1,-1)(10,10)
-\tzticks[blue]{1,...,8}{2,...,7}
-\end{tikzpicture}
-\end{tzcode}
-
-You can change the numbered labels to a different format with slashes and other text, as follows: |<number>/<other text>|.
-
-\begin{tzcode}{.3}
-% \tzticks: tick labels
-\begin{tikzpicture}[scale=.4,font=\scriptsize]
-\tzhelplines(10,10)
-\tzaxes(-1,-1)(10,10)
-\tzticks{1/$x_1$,2,5/$x_2$,8/y}[blue]
- {2/$\sqrt{x}$,3/y,4/m,5,7/$k$}[red]
-\end{tikzpicture}
-\end{tzcode}
-
-\paragraph{Tick marks}
-By specifying the options |(<x-from:x-to>)| for $x$ ticks and/or |(<y-from:y-to>)| for $y$ ticks, you can print tick marks. (The default is |(0pt:0pt)| for both options.)
-
-\begin{tzcode}{.3}
-% \tzticks: tick marks with (<a pt:b pt>)
-\begin{tikzpicture}[scale=.4,font=\scriptsize]
-\tzhelplines(10,10)
-\tzshoworigin
-\tzaxes(-1,-1)(10,10)
-\tzticks[draw=red,thick]
- (-15pt:10pt){1,...,5,6/$k$,7/$\alpha$,8/$\beta$}
- (0pt:3cm) {2,...,6,7/$\gamma$}
-\end{tikzpicture}
-\end{tzcode}
-
-The position of tick labels does not depend on the length of the tick marks.
-You can change the position of tick labels using |[<node opt>]|.
-
-\begin{tzcode}{.3}
-% \tzticks: position of tick labels
-\begin{tikzpicture}[scale=.4,font=\scriptsize]
-\tzhelplines(10,10)
-\tzshoworigin
-\tzaxes(-1,-1)(10,10)
-\tzticks[draw=red,thick]
- (-15pt:10pt){1,...,6,7/$t$,8/$\alpha$}[b=5pt]
- (0pt:3cm) {2,...,6,7/$\beta$}
-\end{tikzpicture}
-\end{tzcode}
-
-\paragraph{Shift}
-
-You can move (or shift) tick marks and labels together by specifying the option |<x-shift,y-shift>|, where |<x-shift>| is for |y-ticks| and |<y-shift>| is for |x-ticks|.
-
-\begin{tzcode}{.3}
-% \tzticks: shift
-\begin{tikzpicture}[scale=.4,font=\scriptsize]
-\tzhelplines(10,10)
-\tzshoworigin
-\tzaxes(-1,-1)(10,10)
-\tzaxes[dashed]<4,2>(-1,-1)(10,10)
-\tzticks[draw=red]<4,2>
- (-5pt:10pt){5,...,8}
- (0pt:3cm) {3,...,7}
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-
-
-%%------------------------------------------------------------
-\section{\protect\cmd{\tzticks*}: Tick marks}
-\label{s:tzticks*}
-
-The starred version \icmd{\tzticks*} always ignores all tick labels and draws tick marks from |0pt| to |3pt|, by default.
-
-\begin{tzdef}{}
-% syntax: minimal
-\tzticks*{<x-ticks pos>}{<y-ticks pos>}
-% syntax: medium
-\tzticks*(<x-from:x-to>){<x-ticks pos>}(<y-from:y-to>){<y-ticks pos>}
-% syntax: full
-\tzticks*[<opt>]<x-shift,y-shift>
- (<x-from:x-to>){<x-ticks pos>}(<y-from:y-to>){<y-ticks pos>}
-% defaults
- []<0,0>(0pt:3pt){<m>}(0pt:3pt){}
-% starred(*) version always suppresses tick labels
-\end{tzdef}
-
-\begin{tzcode}{.3}
-% \tzticks*
-\begin{tikzpicture}[scale=.4,font=\scriptsize]
-\tzhelplines(10,10)
-\tzshoworigin
-\tzaxes(-1,-1)(10,10)
-\tzticks*[draw=red,thick]
- {1,...,7,8/$\alpha$} % label ignored
- (0pt:3cm){2,...,6,7/$\beta$} % label ignored
-\end{tikzpicture}
-\end{tzcode}
-
-\begin{tzcode}{.4}
-% \tzticks(*)
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(10,8)
-\tzaxes(-1,-1)(10,8)
-\tzticks*
- {0,0.2,...,8}
- {0,0.2,...,7} % default (0pt:3pt)
-\tzticks*
- (0pt:10pt){1,...,8}
- (0pt:10pt){1,...,7}
-\tzticks
- {1,...,8}
- {2,...,7} % default (0pt:0pt)
-\end{tikzpicture}
-\end{tzcode}
-
-
-%%------------------------------------------------------------
-\section{\protect\cmd{\tzticksx(*)} and \protect\cmd{\tzticksy(*)}}
-\label{s:tzticksx}
-
-You can handle $x$ ticks and $y$ ticks independently.
-
-\paragraph{X ticks}
-
-\icmd{\tzticksx} only prints x-tick labels but not tick marks, by default.
-To prints tick marks you need to specify |(<x-from>:<x-to>)|.
-
-\begin{tzdef}{}
-% syntax
-\tzticksx[<opt>]<y-shift>(<from>:<to>){<x-tick pos/labels>}[<node opt>]
-% defaults
- []<>(0pt:0pt){<m>}[text height=1.25ex,text depth=.25ex,below]
-\end{tzdef}
-
-
-\icmd{\tzticksx*} only prints x-tick marks from |0pt| to |3pt|, by default, suppressing tick labels.
-
-\begin{tzdef}{}
-% syntax:
-\tzticksx*[<opt>]<y-shift>(<from>:<to>){<xtick pos>}
-% defaults
- *[]<>(0pt:3pt){<m>}
-% starred(*) version always suppresses tick labels
-\end{tzdef}
-
-
-\paragraph{Y ticks}
-
-\icmd{\tzticksy} only prints y-tick labels but not tick marks, by default.
-To prints tick marks you need to specify |(<x-from>:<x-to>)|.
-
-\icmd{\tzticksy*} only prints y-ticks from |0pt| to |3pt| by default, suppressing tick labels.
-
-\begin{tzdef}{}
-% syntax:
-\tzticksy[<opt>]<x-shift>(<from:to>){<y-ticks pos/labels>}[<node opt>]
-% defaults
- []<>(0pt:0pt){<m>}[]
-\end{tzdef}
-
-\begin{tzdef}{}
-% syntax
-\tzticksy*[<opt>]<x-shift>(<from:to>){<yticks pos>}
-% defaults
- []<>(0pt:0pt){<m>}
-% starred(*) version suppresses tick labels
-\end{tzdef}
-
-
-\begin{tzcode}{.3}
-% \tztickx(*), \tzticky(*)
-\begin{tikzpicture}[scale=.4,font=\scriptsize]
-\tzhelplines(10,10)
-\tzshoworigin
-\tzaxes(-1,-1)(10,10)
-\tzticksx [draw=red,thick]
- (-5pt:1cm){1,...,7,8/$\alpha$}
-\tzticksy*[draw=blue,thick]
- (0pt:3cm){2,...,6,7/$\beta$} % labels ignored
-\end{tikzpicture}
-\end{tzcode}
-
-\paragraph{Shift}
-The options |<y-shift>| and |<x-shift>| move x-ticks and y-ticks, respectively.
-
-\begin{tzcode}{.3}
-% \tztickx(*), \tzticky(*): shift
-\begin{tikzpicture}[scale=.4,font=\scriptsize]
-\tzhelplines(10,10)
-\tzshoworigin
-\tzaxes(-1,-1)(10,10)
-\tzaxes[dashed]<2,1>(-1,-1)(10,10)
-\tzticksx*[draw=red,thick]
- <1>(-5pt:10pt){1,...,7,8/$\alpha$} % labels ignored
-\tzticksy [draw=blue,thick]
- <2>(0pt:3cm){2,...,6,7/$\beta$}
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-
-%%==================================
-\chapter{Projections}
-\label{c:projections}
-
-%%------------------------------------------------------------
-\section{\protect\cmd{\tzproj(*)}: Projections on the axes}
-\label{s:tzproj}
-
-\icmd{\tzproj} accepts a mandatory coordinate and draws perpendicular lines onto each axis from the coordinate. The lines are |dotted|, by default.
-
-\begin{tzdef}{}
-% syntax: minimum
-\tzproj(<coor>)
-% syntax: medium
-\tzproj*(<coor>){<x-text>}[<node opt>]{<y-text>}[<node opt>]
-% syntax:
-\tzproj*[<opt>]<x-shift,y-shift>(<coor>)
- {<x-text>}[<node opt>]{<y-text>}[<node opt>](<dot size>)
-% defaults
- *[dotted]<0,0>(<m>){}[text height=1.25ex,text depth=.25ex,below]{}[left](2.4pt)
-\end{tzdef}
-
-\icmd{\tzproj*} additionally prints a `black node dot' of the size |2.4pt|, by default.
-Internally, the node dot is processed by |\tzdot*|.
-The first option |[<opt>]| does not control the node dot.
-
-\paragraph{Dot size}
-You can only control the size of dots by the last optional argument |(<dot size>)| or by the \threeways\ on page \pageref{ss:threeways}.
-If you want to control |fill| or |color| of dots, use |\tzdot*| separately.
-
-\begin{tzcode}{.3}
-%\tzproj(*)
-\begin{tikzpicture}[scale=.5]
-\tzaxes(8,6)
-\tzproj[dashed,blue](2,3)
-\tzcoors(30:7)(A)(50:6)(B);
-\tzproj*(A)
-\tzproj*[dashed,text=blue](B)(5pt)
-\end{tikzpicture}
-\end{tzcode}
-
-\paragraph{Adding text}
-You can also add text around the projection point on each axis by the option |{<text>}|.
-The position and color of the text is controlled by the option |[<node option>]|.
-The default position is (approximately) |[below]| for the x axis and |[left]| for the y axis.
-
-
-\begin{tzcode}{.3}
-% \tzproj(*): adding text
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,6)
-\tzaxes(8,6)
-\tzproj[dashed,blue](2,3){$x$}{$y$}
-\tzcoors(30:7)(A)(50:6)(B);
-\tzproj*[text=blue](A){$a_1$}{$a_2$}
-\tzproj*[dashed](B){$x^*$}[green]{$y^*$}[red]
-\end{tikzpicture}
-\end{tzcode}
-
-\paragraph{Projection shift}
-Specifying the option |<x-shift,y-shift>| moves the projection point and text on each axis.
-
-\begin{tzcode}{.3}
-% \tzproj(*): projection shift
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,6)
-\tzaxes(8,6)
-\tzproj[dashed,blue](2,3){$x$}{$y$}
-\tzcoors(30:7)(A)(50:6)(B);
-\tzaxes[blue]<3,1>(8,6)
-\tzproj*[dashed,text=blue]<3,1>(A){$a_1$}{$a_2$} %%
-\end{tikzpicture}
-\end{tzcode}
-
-
-%%------------------------------------------------------------
-\section{\protect\cmd{\tzprojx(*)} and \protect\cmd{\tzprojy(*)}}
-\label{s:tzprojx}
-
-\icmd{\tzprojx} draws a dotted line, which is perpendicular to the x axis.
-
-\icmd{\tzprojx*} additionally prints a `black node dot' of the size |2.4pt|, by default.
-
-
-\begin{tzdef}{}
-% syntax:
-\tzprojx[<opt>]<x-shift,y-shift>(<coor>){<x-text>}[<node opt>](<dot size>)
-% defaults
- []<0,0>(<m>){}[text height=1.25ex,text depth=.25ex,below](2.4pt)
-\end{tzdef}
-
-
-\icmd{\tzprojy} draws a dotted line, which is perpendicular to the y axis.
-
-\icmd{\tzprojy*} additionally prints a `black node dot' of the size |2.4pt|, by default.
-
-\begin{tzdef}{}
-% syntax:
-\tzprojy[<opt>]<x-shift,y-shift>(<coor>){<y-text>}[<node opt>](<dot size>)
-% defaults
- []<0,0>(<m>){}[left](2.4pt)
-\end{tzdef}
-
-
-You can only control the size of dots by the last option |(<dot size>)|.
-If you want to control |fill| or |color| of dots, use |\tzdot*| separately.
-You can also add text around the projection point on each axis by specifying the option |{<x-text>}| or |{<y-text>}| followed by the option |[<node option>]|.
-
-\begin{tzcode}{.3}
-% \tzprojx(*), \tzprojy(*)
-\begin{tikzpicture}[scale=.5,font=\scriptsize]
-\tzhelplines(-1,-2)(6,6)
-\tzshoworigin
-\tzaxes(-1,-1)(6,6)
-\tzproj*[dashed](4,5){4}{5}(3pt)
-\tzprojx*[green,thick,solid](3,4){$x=3$}[blue]
-\tzprojy*[thick](5,2){2}(5pt)
-\end{tikzpicture}
-\end{tzcode}
-
-Specifying the option |<x-shift,y-shift>| with |\tzprojx(*)| and |\tzprojy(*)| moves the projection point and text accordingly.
-
-\begin{tzcode}{.3}
-% \tzprojx(*), \tzprojy(*): shift
-\begin{tikzpicture}[scale=.5,font=\scriptsize]
-\tzhelplines(-1,-2)(6,6)
-\tzshoworigin
-\tzaxes(-1,-1)(6,6)
-\tzaxes[blue]<2,1>(-1,-1)(6,6)
-\tzprojx*[green,thick,solid]<2,1>(3,4){$x=3$}[blue]
-\tzprojy*[thick]<2,1>(5,2){2}(5pt)
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-%%------------------------------------------------------------
-\section{\protect\cmd{\tzprojs(*)}: Semicolon versions}
-\label{s:tzprojs}
-
-\icmd{\tzprojs} accepts any number of coordinates and draws perpendicular lines onto each axis from the coordinates. The lines are |dotted|, by default.
-|\tzprojs| is a semicolon version of |\tzproj|, so a semicolon is needed to indicate when the coordinate iteration ends. Its repeating pattern is |(<coor>){<x-text>}[<node opt>]{<y-text>}[<node opt>]|.
-
-\begin{tzdef}{}
-% syntax: minimum
-\tzprojs(<coor>)(<coor>) ..repeated.. (<coor>);
-% syntax: medium
-\tzprojs*(<coor>){<x-text>}[<node opt>]{<y-text>}[<node opt>]
- ..repeated.. (){}[]{}[]
-% syntax:
-\tzprojs*[<opt>]<x-shift,y-shift>
- (<coor>){<x-text>}[<node opt>]{<y-text>}[<node opt>]
- ..repeated.. (){}[]{}[];(<dot size>)
-% defaults
- *[dotted]<0,0>
- (<m>){}[text height=1.25ex,text depth=.25ex,below]{}[left]
- ..repeated..(){}[]{}[];(2.4pt)
-\end{tzdef}
-
-\icmd{\tzprojs*} additionally prints |\tzdots*| of the |2.4pt| (by default) on the coordinates.
-The first option |[<opt>]| does not control the node dots.
-
-\begin{tzcode}{.3}
-% \tzprojs(*): adding text
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,6)
-\tzaxes(8,6)
-\tzcoors(30:7)(A)(50:6)(B);
-\tzprojs*[dashed,text=blue]
- (2,3){$x$}{$y$}
- (A){$a_1$}{$a_2$}
- (B){$x^*$}[green]{$y^*$}[red];
-\end{tikzpicture}
-\end{tzcode}
-
-You can move the projection points and text accordingly, using the option |<x-shift,y-shift>| before the first coordinate. The \xem{empty} shift option |<>| is \xem{not allowed}.
-You can also change the dot size with the last option |(<dot size>)| \xem{after} the semicolon, as in |\tzproj(*)|.
-
-\begin{tzcode}{.3}
-% \tzprojs(*): shift and dot size
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,6)
-\tzaxes(8,6)
-\tzaxes[blue]<1,1>(8,6)
-\tzcoors(30:7)(A)(50:6)(B);
-\tzprojs*[dashed,text=blue]<1,1> %%
- (2,3){$x$}{$y$}
- (A){$a_1$}{$a_2$}
- (B){$x^*$}[green]{$y^*$}[red];(4pt) %%
-\end{tikzpicture}
-\end{tzcode}
-
-
-%%------------------------------------------------------------
-\section{\protect\cmd{\tzprojsx(*)} and \protect\cmd{\tzprojsy(*)}: Semicolon versions}
-\label{s:tzprojsx}
-
-\icmd{\tzprojsx} is a semicolon versions of |\tzprojx|. It draws dotted lines, which are perpendicular to the x axis from the specified coordinates, by default.
-
-\icmd{\tzprojsx*} additionally prints |\tzdots*| of the size |2.4pt|, by default.
-
-
-\begin{tzdef}{}
-% syntax:
-\tzprojsx*[<opt>]<x-shift,y-shift>
- (<coor>){<x-text>}[<node opt>]..repeated..(){}[];(<dot size>)
-% defaults
- *[]<0,0> (<m>){}[text height=1.25ex,text depth=.25ex,below]
- ..repeated..(){}[];(2.4pt)
-\end{tzdef}
-
-
-\icmd{\tzprojsy} and \icmd{\tzprojsy*} work simililarly as |\tzprojsx| and |\tzprojsx*| do but to the y axis.
-
-\begin{tzdef}{}
-% syntax:
-\tzprojsy*[<opt>]<x-shift,y-shift>
- (<coor>){<y-text>}[<node opt>]..repeated..(){}[];(<dot size>)
-% defaults
- []<0,0> (<m>){}[left]..repeated..(){}[];(2.4pt)
-\end{tzdef}
-
-
-\begin{tzcode}{.3}
-% \tzprojsx(*), \tzprojsy(*)
-\begin{tikzpicture}[scale=.5,font=\scriptsize]
-\tzhelplines(-1,-2)(6,6)
-\tzshoworigin
-\tzaxes(-1,-1)(6,6)
-\tzprojsx*[thick]
- (4,5){4}
- (3,4){$x=3$}[blue]
- (5,2){2};(5pt)
-\end{tikzpicture}
-\end{tzcode}
-
-Specifying the option |<x-shift,y-shift>| moves the projection points and text accordingly.
-
-\begin{tzcode}{.3}
-% \tzprojsx(*), \tzprojsy(*): shift
-\begin{tikzpicture}[scale=.5,font=\scriptsize]
-\tzhelplines(-1,-2)(6,6)
-\tzshoworigin
-\tzaxes(-1,-1)(6,6)
-\tzaxes[blue]<2,1>(-1,-1)(6,6)
-\tzprojsy*[red,solid]<2,1>
- (3,4){$x=3$}[blue]
- (5,2){2};
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-%%==================================
-\chapter{Plot Functions}
-\label{c:functions}
-
-
-%%------------------------------------------------------------
-\section{\protect\cmd{\tzfn} and \protect\cmd{\tzfn'}: Plot functions and inverse functions}
-\label{s:tzfn}
-
-\subsection{Syntax}
-\label{ss:tzfn}
-
-\icmd{\tzfn} plots a function of |\x|.
-
-\begin{tzdef}{}
-% syntax: minimum
-\tzfn{<fn of \x>}[<domain>]
-% syntax: medium
-\tzfn{<fn of \x>}[<domain>]{<text>}[<pos>]
-% syntax: full
-\tzfn[<opt>]<shift coor>"<path name>"
- {<fn of \x>}[<domain>]{<text>}[<node opt>]<code.append>
-% [<domain>] should be of the form [<from num:to num>]
-% defaults
- [samples=201]<>""{<m>}[<m>]{}[]<>
-\end{tzdef}
-
-|\tzfn| takes two mandatory arguments: |{<fn of \x>}| and |[<domain>]|.
-The domain should be of the form |[<from num:to num>]|, like |[1:5]|.
-
-\begin{tztikz}{}
-\tzfn{.5*(\x)^2-1}[1:5] % works like:
- \draw [samples=201,domain=1:5] plot (\x,{.5*(\x)^2-1});
-\end{tztikz}
-
-\begin{tzcode}{.3}
-% \tzfn : simple example
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(-2,-2)(5,5)
-\tzaxes*(-2,-2)(5,5)
-\tzfn{.5*(\x)^2-1}[0:3.5]
-\end{tikzpicture}
-\end{tzcode}
-
-
-%%------------------------------------------------------------
-\subsection{Inverse functions: \protect\cmd{\tzfn'}}
-\label{ss:tzfn'}
-
-The \iisw{swap version} \icmd{\tzfn'} draws the \iisw{inverse function} of |\tzfn|.
-
-\begin{tztikz}{}
-\tzfn'{.5*(\x)^2-1}[1:5] % works like:
- \draw [samples=201,domain=1:5] plot ({.5*(\x)^2-1},\x);
-\end{tztikz}
-
-\begin{tzcode}{.3}
-% \tzfn' : inverse function (with text)
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(-2,-2)(5,5)
-\tzaxes*(-2,-2)(5,5)
-\tzfn{.5*(\x)^2-1}[0:3.5]
-\tzfn'[red]{.5*(\x)^2-1}[0:3.5]{inversed}[r] %%
-\end{tikzpicture}
-\end{tzcode}
-
-You can add text at the end of the graph,
-by specifying |{<text>}| and |[<node opt>]| immediately after the domain.
-
-\begin{tzcode}{.3}
-% \tzfn(')
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,8)
-\tzaxes(8,8){$Q$}{$P$}
-\tzfn{5-5/7*\x}[0:7]{demand}[a=5mm]
-\tzfn'[blue]{5-5/7*\x}[0:7]
- {inverse\\demand}[r=5mm,align=center]
-\tzticks{5,7}{5,7}
-\end{tikzpicture}
-\end{tzcode}
-
-
-\subsection{Define and name functions}
-
-To use |\tzfn| you need to express a function as a function of |\x|.
-
-\begin{tzcode}{.3}
-% \tzfn
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,8)
-\tzaxes(8,8)
-\def\Dx{7-\x} % define
-\tzfn{\Dx}[0:7]{$D$}[ar]
-\tzfn{1+\x}[0:7]{$S$}[blue,r]
-\end{tikzpicture}
-\end{tzcode}
-
-You can also use the predefined functions of \Tikz\ such as |sin|, |cos|, |ln|, |log10|, |log2|, |exp|, |sqrt|, and so on. (See \Tikz\ manual.)
-
-\begin{tzcode}{.3}
-% \tzfn
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(-2,-2)(8,6)
-\tzaxes*(-2,-2)(8,6)
-\def\Fx{sin(\x r)+3}
-\def\Gx{exp(\x)}
-\def\Hx{ln(\x)}
-\tzfn\Fx[-2:2*pi]{$\sin\,x+3$}[blue,r]
-\tzfn[blue]\Gx[-2:2]{$e^x$}[red,r]
-\tzfn[dashed]\Hx[.2:7]{$\ln\,x$}[r]
-\end{tikzpicture}
-\end{tzcode}
-
-\subsection{Name paths: \texttt{name path}}
-\label{ss:tzfn:namepath}
-
-You can name the path of |\tzfn| by specifying the option |"<path name>"| \xem{immediately before} the mandatory argument |{<fn of \x>}|. You can use the path name to find intersection points.
-
-\begin{tztikz}{}
-\tzfn"mypath"\Fx[1:5] % works like:
- \draw [samples=201,,domain=1:5,name path=mypath] plot (\x,{\Fx});
-\end{tztikz}
-
-\remark Advantage of defining functions:
-
-Suppose that the function's expression |<fn of \x>| consists \xem{only of a macro name}, say, |\Fx|. Then
-
-\begin{itemize}\firmlist
-\item The macro name |Fx| (\xem{without the backslash}) is
- \xem{automatically assigned} to |<path name>|,
- unless you give another name.
-\item That is, |\tzfn\Fx| is equivalent to |\tzfn"Fx"\Fx|.
- (\xem{You don't need to type the same thing twice}.)
-\end{itemize}
-
-\begin{tztikz}{}
-\tzfn\Fx[1:5] % works like:
- \draw [samples=201,domain=1:5,name path=Fx] plot (\x,{\Fx});
-\end{tztikz}
-
-\begin{tzcode}{.3}
-% \tzfn: name path: intersection point
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,8)
-\tzaxes(8,8)
-\def\Dx{7-\x}
-\def\Sx{1+\x}
-\tzfn"Dx"\Dx[0:7]{$D$}[ar] % name path = Dx
-\tzfn \Sx[0:7]{$S$}[blue,r] % name path = Sx
-\tzXpoint*{Dx}{Sx}(E){E} % intersection
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-\subsection{Move graphs: \texttt{shift}}
-
-You can move the graph of |\tzfn| by specifying the option |<shift coor>| \xem{before} the mandatory argument |{<fn of \x>}| or \xem{immediately before} the option |"<path name>"|, if it exists.
-The \xem{empty} shift option |<>| is \xem{not allowed}.
-
-\begin{tzcode}{.3}
-% \tzfn: shift
-\begin{tikzpicture}[scale=.6]
-\tzhelplines(8,8)
-\def\Dx{7-\x}
-\def\Sx{1+\x}
-\tzfn \Dx[0:7]{$D$}[right] % name path = Dx
-\tzfn"supply"\Sx[0:7]{$S$}[right] % name path = supply
-\tzXpoint*{Dx}{supply}(E){$E$}
-\tzfn[dashed]<1,1>"demandA"\Dx[0:7]{$D'$}[r]
-\tzfn[dashed]<1,-1>"supplyA"\Sx[0:7]{$S'$}[r]
-\tzXpoint*{demandA}{supplyA}(E1){$E'$}
-\end{tikzpicture}
-\end{tzcode}
-
-
-\subsection{Extend paths: \texttt{<code.append>}, \protect\cmd{\tzfnAtBegin}, \protect\cmd{\tzfnAtEnd}}
-\label{ss:tzfn:atbegin}
-
-\paragraph{\texttt{<code.append>}}
-You can extend the path of |\tzfn|, by writing \Tikz\ code in the \xem{last optional argument} |<code.append>|. Internally it adds the \Tikz\ code to the path \xem{after} the options |{<text>}| and |[<node opt>]|.
-
-\begin{tzcode}{.3}
-% \tzfn: <code.append>
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(-2,-2)(8,6)
-\tzaxes*(-2,-2)(8,6)
-\def\Fx{sin(\x r)+3}
-\tzfn[->]\Fx[-2:2*pi]{$\sin\,x+3$}[blue,r]
- < to [bend right] ++(2,-2) node [b] {End!} >
-\end{tikzpicture}
-\end{tzcode}
-
-\paragraph{\texttt{\bs tzfnAtEnd}}
-You can also extend the path of |\tzfn| at the end using \icmd{\tzfnAtEnd}.
-Internally it adds \Tikz\ code immediately \xem{before} the options |{<text>}| and |[<node opt>]|. But you have to use |\tzfnAtEnd| (immediately) \xem{before each} |\tzfn|.
-
-\begin{tzcode}{.3}
-% \tzfnAtEnd (before \tzfn)
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(-2,-2)(8,6)
-\tzaxes*(-2,-2)(8,6)
-\def\Fx{sin(\x r)+3}
-\tzfnAtEnd{to [bend right] ++(2,-2) node [b] {End!}}
-\tzfn[->]\Fx[-2:2*pi]{$\sin\,x+3$}[blue,r]
-\end{tikzpicture}
-\end{tzcode}
-
-Specifying the option |<code.append>| extends the path after |\tzfnEnd| if it exists.
-
-\begin{tzcode}{.3}
-% \tzfnAtEnd (before \tzfn)
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(-2,-2)(8,6)
-\tzaxes*(-2,-2)(8,6)
-\def\Fx{sin(\x r)+3}
-\tzfnAtEnd{to [bend right] ++(2,-2) node [b] {End!}}
-\tzfn[->]\Fx[-2:2*pi]{$\sin\,x+3$}[blue,r]
- < arc (0:90:4cm) node [draw,red,left] {Here!!} >
-\end{tikzpicture}
-\end{tzcode}
-
-
-\paragraph{\texttt{\bs tzfnAtBegin}}
-You can use \icmd{\tzfnAtBegin} (immediately) \xem{before each} |\tzfn| to insert \Tikz\ code at the beginning of the path of |\tzfn|.
-
-\begin{tzcode}{.3}
-% \tzfnAtBegin (before \tzfn)
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(-2,-2)(8,6)
-\tzaxes*(-2,-2)(8,6)
-\def\Fx{sin(\x r)+3}
-\tzfnAtBegin{ (1,1) -| }
-\tzfn[->]\Fx[-2:2*pi]{$\sin\,x+3$}[blue,r]
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-
-\remark
-\begin{itemize}\firmlist
-\item |\tzfn| is based on the |plot| operation of \Tikz.
-\item Appending \Tikz\ code at the beginning of |\tzfn| may cause a problem when you use some operations (such as |to| or \verb+|-|+) that expect a coordinate to link.
-\item In the version 2 of the |tzplot| package, this issue is internally taken care of by using
-|(|\ixxw{current subpath start}|)|, which is a special coordinate pre-defined in \Tikz. (See \Tikz\ manual for more details.)
-\end{itemize}
-
-
-
-\section{\protect\cmd{\tzfnofy} and \protect\cmd{\tzfnofy'}: Functions of variable $y$}
-\label{ss:tzfnofy}
-
-\icmd{\tzfnofy} plots a function of |\y|.
-Define a function with the (predefined) variable |\y|.
-
-|\tzfnofy| works just like |\tzfn| but as a function of |\y|.
-
-\begin{tzdef}{}
-% syntax: minimum
-\tzfnofy{<fn of \y>}[<domain>]
-% syntax: medium
-\tzfnofy{<fn of \y>}[<domain>]{<text>}[<pos>]
-% syntax: full
-\tzfnofy[<opt>]<shift coor>"<path name>"
- {<fn of \y>}[<domain>]{<text>}[<node opt>]<code.append>
-% [<domain>] should be of the form [<from num:to num>]
-% defaults
- [samples=201]<>""{<m>}[<m>]{}[]<>
-\end{tzdef}
-
-
-
-\begin{tztikz}{}
-\tzfnofy{2*\y+1}[1:5] % works like:
- \draw [samples=201,domain=1:5,variable=\y] plot ({2*\y+1},\y);
-\end{tztikz}
-
-The \iisw{swap version} \icmd{\tzfnofy'} plots the \iisw{inverse function} of |\tzfnofy|.
-
-\begin{tztikz}{}
-\tzfnofy'{2*\y+1}[1:5] % works like:
- \draw [samples=201,domain=1:5,variable=\y] plot (\y,{2*\y+1});
-\end{tztikz}
-
-
-\begin{tzcode}{.3}
-% \tzfnofy('): variable = \y
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,8)
-\tzaxes(8,8){$x$}{$y$}
-\tzfnofy{5-5/7*\y}[0:7]{$f(y)$}[r=5mm]
-\tzfnofy'[blue]{5-5/7*\y}[0:7]
- {$f^{-1}(y)$}[a=5mm,align=center]
-\tzticks{5,7}{5,7}
-\end{tikzpicture}
-\end{tzcode}
-
-\begin{tzcode}{.3}
-% \tzfnofy: shift: intersections
-\begin{tikzpicture}[scale=.6]
-\tzhelplines(8,8)
-\def\Fy{7-\y}
-\def\Gy{1+\y}
-\tzfnofy\Fy[0:7]{$f(y)$}[right] % name path = Fy
-\tzfnofy\Gy[0:7]{$g(y)$}[right] % name path = Gy
-\tzXpoint*{Fy}{Gy}(E){$E$}
-\tzfnofy[dashed]<1,1>"Fyy"\Fy[0:7]{shifted}[r]
-\tzfnofy[dashed]<1,-1>"Gyy"\Gy[0:7]{shifted}[r]
-\tzXpoint*{Fyy}{Gyy}(E1){$E'$}
-\end{tikzpicture}
-\end{tzcode}
-
-You can also extend the path of |\tzfnofy| using the option |<code.append>| or the macros \icmd{\tzfnofyAtBegin} and \icmd{\tzfnofyAtEnd}.
-
-
-\begin{tzcode}{.3}
-% \tzfnofy: shift: intersections
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,8)
-\def\Fy{7/\y}
-\tzfnofyAtBegin{(6,3) node [a] {Start} to[bend left]}
-\tzfnofyAtEnd{ -| ++(3,-1) }
-\tzfnofy[->]\Fy[1:7]
- < to[bend right] ++(3,2) node [draw,r] {End!} >
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-
-%%%\section{\protect\cmd{\tzdeffn} and \protect\cmd{\tzdeffnofy}}
-%%%\label{ss:tzdeffn}
-%%%
-%%%\icmd{\tzdeffn} defines a functions of |\x|.
-%%%
-%%%\icmd{\tzdeffnofy} defines a function of |\y|.
-%%%%%
-%%%%%These are redundant, since one can use |\def|.
-%%%%%
-
-
-
-
-
-%%------------------------------------------------------------
-\section{Horizontal lines}
-\label{s:tzhfn}
-
-\subsection{\protect\cmd{\tzhfnat}}
-\label{ss:tzhfnat}
-
-\icmd{\tzhfnat} draws a horizontal line at a specified value of $y$.
-
-
-\begin{tzdef}{}
-% syntax: minimal
-\tzhfnat{<y-val>}[<domain>]
-% syntax: full
-\tzhfnat[<opt>]<shift coor>"<path name>"
- {<y-val>}[<domain>]{<text>}[<node opt>]<code.append>
-% defaults
- []<>""{<m>}[west:east (of current bounding box)]{}[]<>
-\end{tzdef}
-
-
-|\tzhfnat| accepts only one mandatory argument |{<y-val>}|.
-The domain is optional and should be of the form |[<from num:to num>]|.
-The default domain is from left to right of the \ixxw{current bounding box}.
-
-\remark Internally, the default domain of |\tzhfnat| depends on the \ixxw{current bounding box}.
-\begin{itemize}
-\item Each |\tzhfnat| may draw a line with a (slightly) different length.
-\item If an appropriate |current bounding box| is not formed before |\tzhfnat| is executed, you will probably get an unexpected result.
-\item In that case, you can fix a |bounding box| in the beginning of the |tikzpicture| environment using macros such as \icmd{\tzbbox}, \icmd{\tzhelplines*}, \icmd{\tzaxes*}, or \Tikz's \icmd{\useasboundingbox}.
-\end{itemize}
-
-
-\begin{tzcode}{.3}
-% \tzhfnat
-\begin{tikzpicture}
-\tzhelplines(4,3)
-\tzhfnat[blue,thick]{0}
-\tzhfnat[red,thick]{1}[1:3]{line A}[r]
-\tzhfnat[blue]{2}{line B}[r]
-\tzhfnat[red]{3}[0:3]{line C}[draw=blue,red,r]
-\end{tikzpicture}
-\end{tzcode}
-
-You can name the path of |\tzhfnat| by the option |"<path name>"|.
-You can move the line by the option |<shift coor>|.
-You can also extend the path from the end of the line by writing \Tikz\ code in the last optional argument |<code.append>|.
-
-\begin{tzcode}{.3}
-% \tzhfnat: shift, <code.append>
-\begin{tikzpicture}
-\tzhelplines*(4,4) % bounding box
-\tzfn"XX"{\x}[0:4]
-\tzhfnat[blue,thick]{0}
-\tzhfnat[red,thick]<-.5,.5>"AA"{1}[1:3]{line A'}[r]
-\tzXpoint*{AA}{XX}
-\tzhfnat[blue,->]{2}{line B}[r]
- < arc (0:120:1.5) node [b,draw] {End!} >
-\end{tikzpicture}
-\end{tzcode}
-
-You can also use \icmd{\tzhfnatAtBegin} and \icmd{\tzhfnatAtEnd} to extend the path of |\tzhfnat| at the beginning and at the end. Specifying the option |<code.append>| extends the path after |\tzhfnatAtEnd| if it exists.
-(See other examples of using |\tz<..>AtBegin| and |\tz<...>AtEnd|.)
-
-
-\subsection{\protect\cmd{\tzhfn}}
-\label{ss:tzhfn}
-
-\icmd{\tzhfn} accepts a coordinate as a mandatory argument and draws a horizontal line at the $y$ value of the coordinate.
-For example, |\tzhfn(<x>,3)|, ignoring |<x>|, is equivalent to |\tzhfnat{3}|.
-
-\xem{Everything else is the same as in} |\tzhfnat|.
-
-\begin{tzdef}{}
-% syntax: minimal
-\tzhfn(<coor>)[<domain>]
-% syntax: full
- \tzhfn[<opt>]<shift coor>"<path name>"
- (<coor>)[<domain>]{<text>}[<node opt>]<code.append>
-% defaults
- []<>""(<m>)[west:east (of current bounding box)]{}[]<>
-\end{tzdef}
-
-\begin{tzcode}{.3}
-% \tzhfn: shift, <code.append>
-\begin{tikzpicture}
-\tzhelplines*(4,4) % bounding box
-\tzcoors(0,1)(A)(0,2)(B);
-\tzfn"XX"{\x}[0:4]
-\tzhfn[blue,thick](5,0) % x=5 ignored
-\tzhfn[red,thick]<-.5,.5>"AA"(A)[1:3]{line A'}[r]
-\tzXpoint*{AA}{XX}
-\tzhfn[blue,->](B){line B}[r]
- < arc (0:120:1.5) node [b,draw] {End!} >
-\end{tikzpicture}
-\end{tzcode}
-
-You can also use \icmd{\tzhfnAtBegin} and \icmd{\tzhfnAtEnd} to extend the path of |\tzhfn| at the beginning and at the end. Specifying the option |<code.append>| extends the path after |\tzhfnAtEnd| if it exists.
-(See other examples of using |\tz<..>AtBegin| and |\tz<...>AtEnd|.)
-
-
-%%------------------------------------------------------------
-\section{Vertical lines}
-\label{s:tzvfn}
-
-\subsection{\protect\cmd{\tzvfnat}}
-\label{ss:tzvfnat}
-
-\icmd{\tzvfnat} draws a vertical line at a specified value of $x$.
-
-\begin{tzdef}{}
-% syntax: minimal
-\tzvfnat{<x-val>}[<domain>]
-% syntax: full
-\tzvfnat[<opt>]<shift coor>"<path name>"
- {<x-val>}[<domain>]{<text>}[<node opt>]<code.append>
-% defaults
- []<>""{<m>}[south:north (of current bounding box)]{}[]<>
-\end{tzdef}
-
-|\tzvfnat| accepts only one mandatory argument |{<x-val>}|.
-The domain is optional and should be of the form |[<from num:to num>]|.
-The default domain is from bottom to top of the \ixxw{current bounding box}.
-
-
-\begin{tzcode}{.3}
-% \tzvfnat
-\begin{tikzpicture}
-\tzhelplines(4,4)
-\tzvfnat[blue,thick]{0}
-\tzvfnat[red,thick]{1}[1:3]{line A}[a]
-\tzvfnat[blue]{2}{line B}[a]
-\tzvfnat[red]{3}[0:3]{line C}[draw=blue,red,a]
-\end{tikzpicture}
-\end{tzcode}
-
-You can name the path of |\tzvfnat| by the option |"<path name>"|.
-You can move the line by the option |<shift coor>|.
-You can also extend the path from the end of the line by writing \Tikz\ code in the last optional argument |<code.append>|.
-
-\begin{tzcode}{.3}
-% \tzvfnat: shift, <code.append>
-\begin{tikzpicture}
-\tzhelplines*(4,4) % bounding box
-\tzfn"XX"{\x}[0:4]
-\tzvfnat[blue,thick]{0}
-\tzvfnat[red,thick]<.5,-.5>"AA"{1}[1:3]{line A'}[a]
-\tzXpoint*{AA}{XX}
-\tzvfnat[blue,->]{2}{line B}[a]
- < arc (90:-30:1.5) node [b,draw] {End!} >
-\end{tikzpicture}
-\end{tzcode}
-
-In the previous example, |\tzhelplines*| is used to fix a bounding box. (See Section \ref{s:tzhelplines} on page \pageref{s:tzhelplines}, for more details.)
-
-You can also use \icmd{\tzvfnatAtBegin} and \icmd{\tzvfnatAtEnd} to extend the path of |\tzvfnat| at the beginning and at the end. Specifying the option |<code.append>| extends the path after |\tzvfnatAtEnd| if it exists.
-(See other examples of using |\tz<..>AtBegin| and |\tz<...>AtEnd|.)
-
-\subsection{\protect\cmd{\tzvfn}}
-\label{ss:tzvfn}
-
-\icmd{\tzvfn} accepts a coordinate as a mandatory argument and draws a horizontal line at the $x$ value of the coordinate.
-For example, |\tzvfn(3,<y>)|, ignoring |<y>|, is equivalent to |\tzvfnat{3}|.
-
-\xem{Everything else is the same as in} |\tzvfnat|.
-
-\begin{tzdef}{}
-% syntax: minimal
-\tzvfn(<coor>)[<domain>]
-% syntax: full
-\tzvfn[<opt>]<shift coor>"<path name>"
- (<coor>)[<domain>]{<text>}[<node opt>]<code.append>
-% defaults
- []<>""(<m>)[south:north (of current bounding box)]{}[]<>
-\end{tzdef}
-
-\begin{tzcode}{.3}
-% \tzvfn: shift, <code.append>
-\begin{tikzpicture}
-\tzhelplines*(4,4) % bounding box
-\tzcoors(1,0)(A)(2,0)(B);
-\tzfn"XX"{\x}[0:4]
-\tzvfn[blue,thick](0,5) % y=5 ignored
-\tzvfn[red,thick]<.5,-.5>"AA"(A)[1:3]{line A'}[a]
-\tzXpoint*{AA}{XX}
-\tzvfn[blue,->](B){line B}[a]
- < arc (90:-30:1.5) node [b,draw] {End!} >
-\end{tikzpicture}
-\end{tzcode}
-
-You can also use \icmd{\tzvfnAtBegin} and \icmd{\tzvfnAtEnd} to extend the path of |\tzvfn| at the beginning and at the end. Specifying the option |<code.append>| extends the path after |\tzvfnAtEnd| if it exists.
-(See other examples of using |\tz<..>AtBegin| and |\tz<...>AtEnd|.)
-
-
-
-
-
-%%==================================
-\chapter{Plot Linear Functions}
-\label{c:linearfunctions}
-
-
-%%------------------------------------------------------------
-\section{\protect\cmd{\tzLFn}: Plot linear functions}
-\label{s:tzLFn}
-
-\subsection{\protect\cmd{\tzLFn} and \protect\cmd{\tzLFn'}}
-
-Knowing two coordinates or one coordinate with a slope, you can draw a linear function with the macro \icmd{\tzLFn}, \xem{without writing the explicit definition of a linear function}.
-
-\begin{itemize}\firmlist
-\item
-|\tzLFn(<coor1>)(<coor2>)| is prepared for when you know two coordinates on a line.
- \begin{itemize}
- \item
- If you provide two points $(x_1,y_1)$ and $(x_2,y_2)$, |\tzLFnofy(x1,x2)(y1,y2)| draws the graph of $f(x)=\frac{y_2-y_1}{x_2-x_1}(x-x_1)+y_1$.
- \end{itemize}
-
-\item
-|\tzLFn(<coor1>){<slope>}| is prepared for when you know one coordinate and the slope of a line.
-\item
-If you specify all the three arguments |(<coor1>)(<coor2>){<slope>}|, then the slope is ignored.
-\end{itemize}
-
-For example, |\tzLFn(1,1)(2,3)[0:4]| draws a line passing through two points: $(1,1)$ and $(2,3)$, over $0\leq x\leq 4$.
-|\tzLFn(1,1){.5}[0:4]| draws a line passing through a point $(1,1)$ with a slope |.5|, over $0\leq x\leq 4$.
-
-|\tzLFn| accepts two mandatory arguments: |(<coor1>)| and |[<domain>]|.
-\begin{itemize}\firmlist
-\item The \xem{domain} should be of the form |[<from num:to num>]|.
-\item If just one coordinate is specified without a slope, the slope is regarded as |1|, by default.
-\end{itemize}
-
-\begin{tzdef}{}
-% syntax: minimum
-\tzLFn(<coor1>)(<coor2>)[<domain>]
-\tzLFn(<coor1>){<slope>)[<domain>]
-% syntax: full
-\tzLFn[<opt>]<shift coor>"<path name>"
- (<coor1>)(<coor2>){<slope>}[<domain>]{<text>}[<node opt>]<code.append>
-% defaults
- []<>""(<m>)(){1}[<m>]{}[]<>
-\end{tzdef}
-
-
-
-You can add text at the end of the line of |\tzLFn| by the options |{<text>}| and |[<node opt>]|. You can also name the path of |\tzLFn| by specifying the option |"<path name>"| \xem{immediately before} the mandatory coordinate.
-
-\begin{tzcode}{.3}
-% \tzLFn: two coordinates
-\begin{tikzpicture}
-\tzhelplines(4,3)
-\tzcoors*(1,1)(A)(1,2)(B)(3,1)(C);
-\tzLFn[blue]"Gx"(B)(C)[0:4]{$g(x)$}[r]
-\end{tikzpicture}
-\end{tzcode}
-
-
-\begin{tzcode}{.3}
-% \tzLFn: one coordinate and slope
-\begin{tikzpicture}
-\tzhelplines(4,3)
-\tzcoors*(1,1)(A)(1,2)(B)(3,1)(C);
-\tzLFn[red]"Fx"(A){.5}[0:4]{$f(x)$}[a]
-\end{tikzpicture}
-\end{tzcode}
-
-\remark If you inadvertently try an \xem{infinite slope}, you will get an \iisw{error message}.
-
-\paragraph{Inverse function} The \iisw{swap version} \icmd{\tzLFn'} draws the \iisw{inverse function} of |\tzLFn|.
-
-\begin{tzcode}{.3}
-% \tzLFn' : inverse function : intersection
-\begin{tikzpicture}[scale=.8]
-\tzhelplines(4,4)
-\tzcoors*(1,1)(A)(1,2)(B)(3,1)(C);
-\tzLFn'[red]"Fx"(A){.5}[0:4]{$f(x)$}[a]
-\tzLFn'[blue]"Gx"(B)(C)[0:4]{$g(x)$}[r]
-\tzXpoint*[fill=none]{Fx}{Gx}(E){$E$}(3pt)
-\end{tikzpicture}
-\end{tzcode}
-
-
-You can move the line of |\tzLFn| by specifying the option |<shift coor>| immediately before the option |"<path name>"|. (The \xem{empty} shift option |<>| is \xem{not allowed}.)
-You can also extend the path of |\tzLFn| by writing \Tikz\ code in the last optional argument |<code.append>|.
-
-\begin{tzcode}{.3}
-% \tzLFn: shift, extending path
-\begin{tikzpicture}[scale=.8]
-\tzhelplines(4,4)
-\tzcoors*(1,1)(A)(1,2)(B)(3,1)(C);
-\tzLFn[blue]"Gx"(B)(C)[0:4]{$g(x)$}[r]
-\tzLFn[dashed,red,->]<1,1>"Gx"(B)(C)[0:4]{$g(x)$}[r]
- < arc (0:140:2) node [below] {End!} >
-\end{tikzpicture}
-\end{tzcode}
-
-\icmd{\tzLFnAtBegin} and \icmd{\tzLFnAtEnd} are available to extend a path of |\tzLFn| at the beginning and and the end, respectively.
-Specifying the option |<code.append>| extends the path after |\tzLFnAtEnd|, if it exist.
-
-\begin{tzcode}{.3}
-% \tzLFnAtBegin and \tzLFnAtEnd
-\begin{tikzpicture}[scale=.8]
-\tzhelplines*(4,4)
-\tzcoors*(1,1)(A)(1,2)(B)(3,1)(C);
-\tzLFn[blue]"Gx"(B)(C)[0:4]{$g(x)$}[r]
-\tzLFnAtBegin{(1,0) to [bend left] }
-\tzLFnAtEnd{ arc (0:140:2) }
-\tzLFn[dashed,red,->]<1,1>"Gx"(B)(C)[0:4]{$g(x)$}[r]
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-\subsection{\protect\cmd{\tzLFnofy} and \protect\cmd{\tzLFnofy'}}
-
-\icmd{\tzLFnofy} draws a line as a function of |\y|.
-|\tzLFnofy| works just like |\tzLFn| but for the variable $y$.
-If you provide two points $(x_1,y_1)$ and $(x_2,y_2)$, |\tzLFnofy(x1,x2)(y1,y2)| draws the graph of $f(y)=\frac{x_2-x_1}{y_2-y_1}(y-y_1)+x_1$.
-
-Everything else is the same as |\tzLFn|.
-
-\begin{tzdef}{}
-% syntax: minimum
-\tzLFnofy(<coor1>)(<coor2>)[<domain>]
-\tzLFnofy(<coor1>){<slope>)[<domain>]
-% syntax: full
-\tzLFnofy[<opt>]<shift coor>"<path name>"
- (<coor1>)(<coor2>){<slope>}[<domain>]{<text>}[<node opt>]<code.append>
-% defaults
- []<>""(<m>)(){1}[<m>]{}[]<>
-\end{tzdef}
-
-\begin{tzcode}{.3}
-% \tzLFnofy
-\begin{tikzpicture}[scale=.8]
-\tzhelplines(4,4)
-\tzcoors*(1,1)(A)(1,2)(B)(3,1)(C);
-\tzLFnofy[red]"Fx"(A){.5}[0:4]{$f(y)$}[a]
-\tzLFnofy[blue]"Gx"(B)(C)[.5:2.5]{$g(y)$}[a]
-\tzXpoint*[fill=none]{Fx}{Gx}(E){$E$}(3pt)
-\end{tikzpicture}
-\end{tzcode}
-
-\remark If you inadvertently try an \xem{infinite slope}, you will get an \xem{error message}.
-
-\paragraph{Inverse function}
-The \iisw{swap version} \icmd{\tzLFnofy'} draws the \iisw{inverse function} of |\tzLFnofy|.
-
-\begin{tzcode}{.3}
-% \tzLFnofy'
-\begin{tikzpicture}[scale=.8]
-\tzhelplines(4,4)
-\tzcoors*(1,1)(A)(1,2)(B)(3,1)(C);
-\tzLFnofy'[red]"Fx"(A){.5}[0:4]{$f(y)$}[a]
-\tzLFnofy'[blue]"Gx"(B)(C)[.5:2.5]{$g(y)$}[r]
-\tzXpoint*[fill=none]{Fx}{Gx}(E){$E$}(3pt)
-\end{tikzpicture}
-\end{tzcode}
-
-
-You can use \icmd{\tzLFnofyAtBegin} and \icmd{\tzLFnofyAtEnd} to extend the path of |\tzLFnofy| at the beginning and at the end. Specifying the optional argument |<code.append>| extends the path after |\tzLFnofyAtEnd| if it exists.
-(See other examples of using |\tz<..>AtBegin| and |\tz<...>AtEnd|.)
-
-
-\section{\protect\cmd{\tzdefLFn}}
-\label{s:tzdefLFn}
-
-\icmd{\tzdefLFn} simply defines a linear function $ax+b$ and saves it to a macro.
-You can use |\tzdefLFn| together with |\tzfn| to graph a linear function, \xem{without writing an explicit definition} of a linear function.
-(Of course, you can directly use |\tzLFn|.)
-
-\begin{tzdef}{}
-% syntax
-\tzdefLFn{<fn csname>}(<coor1>)(<coor2>){<slope>}
-% defaults
-{<m>}(<m>)(){1}
-\end{tzdef}
-
-If |(<coor2>)| is specified, |{<slope>}| is ignored.
-If |(<coor2>)| is missing |<slope>| is considered as the slope of the line (by default 1).
-
-For example, |\tzdefLFn{\Gx}(1,1){.5}| defines as |\Gx| a linear function passing through the point $(1,2)$ with a slope |.5|.
-That is, it defines a function as $f(x)=.5(x-1)+1$.
-Knowing two coordinates, linear function |\Gx| passing through the two points can be defined, for example, by |\tzdefLFn{\Gx}(1,2)(3,1)|.
-That is, it defines a function as $g(x)=\frac{1-2}{3-1}(x-1)+2$.
-
-
-\begin{tztikz}{}
-\tzdefLFn{\Fx}(1,2)(3,1) % works like
- \def\Fx{-1/2*(\x-1)+2}
-\end{tztikz}
-
-\remark If you inadvertently try an \xem{infinite slope}, you will get an \iisw{error message}.
-
-
-\begin{tzcode}{.3}
-% \tzdefLFn and \tzfn
-\begin{tikzpicture}
-\tzhelplines(4,4)
-\tzaxes(4,4){$x$}{$y$}
-\tzticks{1,2,3,4}{1,2,3,4}
-\tzcoors*(1,1)(A){A}[-90](1,2)(B){B}(3,1)(C){C}[45];
-\tzdefLFn\Fx(A){.5}
-\tzdefLFn\Gx(B)(C)
-\tzfn[red]\Fx[0:4]{$f(x)$}[r]
-\tzfn[blue]\Gx[0:4]{$g(x)$}[r]
-\end{tikzpicture}
-\end{tzcode}
-
-\remark The swap version |\tzfn'| simply draws the graph of the inverse function of |\tzfn|.
-So |\tzfn'(A)(B)| and |\tzfn'(A){.5}| do not guarantee passing through the coordinate |(A)| or |(B)|.
-
-\begin{tzcode}{.3}
-% \tzdefLFn and \tzfn'
-\begin{tikzpicture}
-\tzhelplines(4,4)
-\tzaxes(4,4){$x$}{$y$}
-\tzticks{1,2,3,4}{1,2,3,4}
-\tzcoors*(1,1)(A){A}[-90](1,2)(B){B}(3,1)(C){C}[45];
-\tzdefLFn\Fx(A){.5}
-\tzdefLFn\Gx(B)(C)
-\tzfn'[red]\Fx[0:4]{$f(x)$}[r]
-\tzfn'[blue]\Gx[0:4]{$g(x)$}[r]
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-
-
-%%------------------------------------------------------------
-\section{\protect\cmd{\tzdefLFnofy}}
-\label{s:tzdefLFnofy}
-
-\icmd{\tzdefLFnofy} defines a function of |\y|.
-
-\begin{tzdef}{}
-% syntax
-\tzdefLFnofy{<fn csname>}(<coor1>)(<coor2>){<slope>}
-% defaults
-{<m>}(<m>)(){1}
-\end{tzdef}
-
-If |(<coor2>)| is specified, |{<slope>}| is ignored.
-If |(<coor2>)| is missing |<slope>| is considered as the slope of the line (by default 1).
-
-
-\begin{tztikz}{}
-\tzdefLFnofy\Fy(1,1){.5} % works like
- \def\Fy{.5*(\y-1)+1}
-\end{tztikz}
-
-\begin{tztikz}{}
-\tzdefLFnofy\Gy(1,2)(3,1) % works like
- \def\Gy{-2*(\y-2)+1}
-\end{tztikz}
-
-
-\remark If you inadvertently try an \xem{infinite slope}, you will get an \xem{error message}.
-
-You can use |\tzdefLFnofy| together with |\tzFnofy| to graph a linear function of $y$.
-(Of course, you can directly use |\tzLFnofy|.)
-
-
-\begin{tzcode}{.3}
-% \tzdefLFnofy and \tzLFnofy
-\begin{tikzpicture}
-\tzhelplines(4,4)
-\tzaxes*(4,4){$x$}{$y$}
-\tzticks{1,2,3,4}{1,2,3,4}
-\tzcoors*(1,1)(A){A}[-90](1,2)(B){B}(3,1)(C){C}[45];
-\tzdefLFnofy\Fy(A){.5}
-\tzdefLFnofy\Gy(B)(C)
-\tzfnofy[red]\Fy[0:4]{$f(y)$}[r]
-\tzfnofy[blue]\Gy[0:4]{$g(y)$}[a]
-\end{tikzpicture}
-\end{tzcode}
-
-\remark The swap version |\tzfnofy'| simply draws the graph of the inverse function of |\tzfnofy|.
-So |\tzfnofy'(A)(B)| and |\tzfnofy'(A){.5}| do not guarantee passing through the coordinate |(A)| or |(B)|.
-
-\begin{tzcode}{.3}
-% \tzdefLFnofy and \tzLFnofy'
-\begin{tikzpicture}
-\tzhelplines(4,4)
-\tzaxes*(4,4){$x$}{$y$}
-\tzticks{1,2,3,4}{1,2,3,4}
-\tzcoors*(1,1)(A){A}[-90](1,2)(B){B}(3,1)(C){C}[45];
-\tzdefLFnofy\Fy(A){.5}
-\tzdefLFnofy\Gy(B)(C)
-\tzfnofy'[red]\Fy[0:4]{$f(y)$}[r]
-\tzfnofy'[blue]\Gy[0:4]{$g(y)$}[r]
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-%%==================================
-\chapter{Some More Functions}
-\label{c:more-fn}
-
-%%------------------------------------------------------------
-\section{\texttt{\bs tzpdfN(*)} and \texttt{\bs tzpdfZ}: Normal distributions}
-\label{s:normaldistributions}
-
-
-|\tzpdfN|, |\tzpdfN*| and |\tzpdfZ| are predefined functions to plot the \iisw{probability density function} (pdf) of a \iisw{normal distribution} $N(\mu,\sigma^2)$.
-
-\paragraph{Normal distributions}
-\icmd{\tzpdfN} accepts two mandatory arguments, |{<mean>}| $\mu$ and |{<variance>}| $\sigma^2$, to define the pdf function:
-\[
- \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac12\big(\frac{x-\mu}{\sigma}\big)^2}
-\]
-
-\icmd{\tzpdfN*} uses a standard deviation $\sigma$ instead of a variance.
-
-With these predefined functions together with |\tzfn| you can plot the pdf of normal distributions.
-
-\begin{tzcode}{.3}
-% \tzpdfN, \tzpdfN*
-\begin{tikzpicture}[xscale=0.5,yscale=5,font=\scriptsize]
-\tzhelplines[ystep=0.1](-4,0)(6,.5)
-\tzaxes(-4,0)(6,.5)
-\tzticks{1,...,5}{0.1,0.2,0.3,0.4}
-\tzfn[blue]{\tzpdfN{0}{1.5}}[-4:4] % \tzpdfN
-\tzfn[red]{\tzpdfN*{2}{sqrt(1.5)}}[-2:6] % \tzpdfN*
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-\begin{tzcode}{.3}
-% \tzpdfN, \tzpdfN*
-\begin{tikzpicture}[x=0.5cm,y=5cm,font=\scriptsize]
-\tzhelplines[step=.5cm](-4,0)(6,.5)
-\tzaxes(-4,0)(6,.5)
-\tzticks{1,2,3}{0.1,0.2,0.3,0.4}
-\tzfn[blue]{\tzpdfN{0}{1.5}}[-4:4]
-\tzfn[red]{\tzpdfN*{2}{sqrt(1.5)}}[-2:6]
-\end{tikzpicture}
-\end{tzcode}
-
-\remark
-If |tikzpicture| is scaled when plotting the inverse function using |\tzfn'|, you may want to change the scale accordingly.
-
-\begin{tzcode}{.3}
-% \tzpdfN and \tzfn'
-\begin{tikzpicture}[x=0.5cm,y=5cm,,font=\scriptsize]
-\tzhelplines[step=0.5cm](-4,-.3)(6,.5)
-\tzaxes(-4,-.3)(6,.5)
-% scale change
-\begin{scope}[xscale={1/0.5*5},yscale={1/5*0.5}]
-\tzfn'[blue]{\tzpdfN{0}{1.5}}[-4:4]
-\tzfn'[red]{\tzpdfN*{2}{sqrt(1.5)}}[-2:6]
-\end{scope}
-\end{tikzpicture}
-\end{tzcode}
-
-\paragraph{Standard normal distributions}
-\icmd{\tzpdfZ} (with no arguments) represents the pdf of the \xem{standard normal distribution}.
-
-\begin{tzcode}{.3}
-% \tzpdfZ: shift
-\begin{tikzpicture}[x=0.5cm,y=5cm,font=\scriptsize]
-\tzhelplines[step=0.5cm](-4,0)(6,.5)
-\tzaxes(-4,0)(6,.5)
-\tzticks{1,...,5}{0.1,0.2,0.3,0.4}
-\tzfn[blue]{\tzpdfZ}[-3:3] % \tzpdfZ
-\tzfn[red]<2,0>{\tzpdfZ}[-3:3] % shift
-\tzfn[dashed]<3,.1>{\tzpdfZ}[-3:3] % shift
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-
-%%------------------------------------------------------------
-\section{\protect\cmd{\tzfnarea(*)}: Fill under the graph}
-
-
-\subsection{\protect\cmd{\tzfnarea(*)}}
-
-\icmd{\tzfnarea} fills the area between the graph of a function and the x axis, with a color or pattern, on the |behind| layer by default.
-With \icmd{\settzfnarealayer}, you can change the layer of |\tzfnarea|.
-
-|\tzfnarea| accepts two mandatory arguments: |{<fn of \x>}| and |[<domain>]|, like |\tzfn|.
-
-\begin{tzdef}{}
-% syntax: minimum
-\tzfnarea{<fn of \x>}[<domain>]
-% syntax: full
-\tzfnarea[<opt>]{<fn of \x>}[<domain>]{<fill opacity>}<code.append>
-% defaults
- []{<m>}[<m>]{.3}<>
-\end{tzdef}
-
-\begin{tzcode}{.3}
-% \tzfnarea
-\begin{tikzpicture}
-\tzhelplines(0,-1)(4,2)
-\tzaxes(0,-1)(4,2)
-\def\Fx{.5*(\x-1)^2-1}
-\tzfn[very thick]\Fx[0:3.5]
-\tzfnarea[fill=red]\Fx[1:3] %
-\end{tikzpicture}
-\end{tzcode}
-
-\begin{tzcode}{.3}
-% \tzfnarea: layer: comparison
-\begin{tikzpicture}
-\tzhelplines(0,-1)(4,2)
-\settzfnarealayer{main} %%
-\tzaxes(0,-1)(4,2)
-\def\Fx{.5*(\x-1)^2-1}
-\tzfn[very thick]\Fx[0:3.5]
-\tzfnarea[fill=red]\Fx[1:3] %
-\end{tikzpicture}
-\end{tzcode}
-
-
-The starred version \icmd{\tzfnarea*} fills the area with |fill=black!50| and |fill opacity=.3|, and |text opacity=1|, by default. The default values can be changed by |\settzfillcolor|, |\settzfillopacity|, and the option |{<fill opacity>}|.
-
-\begin{tzcode}{.3}
-% \tzfnarea*
-\begin{tikzpicture}[x=0.5cm,y=5cm,font=\scriptsize]
-\tzhelplines[step=.5cm](-4,-.1)(6,.5)
-\tzaxes(-4,-.1)(6,.5)
-\tzfn[blue]{\tzpdfZ}[-3:3]
-\tzfnarea*{\tzpdfZ}[-3:1]
-\tzfn"AA"{\tzpdfN21}[-1:5]
-\tzfnarea*[pattern=north east lines]{\tzpdfN21}[1:3]
-\end{tikzpicture}
-\end{tzcode}
-
-
-\begin{tzcode}{.3}
-% \tzfnarea*: default change and <code.append>
-\begin{tikzpicture}
-\settzfnarealayer{main} %
-\tzhelplines(0,-1)(4,2)
-\tzaxes(0,-1)(4,2)
-\def\Fx{.5*(\x-1)^2-1}
-\tzfn\Fx[0:3.5]
-\tzfnarea*[red]\Fx[1:3]{.1}<(2,-1) node[black]{Area}>
-\end{tikzpicture}
-\end{tzcode}
-
-
-\begin{tzcode}{.3}
-% \tzfnarea*
-\begin{tikzpicture}
-\tzhelplines(0,-2)(4,3)
-\tzaxes(0,-2)(4,3){$x$}{$y$}
-\def\Fx{.5*(\x-1)^2-1}
-\tzfn\Fx[0:3.5]{$f(x)$}[a]
-\tzfnarea*[red]\Fx[1:3]
-\tzvXpointat{Fx}{1}(A)
-\tzvXpointat{Fx}{3}(B)
-\tzline[blue,thick](A)(A|-0,0)
-\tzline[very thick](B)(B|-0,0)
-\end{tikzpicture}
-\end{tzcode}
-
-
-\subsection{\protect\cmd{\tzfnarealine(')}}
-
-\icmd{\tzfnarealine} draws one or two boundary lines of |\tzfnarea| using |\tzto|, on the |behind| layer by default. The layer can be changed by \icmd{\settzfnarealayer}.
-It takes two mandatory arguments: |{<path>}| and |{<x1>}|.
-\begin{itemize}\firmlist
-\item |\tzfnarealine{<path>}{<x1>}| draws a vertical line to the $x$ axis at |<x1>|.
-\item |\tzfnarealine{<path>}{<x1>}{<x2>}| draws two vertical lines at |<x1>| and |<x2>|.
-\end{itemize}
-
-The first option |[<opt>]| controls both lines and is overwritten by each of |[<opt1>]| (for the line at |<x1>|) and |[<opt2>]| (for the line at |<x2>|).
-The line width is |very thin| by default, which can be changed by \icmd{\settzfnarealinewith}.
-
-You can change the length of lines by specifying |(<coor1>)| and |(<coor2>)|.
-
-\begin{tzdef}{}
-% syntax: minimum
-\tzfnarealine{<path>}{<x1>}
-% syntax: minimal
-\tzfnarealine{<path>}{<x1>}{<x2>}
-% syntax: medium
-\tzfnarealine{<path>}{<x1>}[<opt1>]{<x2>}[<opt2>]
-% syntax: full
-\tzfnarealine[<opt>]{<path>}{<x1>}[<opt1>](<coor1>)
- {<x2>}[<opt2>](<coor2>)
-% defaults:
- [very thin]{<m>}{<m>}[](0,0){}[](0,0)
-\end{tzdef}
-
-
-\begin{tzcode}{.3}
-% \tzfnarealine
-\begin{tikzpicture}[x=0.5cm,y=5cm,font=\scriptsize]
-\tzhelplines[step=.5cm](-4,-.1)(6,.5)
-\tzaxes(-4,-.1)(6,.5)
-\tzfn"AA"{\tzpdfN21}[-1:5]
-\tzfnarea*[pattern=north east lines]{\tzpdfN21}[1:3]
-\tzfnarealine[red]{AA}{1}{3} %%
-\end{tikzpicture}
-\end{tzcode}
-
-\begin{tzcode}{.3}
-% \tzfnarealine: one line
-\begin{tikzpicture}
-\tzhelplines(0,-1)(4,2)
-\tzaxes(0,-1)(4,2)
-\def\Fx{.5*(\x-1)^2-1}
-\tzfn\Fx[0:3.5]
-\tzfnarea*[red]\Fx[1:3]{.1}<(2,-1) node[black]{Area}>
-\tzfnarealine[blue]{Fx}{3}
-\end{tikzpicture}
-\end{tzcode}
-
-\begin{tzcode}{.3}
-% \tzfnarealine: two lines
-\begin{tikzpicture}
-\tzhelplines(0,-1)(4,2)
-\tzaxes(0,-1)(4,2)
-\def\Fx{.5*(\x-1)^2-1}
-\tzfn\Fx[0:3.5]
-\tzfnarea*[red]\Fx[1:3]{.1}<(2,-1) node[black]{Area}>
-\tzfnarealine[blue,thick]{Fx}{1}{3}[very thick,red]
-\end{tikzpicture}
-\end{tzcode}
-
-The \xem{swap version} \icmd{\tzfnarealine'} draws one or two horizontal lines from the point $(x_i,f(x_i))$ to the $y$ axis.
-Everything else is the same as in |\tzfnarealine|.
-
-\begin{tzcode}{.3}
-% \tzfnarealine'
-\begin{tikzpicture}
-\tzhelplines(0,-1)(4,2)
-\settzfnarealinestyle{thick}
-\tzaxes(0,-1)(4,2)
-\def\Fx{.5*(\x-1)^2-1}
-\tzfn\Fx[0:3.5]
-\tzfnarea*[blue]\Fx[1:3]{.1}<(2,-1) node[black]{Area}>
-\tzfnarealine'[->,red]{Fx}{2}{3}
-\end{tikzpicture}
-\end{tzcode}
-
-
-%%------------------------------------------------------------
-\section{Envelope curves}
-\label{s:evelopecurves}
-
-
-%%------------------------------------------------------------
-\subsection{\protect\cmd{\tzfnmax}: Upper envelope curves}
-\label{ss:tzfnmax}
-
-\icmd{\tzfnmax} plots the maximum of a list of functions, that is, an \xem{upper} \iisw{envelope curve}. The first mandatory argument should be a comma separated list of functions.
-The second mandatory argument |[<domain>]| should be colon separated.
-
-The default line width of |\tzfnmax| is |thick|.
-
-\begin{tzdef}{}
-% syntax: minimum
-\tzfnmax{<fn list>}[<domain>]
-% syntax: full
-\tzfnmax[<opt>]<shift coor>"<path name>"
- {<fn list>}[<domain>]{<text>}[<node opt>]<code.append>
-% remark:
- - {<fn list>} should be comma separated
- - [<domain>] should be of the form [<from num:to num>]
-% defaults
- [thick,samples=201]<>""{<m>}[<m>]{}[]<>
-\end{tzdef}
-
-
-\begin{tzcode}{.3}
-% \tzfnmax
-\begin{tikzpicture}
-\tzhelplines*(4,4)
-\def\Fx{\x}
-\def\Gx{.5*\x+.5}
-\def\Hx{-.25*\x+2}
-\tzfn{\Fx}[0:4] \tzfn{\Gx}[0:4] \tzfn{\Hx}[0:4]
-\tzfnmax[blue]{\Fx,\Gx,\Hx}[0:4]{upper envelope}[l]
-\end{tikzpicture}
-\end{tzcode}
-
-\remark If you want sharp corners at kinked points, you may need to select an appropriate number of sample points, which is |samples=201|, by default. You can try an odd number.
-\remarkafterskip
-
-If you want, you can shift and extend path using the option |<shift coor>| and |<code.append>|, respectively.
-You can also use \icmd{\tzfnmaxAtBegin} and \icmd{\tzfnmaxAtEnd} to extend the path of |\tzfnmax| at the beginning and at the end, respectively.
-
-\begin{tzcode}{.3}
-% shift, <code.append>
-\begin{tikzpicture}
-\tzhelplines*(4,5)
-\def\Fx{\x}
-\def\Gx{.5*\x+.5}
-\def\Hx{-.25*\x+2}
-\tzfn{\Fx}[0:4] \tzfn{\Gx}[0:4] \tzfn{\Hx}[0:4]
-\tzfnmax[blue,thick]{\Fx,\Gx,\Hx}[0:4]
-\tzfnmaxAtBegin{ (-1,0) -- }
-\tzfnmax[dashed]<0,.5>{\Fx,\Gx,\Hx}[0:4]
- < to[bend left] ++ (1,-1) >
-\end{tikzpicture}
-\end{tzcode}
-
-
-%%------------------------------------------------------------
-\subsection{\protect\cmd{\tzfnmin}: Lower envelope curves}
-\label{ss:tzfnmin}
-
-\icmd{\tzfnmin} plots the minimum of a list of functions.
-|\tzfnmin| works just like |\tzfnmax|, but it draws \xem{lower} a \iisw{envelope curve}.
-
-\begin{tzdef}{}
-% syntax: minimum
-\tzfnmin{<fn list>}[<domain>]
-% syntax: full
-\tzfnmin[<opt>]<shift coor>"<path name>"
- {<fn list>}[<domain>]{<text>}[<node opt>]<code.append>
-% remark:
- - {<fn list>} should be comma separated
- - [<domain>] should be of the form [<from num:to num>]
-% defaults
- [thick,samples=201]<>""{<m>}[<m>]{}[]<>
-\end{tzdef}
-
-
-\begin{tzcode}{.3}
-% \tzfnmin, \tzfnmax: name path: intersection
-\begin{tikzpicture}
-\tzhelplines(4,6)
-\def\Fx{\x}
-\def\Gx{.5*(\x)^2}
-\def\Hx{-.25*\x+2}
-\tzfn{\Fx}[0:4]{$f(x)$}[r]
-\tzfn{\Gx}[0:3]{$g(x)$}[r]
-\tzfn{\Hx}[0:4]{$h(x)$}[r]
-\tzfnmin[samples=2001,blue,thick]"XX" %%
- {\Fx,\Gx,\Hx}[0:4]{lower evelope}[bl]
-\tzvXpointat*{XX}{3}
-\tzfnmax[samples=201,red,thick]"YY" %%
- {\Fx,\Gx,\Hx}[0:3]{upper envelope}[bl]
-\tzvXpointat*[fill=none]{YY}{2.5} %%
-\end{tikzpicture}
-\end{tzcode}
-
-\remark If you want sharp corners at kinked points, you may need to select appropriate number of sample points. Try odd number.
-
-To extend the path of |\tzfnmin|, you can use the option |<code.append>| or the macros \icmd{\tzfnminAtBegin} and \icmd{\tzfnminAtEnd}.
-
-
-
-
-
-
-%%==================================
-\chapter{Intersections}
-\label{c:intersections}
-
-%%------------------------------------------------------------
-\section{\protect\cmd{\tzXpoint(*)}: Intersection points}
-\label{s:intersections}
-
-\icmd{\tzXpoint} finds \iisw{intersection} points of two paths and saves them as coordinate names for later use.
-
-\begin{tzdef}{}
-% syntax: minimal
-\tzXpoint{<path>}{<path>}(<coor name>)
-% syntax: medium
-\tzXpoint{<path>}{<path>}(<coor name>){<label>}[<angle>]
-% syntax: full
-\tzXpoint[<opt>]{<path>}{<path>}(<coor name>)[<nth>]{<label>}[<[label opt]angle>]
-% defaults
- []{<m>}{<m>}(intersection)[1]{}[]
-\end{tzdef}
-
-For example, |\tzXpoint{path1}{path2}(A)| determines an intersection of the two paths and names the point |(A)| or |(A-1)|. (By default, the name is |(intersection)| as in \Tikz.)
-If there are two or more intersection points, they are named as follows: |(A)=(A-1)|, |(A-2)|, |(A-3)|, etc.
-
-You can determine which intersection point is named directly by specifying the option |[<nth>]|. If you select the second intersection point to be named |(A)| out of multiple intersections, they are named as follows: |(A-1)|, |(A)=(A-2)|, |(A-3)|, |(A-4)|, etc.
-You can label intersection points by specifying the option |{<label>}| and |[<angle>]|.
-
-
-\begin{tzcode}{.3}
-% \tzXpoint
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,8)
-\tzaxes(8,8)
-\tzto[red,bend right]"AA"(1,8)(8,1)
-\tzto[blue,bend right]"BB"(0,2)(8,6)
-\tzXpoint{AA}{BB}(A)
-\tzdot*(A)
-\end{tikzpicture}
-\end{tzcode}
-
-\warning When using |\tzXpoint|, the intersection of two paths must actually exist.Otherwise, an error will occur when using coordinates.
-
-\paragraph{\icmd{\tzXpoint*}} The starred version |\tzXpoint*| simply adds a node dot to |\tzXpoint|.
-The default dot size is |2.4pt| and it can be changed by the last option |(<dot size>)| or the \threeways\ (on page \pageref{ss:threeways}).
-
-\begin{tzdef}{}
-% syntax: minimum
-\tzXpoint*{<path>}{<path>}
-% syntax: medium
-\tzXpoint*{<path>}{<path>}(<coor name>){<label>}[<angle>]
-% syntax:
-\tzXpoint*[<opt>]{<path>}{<path>}
- (<coor name>)[<nth>]{<label>}[<[label opt]angle>](<dot size>)
-% defaults
- [tzdot=2.4pt]{<m>}{<m>}(intersection)[1]{}[](2.4pt)
-\end{tzdef}
-
-
-\begin{tzcode}{.3}
-% \tzXpoint*
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(10,10)
-\tzaxes(10,10)
-\def\bgt{8-\x}
-\def\Fx{\x}
-\def\IC{7/\x}
-\tzfn\bgt[0:8] % name path=bgt
-\tzfn\Fx[0:8]{ray}[red,ar] % name path=Fx
-\tzfn"IC"\IC[.8:8]{$u$}[r] % name path=IC
-\tzXpoint*{bgt}{Fx}(A){A}[[draw,red] r] % <angle> or % abb
-\tzproj[dashed](A){$x$}{$y$}
-\tzXpoint*[fill=none,red]{bgt}{IC}(B){B}[0](4pt)
-\tzdot*[blue](B-2){B2}[[red]45](4pt)
-\end{tikzpicture}
-\end{tzcode}
-
-
-%%------------------------------------------------------------
-\section{Vertical intersection points}
-\label{s:tzvXpointat}
-
-\subsection{\protect\cmd{\tzvXpointat(*)}}
-\label{ss:tzvXpointat}
-
-\icmd{\tzvXpointat} determines \iisw{vertical intersection} points of a path at a specified value of $x$.
-So it takes |{<path>}| and |{<x-val>}| as mandatory arguments.
-
-\remark
-Internally, |\tzvXpointat| depends on the |current bounding box|, which generally does not cause a problem because it is used after paths to be intersected are formed.
-In case of any problem of no intersection point, you may want to fix a bounding box using |\tzbbox| or |\tzaxes*| or \Tikz's |\useasboundingbox|.
-
-\begin{tzdef}{}
-% syntax: minimal
-\tzvXpointat{<path>}{<x-val>}(<coor name>)
-% syntax: medium
-\tzvXpointat{<path>}{<x-val>}(<coor name>){<label>}[<angle>]
-% syntax: full
-\tzvXpointat[<opt>]{<path>}{<x-val>}(<coor name>)[<n>]
- {<label>}[<[label opt]angle>]
-% defaults
- []{<m>}{<m>}(intersection)[1]{}[]
-\end{tzdef}
-
-The starred version \icmd{\tzvXpointat*} additionally prints a node dot of the size |2.4pt|, by default, at the (first) intersection point.
-
-\begin{tzdef}{}
-% syntax: minimum
-\tzvXpointat*{<path>}{<x-val>}
-% syntax: medium
-\tzvXpointat*{<path>}{<x-val>}(<coor name>){<label>}[<angle>]
-% syntax: full
-\tzvXpointat*[<opt>]{<path>}{<x-val>}(<coor name>)[<n>]
- {<label>}[<[label opt]angle>](<dot size>)
-% defaults
- [tzdot=2.4pt]{<m>}{<m>}(intersection)[1]{}[](2.4pt)
-\end{tzdef}
-
-\begin{tzcode}{.3}
-% \tzvXpointat(*)
-\begin{tikzpicture}
-\tzhelplines(4,3)
-\def\Fx{.5*(\x-2)^2}
-\tzfn\Fx[0:4] % name path=Fx
-\tzvfnat[dashed]{1}
-\tzvXpointat{Fx}{1}(A)
-\tzdot(A){A}[45]
-\tzvXpointat*{Fx}{3.2}{B}[[red,draw]br]
-\end{tikzpicture}
-\end{tzcode}
-
-
-%%------------------------------------------------------------
-\subsection{\protect\cmd{\tzvXpoint(*)}}
-\label{ss:tzvXpoint}
-
-\icmd{\tzvXpoint} accepts |{<path>}| and |(<coor>)| as mandatory arguments to find \iisw{vertical intersection} points of a path at the $x$ value of the coordinate, ignoring the $y$ value.
-For example, |\tzvXpoint{mypath}(3,<y>)|, ignoring |<y>|, is equivalent to |\tzvXpointat{mypath}{3}|.
-
-\xem{Everything else is the same as in} |\tzvXpointat|.
-
-\begin{tzdef}{}
-% syntax
-\tzvXpoint[<opt>]{<path>}(<coor>)(<coor name>)[<n>]
- {<label>}[<[label opt]angle>]
-% defaults
- []{<m>}(<m>)(intersection)[1]{}[]
-\end{tzdef}
-
-The starred version \icmd{\tzvXpoint*} just adds a node dot to |\tzvXpoint|.
-
-\begin{tzdef}{}
-% syntax
-\tzvXpoint*[<opt>]{<path>}(<coor>)(<coor name>)[<n>]
- {<label>}[<[<opt>]angle>](<dot size>)
-% defaults
- [tzdot=2.4pt]{<m>}(<m>)(intersection)[1]{}[](2.4pt)
-\end{tzdef}
-
-|\tzvXpoint*| prints, at the first intersection point, a node dot of the size 2.4pt by default.
-
-\begin{tzcode}{.3}
-% \tzvXpoint(*)
-\begin{tikzpicture}
-\tzhelplines(4,3)
-\def\Fx{.5*(\x-2)^2}
-\tzfn\Fx[0:4] % name path=Fx
-\tzcoors(1,0)(A)(3.2,0)(B);
-\tzvfn[dashed](1,0)
-\tzvXpoint{Fx}(A)(Ax)
-\tzdot(Ax){Ax}[45]
-\tzvXpoint*{Fx}(B){B}[[red,draw]br] % abb
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-%%------------------------------------------------------------
-\section{Horizontal intersection points}
-\label{s:tzhXpointat}
-
-\subsection{\protect\cmd{\tzhXpointat(*)}}
-\label{ss:tzhXpointat}
-
-\icmd{\tzhXpointat} determines \iisw{horizontal intersection} points of a path at a specified value of $y$.
-So it takes |{<path>}| and |{<y-val>}| as mandatory arguments.
-
-\begin{tzdef}{}
-% syntax: minimal
-\tzhXpointat{<path>}{<y-val>}(<coor name>)
-% syntax: medium
-\tzhXpointat{<path>}{<y-val>}(<coor name>){<label>}[<angle>]
-% syntax: full
-\tzhXpointat[<opt>]{<path>}{<y-val>}(<coor name>)[<n>]{<label>}[<[label opt]angle>]
-% defaults
- []{<m>}{<m>}(intersection)[1]{}[]
-\end{tzdef}
-
-
-The starred version \icmd{\tzhXpointat*} additionally prints a node dot of the size 2.4pt, by default, at the intersection point.
-
-\begin{tzdef}{}
-% syntax: minimum
-\tzhXpointat*{<path>}{<y-val>}
-% syntax: medium
-\tzhXpointat*{<path>}{<y-val>}(<coor name>){<label>}[<angle>]
-% syntax: full
-\tzhXpointat*[<opt>]{<path>}{<y-val>}(<coor name>)[<n>]
- {<label>}[<[label opt]angle>](<dot size>)
-% defaults
- [tzdot=2.4pt]{<m>}{<m>}(intersection)[1]{}[](2.4pt)
-\end{tzdef}
-
-\begin{tzcode}{.3}
-% \tzhXpointat(*)
-\begin{tikzpicture}
-\tzhelplines*(4,3)
-\def\Fx{.5*(\x-2)^2}
-\tzfn\Fx[0:4] % name path=Fx
-\tzhXpointat{Fx}{1}(A)
-\tzdot(A){A}[0]
-\tzhfnat[dashed]{1.5}
-\tzhXpointat*{Fx}{1.5}(X){X}[45]
-\tzdot*(X-2){Y}[[red,draw]al] % abb
-\end{tikzpicture}
-\end{tzcode}
-
-%%------------------------------------------------------------
-\subsection{\protect\cmd{\tzhXpoint(*)}}
-\label{ss:tzhXpoint}
-
-\icmd{\tzhXpoint} accepts |{<path>}| and |(<coor>)| as mandatory arguments to find \iisw{horizontal intersection} points of a path at the $y$ value of the coordinate, ignoring the $x$ value.
-For example, |\tzhXpoint{mypath}(<x>,3)|, ignoring |<x>|, is equivalent to |\tzhXpointat{mypath}{3}|.
-
-\xem{Everything else is the same as in} |\tzhXpointat|.
-
-\begin{tzdef}{}
-% syntax
-\tzhXpoint[<opt>]{<path>}(<coor>)(<coor name>)[<n>]{<label>}[<[label opt]angle>]
-% defaults
- []{<m>}(<m>)(intersection)[1]{}[]
-\end{tzdef}
-
-The starred version \icmd{\tzhXpoint*} just adds a node dot to |\tzhXpoint|.
-
-\begin{tzdef}{}
-% syntax
-\tzhXpoint*[<opt>]{<path>}(<coor>)(<coor name>)[<n>]
- {<label>}[<[label opt]angle>](<dot size>)
-% defaults
- [tzdot=2.4pt]{<m>}(<m>)(intersection)[1]{}[](2.4pt)
-\end{tzdef}
-
-|\tzhXpoint*| prints, at the (first) intersection point, a node dot of the size |2.4pt|, by default.
-
-\begin{tzcode}{.3}
-% \tzhXpoint(*)
-\begin{tikzpicture}
-\tzhelplines*(4,3)
-\def\Fx{.5*(\x-2)^2}
-\tzfn\Fx[0:4] % name path=Fx
-\tzcoors(0,1)(A)(0,1.5)(B);
-\tzhXpoint{Fx}(A)(A)
-\tzdot(A){A}[0]
-\tzhfn[dashed](0,1.5)
-\tzhXpoint*{Fx}(X){X}[45]
-\tzdot*(X-2){Y}[ [red,draw] al ] % abb
-\end{tikzpicture}
-\end{tzcode}
-
-
-%%------------------------------------------------------------
-\section{\protect\cmd{\tzLFnXpoint(*)}: Intersection point of linear functions}
-\label{s:tzLFnXpoint}
-
-Sometimes you may want to \xem{quickly} find intersection point of two linear functions \xem{without specifying path names}.
-
-\icmd{\tzLFnXpoint} finds the solution of two linear functions without printing anything by default.
-You can name it and use it.
-
-\begin{tzdef}{}
-% syntax: minimum
-\tzLFnXpoint{<\fn>}{<\fn>}
-% syntax: medium
-\tzLFnXpoint{<\fn>}{<\fn>}(<coor name>){<label>}[<[<label opt>]angle>]
-% syntax: full
-\tzLFnXpoint[<opt>]{<\fn>}{<\fn>}(<coor name>)
- {<label>}[<[<label opt>]angle>](<dot size>)
-% defaults
- [tzdot=2.4pt,draw=none,mimumum size=0pt]{<m>}{<m>}[0](){}[](2.4pt)
-\end{tzdef}
-
-
-\begin{tzcode}{.3}
-\begin{tikzpicture}[scale=.8]
-\tzhelplines(4,3)
-\tzfn[yellow]{\x}[0:3]
-\tzfn[yellow]{3-\x}[0:3]
-\tzLFnXpoint{\x}{3-\x}(A) % invisible
-\tzto+[->,bend left](A)(2,-1)
-\end{tikzpicture}
-\end{tzcode}
-
-\icmd{\tzLFnXpoint*} additionally print a filled node dot at the solution of two linear equations.
-
-\begin{tzdef}{}
-% syntax: minimum
-\tzLFnXpoint{<\fn>}{<\fn>}
-% syntax: medium
-\tzLFnXpoint{<\fn>}{<\fn>}(<coor name>){<label>}[<[<label opt>]angle>]
-% syntax: full
-\tzLFnXpoint[<opt>]{<\fn>}{<\fn>}(<coor name>)
- {<label>}[<[<label opt>]angle>](<dot size>)
-% defaults
- [tzdot=2.4pt,fill]{<m>}{<m>}[0](){}[](2.4pt)
-\end{tzdef}
-
-\begin{tzcode}{.3}
-\begin{tikzpicture}[scale=.8]
-\tzhelplines(4,3)
-\tzLFnXpoint*{\x}{3-\x}(A){A}[[red,draw]90] %%
-\tzline+[blue](A)(2,-1)
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-%%==================================
-\chapter{Secant and Tangent Lines}
-\label{c:tangent}
-
-%%------------------------------------------------------------
-\section{Secant lines}
-\label{s:secant}
-
-\subsection{\protect\cmd{\tzsecantat}}
-\label{ss:tzsecantat}
-
-\icmd{\tzsecantat} draws a line segment or a \iisw{secant} line of a curve, on the |behind| layer by default.
-|\tzsecantat| accepts three mandatory arguments:
-Three mandatory arguments are a path name and two values of $x$: |{<path>}|, |{<from->x}|, and |{<to-x>}|.
-With \icmd{\settzsecantlayer}, you can change the layer, like |\settzsecantlayer{main}|.
-
-\begin{tzdef}{}
-% syntax: minium
-\tzsecantat{<path>}{<from-x>}{<to-x>}
-% syntax: medium
-\tzsecantat{<path>}{<from-x>}{<to-x>}[<domain>]{<text>}[<node opt>]
-% syntax: full
-\tzsecantat[<opt>]<shift coor>"path name"
- {<path>}{<from-x>}{<to-x>}[<domain>]{<text>}[<node opt>]<code.append>
-% defaults
- []<>""{<m>}{<m>}{<m>}[]{}[]<>
-\end{tzdef}
-
-\paragraph{Domain}
-The domain of the form |[<from num:to num>]| is optional.
-Without specifying the optional domain, |\tzsecantat| draws a line segment connecting two points on the (curved) path.
-
-\begin{tzcode}{.3}
-% \tzsecantat
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,6)
-\tzaxes(-1,-1)(8,6){$x$}{$y$}
-\tzbezier+"curve"(.5,1)(1,6)(-1,-4)(7,5)
-\tzsecantat[thick,red]{curve}{1}{3}
-\tzsecantat[thick,red]{curve}{1}{4}
-\tzsecantat[thick,red]{curve}{1}{7}
-\end{tikzpicture}
-\end{tzcode}
-
-Specifying the option |[<domain>]|, you can extend (or shorten) the line of |\tzsecantat|.
-
-\begin{tzcode}{.3}
-% \tzsecantat: domain
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,6)
-\tzaxes(-1,-1)(8,6){$x$}{$y$}
-\tzbezier+"curve"(.5,1)(1,6)(-1,-4)(7,5){curve}[ar]
-\tzsecantat[blue]{curve}{1}{3}[0:5]{secant line}[a]
-\tzsecantat[thick,red]{curve}{1}{4}
-\tzsecantat[blue,dashed]{curve}{1}{7}[0:8]
-\end{tikzpicture}
-\end{tzcode}
-
-\paragraph{Shift}
-You can move |\tzsecantat| by specifying the option |<shift coor>|.
-
-\begin{tzcode}{.3}
-% \tzsecantat: domain
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,6)
-\tzaxes(-1,-1)(8,6){$x$}{$y$}
-\tzbezier+"curve"(.5,1)(1,6)(-1,-4)(7,5){curve}[ar]
-\tzsecantat[thick,red]{curve}{1}{4}
-\tzsecantat[blue]{curve}{1}{3}[0:5]{secant line}[a]
-\tzsecantat[blue]<1,-1>{curve}{1}{3}[0:5]{shifted}[r]
-\end{tikzpicture}
-\end{tzcode}
-
-\paragraph{Naming paths}
-By specifying the option |"<path name>"| you can name a path of |\tzsecantat|, and use it to find intersection points.
-
-\begin{tzcode}{.3}
-% \tzsecantat: shift, name path (intersection)
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,6)
-\tzaxes(-1,-1)(8,6){$x$}{$y$}
-\tzbezier+"curve"(.5,1)(1,6)(-1,-4)(7,5){curve}[ar]
-\tzsecantat[thick,red]{curve}{1}{4}
-\tzsecantat[blue]{curve}{1}{3}[0:5]{secant line}[a]
-\tzsecantat[blue]<1,-1>"shift"{curve}{1}{3}[0:5] %%
-\tzvXpointat*{shift}{3}{X}[-45]
-\tzticksx(-1mm:2mm){3/$x=3$}[scale=.7]
-\end{tikzpicture}
-\end{tzcode}
-
-\paragraph{\texttt{<code.append>}}
-You can extend the path of |\tzsecantat| by writing \Tikz\ code in the last optional argument |<code.append>|.
-
-\begin{tzcode}{.3}
-% \tzsecantat: <code.append>
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,6)
-\tzaxes(-1,-1)(8,6){$x$}{$y$}
-\tzbezier+"curve"(.5,1)(1,6)(-1,-4)(7,5){curve}[ar]
-\tzsecantat[blue]{curve}{1}{3}[0:5]{secant line}[a]
-\tzsecantat[blue,->]<1,-1>"shift"{curve}{1}{3}[0:5] %%
- < --++(1,-3) node [red,b] {Ends!} >
-\tzvXpointat*{shift}{3}{X}[-45]
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-\subsection{\protect\cmd{\tzsecant}}
-\label{ss:tzsecant}
-
-\icmd{\tzsecant} uses two \xem{coordinates instead of two values of $x$} to draw a line segment or a \iisw{secant} line of a curve, on the |behind| layer by default.
-You need to specify a path name and two coordinates, then |\tzsecant| uses the $x$ values of the two coordinates.
-
-\xem{Everything else is the same as in} |\tzsecantat|.
-
-You can change the layer with \icmd{\settzsecantlayer}.
-
-\begin{tzdef}{}
-% syntax: minium
-\tzsecant{<path>}(<coor>)(<coor>)
-% syntax: medium
-\tzsecant{<path>}(<coor>)(<coor>)[<domain>]{<text>}[<node opt>]
-% syntax: full
-\tzsecant[<opt>]<shift coor>"path name"
- {<path>}(<coor>)(<coor>)[<domain>]{<text>}[<node opt>]<code.append>
-% defaults
- []<>""{<m>}(<m>)(<m>)[]{}[]<>
-\end{tzdef}
-
-The domain should be of the form |[<from num:to num>]|.
-Without specifying the optional domain, |\tzsecant| draws a line segment connecting two points on the (curved) path.
-
-\begin{tzcode}{.3}
-% \tzsecant: domain
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,6)
-\tzaxes(-1,-1)(8,6){$x$}{$y$}
-\tzbezier+"curve"(.5,1)(1,6)(-1,-4)(7,5){curve}[ar]
-\tzcoor(1,0)(K)
-\tzsecant[blue]{curve}(K)(3,0)[0:5]{secant line}[a]
-\tzsecant[thick,red]{curve}(K)(4,0)
-\tzsecant[blue,dashed]{curve}(K)(7,0)[0:8]
-\end{tikzpicture}
-\end{tzcode}
-
-With the last option |<code.append>| you can extend the path of |\tzsecantat|.
-
-\begin{tzcode}{.3}
-% \tzsecant: shift, <code.append>
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(8,6)
-\tzaxes(-1,-1)(8,6){$x$}{$y$}
-\tzbezier+"curve"(.5,1)(1,6)(-1,-4)(7,5){curve}[ar]
-\tzcoor(1,0)(K)
-\tzsecant[blue]{curve}(K)(3,0)[0:5]{secant line}[a]
-\tzsecant[blue,->]<1,-1>"shift"{curve}(K)(3,0)[0:5] %%
- < --++(1,-3) node [red,b] {Ends!} >
-\tzvXpointat*{shift}{3}{X}[-45]
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-%%------------------------------------------------------------
-\section{Tangent lines}
-\label{s:tangent}
-
-\subsection{\protect\cmd{\tztangentat}}
-\label{ss:tztangentat}
-
-\icmd{\tztangentat} draws a \iisw{tangent} line to a curve at a specified value of $x$. Three mandatory arguments are a curve name, a value of $x$, and a domain: |{<path>}|, |{<x-val>}|, and |[domain]|.
-By default, the tangent line is drawn on the |behind| layer, which can be changed by \icmd{\settztangentlayer}, like |\settztangentlayer{main}|.
-
-\remark
-To calculate the slope at $x$, $x$ varies over the interval $(x-\varepsilon_1,x+\varepsilon_2)$ and $\varepsilon_1=\varepsilon_2=0.01$, by default.
-So the slope of tangent line is only \xem{approximate}.
-
-\begin{tzdef}{}
-% syntax: minimum
-\tztangentat{<path>}{<x-val>}[<domain>]
-% syntax: medium
-\tztangentat{<path>}{<x-val>}[<domain>]{<text>}[<node opt>]
-% syntax: full
-\tztangentat[<opt>]<shift coor>"<path name>"
- {<path>}{<x-val>}(<epsilon1>,<epsilon2>)[<domain>]
- {<text>}[<node opt>]<code.append>
-% The domain should be of the form [<from:to>]
-% defaults
- []<>""{<m>}{<m>}(.01,.01)[<m>]{}[]<>
-\end{tzdef}
-
-
-\paragraph{Domain}
-The mandatory argument |[<domain>]| should be of the form |[<from num:to num>]|.
-
-\begin{tzcode}{.3}
-% \tztangentat
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(9,8)
-\tzaxes(9,8)
-\tzplotcurve"AA"(1,7)(3,3)(8,1);
-\tztangentat{AA}{2}[.5:4]
-\tztangentat[blue]{AA}{4}[1:7]{tangent}[red,b]
-\tzvXpointat*{AA}{4}
-\tzticksx{1,7}
-\end{tikzpicture}
-\end{tzcode}
-
-
-\paragraph{Shift}
-You can move the tangent line by specifying the option |<shift coor>|.
-
-\begin{tzcode}{.3}
-% \tztangentat: shift
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(9,8)
-\tzaxes(9,8)
-\tzplotcurve"AA"(1,7)(3,3)(8,1);
-\tztangentat{AA}{2}[.5:4]
-\tztangentat[blue]{AA}{4}[1:7]{tangent}[red,b]
-\tztangentat[blue]<2,1>{AA}{4}[1:7]{tangent'}[r]
-\tzvXpointat*{AA}{4}
-\tzticksx{1,7}
-\end{tikzpicture}
-\end{tzcode}
-
-\paragraph{Naming paths}
-By specifying the option |"<path name>"|, you can name the path of |\tztangentat| and use it to find intersection points.
-
-\begin{tzcode}{.3}
-% \tztangentat: name path (intersection)
-\begin{tikzpicture}[scale=.5,font=\footnotesize]
-\tzhelplines(9,8)
-\tzaxes(9,8)
-\tzplotcurve"AA"(1,7)(3,3)(8,1);
-\tztangentat{AA}{2}[.5:4]
-\tztangentat[blue]{AA}{4}[1:7]{tangent}[red,b]
-\tzvXpointat*{AA}{4}
-\tztangentat[blue]
- <2,1>"BB"{AA}{4}[1:7]{tangent'}[r]
-\tzvXpointat*[blue]{BB}{5}(X){$x$}[45](3pt)
-\tzproj(X){$x_1$}{$x_2$}
-\end{tikzpicture}
-\end{tzcode}
-
-\paragraph{\texttt{<code.append>}}
-You can extend the path of |\tztangentat| by writing \Tikz\ code in the last optional argument |<code.append>|.
-
-\begin{tzcode}{.3}
-% \tztangentat: <code.append>
-\begin{tikzpicture}[scale=.5,font=\footnotesize]
-\tzhelplines(9,8)
-\tzaxes(9,8)
-\tzplotcurve"AA"(1,7)(3,3)(8,1);
-\tztangentat{AA}{2}[.5:4]
-\tztangentat[blue]{AA}{4}[1:7]{tangent}[red,b]
-\tzvXpointat*{AA}{4}
-\tztangentat[blue,->]<2,1>"BB"{AA}{4}[1:7]
- < to [bend right] ++(-1,3) node [a] {Ends!} >
-\tzvXpointat*[blue]{BB}{5}(X){$x$}[45](3pt)
-\tzproj(X){$x_1$}{$x_2$}
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-\paragraph{Variations}
-Since the slope of the tangent line is \xem{approximate}, sometimes you may want to change the variation interval to get better results.
-You can change $\varepsilon_1$ and $\varepsilon_2$ by specifying the option |(<epsilon1,epsilon2>)| \xem{immediately after} the mandatory argument |{<x-val>}|.
-Or you can change the variations by the macro \icmd{\settztangentepsilon},
-like |\settztangentepsilon{|$\varepsilon_1$|}{|$\varepsilon_2$|}|. The effect remains until the end of |tikzpicture| environment unless changed again.
-
-\begin{tzcode}{.3}
-% \tztangentat: variations, shift
-\begin{tikzpicture}[scale=.5,font=\footnotesize]
-\tzhelplines(9,8)
-\tzaxes(9,8)
-\tzplotcurve"AA"(1,7)(3,3)(8,1);
-\tztangentat[blue!50,dashed]{AA}{4}[1:7]
-\settztangentepsilon{0.005}{0.01} %%
-\tztangentat[blue]{AA}{4}[1:7]{corrected}[r]
-\tztangentat[red]<0,1>{AA}{4}[1:7]{shifted}[r] %%
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-
-\subsection{\protect\cmd{\tztangent}}
-\label{ss:tztangent}
-
-
-\icmd{\tztangent} uses a \xem{coordinate instead of a value} of $x$ to draw a \iisw{tangent} line to a curve.
-|\tztangent| accepts three mandatory arguments: |{<path>}|, |(<coor>)|, and |[<domain>]|.
-
-The value of $y$ in |(<coor>)| is ignored.
-For example, |\tztangent{curve}(4,|$y$|)| is equivalent to |\tztangentat{curve}{4}| for any $y$.
-
-\xem{Everything else is the same as in} |\tztangentat|.
-
-\begin{tzdef}{}
-% syntax: minimum
-\tztangent{<path>}(<coor>)[<domain>]
-% syntax: medium
-\tztangent{<path>}(<coor>)[<domain>]{<text>}[<node opt>]
-% syntax: full
-\tztangent[<opt>]<shift coor>"<path name>"
- {<path>}(<coor>)(<epsilon1>,<epsilon2>)[<domain>]
- {<text>}[<node opt>]<code.append>
-% The domain should be of the form [<from:to>]
-% defaults
- []<>""{<m>}(<m>)(.01,.01)[<m>]{}[]<>
-\end{tzdef}
-
-
-
-\begin{tzcode}{.3}
-% \tztangent
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(9,8)
-\tzaxes(9,8)
-\tzplotcurve"AA"(1,7)(3,3)(8,1);
-\tzcoors(2,0)(K2)(4,0)(K4);
-\tztangent{AA}(K2)[.5:4]
-\tztangent[blue]{AA}(K4)[1:7]{tangent}[red,b]
-\tzvXpoint*{AA}(K4)
-\tzticksx{1,7}
-\end{tikzpicture}
-\end{tzcode}
-
-You can shift the tangent line and extend its path.
-
-\begin{tzcode}{.3}
-% \tztangent: shift, <code.append>
-\begin{tikzpicture}[scale=.5,font=\footnotesize]
-\tzhelplines(9,8)
-\tzaxes(9,8)
-\tzplotcurve"AA"(1,7)(3,3)(8,1);
-\tzcoors(2,0)(K2)(4,0)(K4);
-\tztangent{AA}(K2)[.5:4]
-\tztangent[blue]{AA}(K4)[1:7]{tangent}[red,b]
-\tzvXpoint*{AA}(K4)
-\tztangent[blue,->]<2,1>"BB"{AA}(K4)[1:7]
- < to [bend right] ++(-1,3) node [a] {Ends!} >
-\tzvXpoint*[blue]{BB}(5,0)(X){$x$}[45](3pt)
-\tzproj(X){$x_1$}{$x_2$}
-\end{tikzpicture}
-\end{tzcode}
-
-You can control the interval of variations of $x$ by the option |(<epsilon1,epsion2>)| or the marcro |\settztangentepsilon|.
-
-\begin{tzcode}{.3}
-% \tztangent: variations, shift
-\begin{tikzpicture}[scale=.5,font=\footnotesize]
-\tzhelplines(9,8)
-\tzaxes(9,8)
-\tzplotcurve"AA"(1,7)(3,3)(8,1);
-\tztangent[blue!50,dashed]{AA}(4,0)[1:7]
-\settztangentepsilon{0.005}{0.01} %%
-\tztangent[blue]{AA}(4,0)[1:7]{corrected}[r]
-\tztangent[red]<0,1>{AA}(4,0)[1:7]{shifted}[r] %%
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-%%------------------------------------------------------------
-\section{Slope lines and normal lines}
-\label{s:slopes}
-
-\subsection{\protect\cmd{\tzslopeat}}
-\label{ss:tzslopeat}
-
-\icmd{\tzslopeat} draws a slope line to a path with a specified length.
-The mandatory arguments are |{<path>}|, |{<x>}|, and |{<length>}|.
-The tangent point is the middle point of the slope line.
-
-By default, the slope lines are drawn on the |behind| layer.
-You can change the layer, like \icmd{\settztslopelayer}|{main}|.
-
-\remark The slope is \xem{approximate} and you can manipulate the slope by changing the variation interval with the option |(<epsilon1,epsilon2>)|. The default is $(\varepsilon_1,\varepsilon_2)=(0.01,0.01)$.
-You can also use
-\icmd{\tzslopeepsilon}|{<epsilon1>}{<epsilon2>}| \xem{immediately after} the option |{<x-val>}|, which is valid until the end of the |tikzpicture| environment, unless changed again.
-
-
-\begin{tzdef}{}
-% syntax: minimum
-\tzslopeat{<path>}{<x-val>}{<length>}
-% syntax: minimum (for normal line)
-\tzslopeat{<path>}{<x-val>}{<length>}[90] % or [-90]
-% syntax: full
-\tzslopeat[<opt>]{<path>}{<x-val>}(<epsilon1>,<epsilon2>){<length>}[<rotate>]
- {<text>}[<node opt>]<code.append>
-% defaults
- []{<m>}{<m>}(.01,.01){<m>}[0]{}[]<>
-\end{tzdef}
-
-\begin{tzcode}{.3}
-% \tzslopeat
-\begin{tikzpicture}
-\tzhelplines*(5,4)
-\tzcoors*(0,0)(A)(2,2)(B){B}[b](3.5,1)(C)(5,3)(D);
-\tztos[thick]"AA"(A)[out=0,in=180]
- (B)[out=0,in=180]
- (C)[out=0,in=255]
- (D);
-\tzslopeat[red,thick]{AA}{1}{3cm}
-\tzslopeat[blue]{AA}{2}{2cm}
-\tzslopeat{AA}{4}{2cm}{$f'(4)$}[r]
-\end{tikzpicture}
-\end{tzcode}
-
-
-\begin{tzcode}{.3}
-% \tzslopeat: repeated
-\begin{tikzpicture}
-\tzhelplines(1,1)(5,4)
-\tzcoors*(0,1)(A)(2,3)(B)(4,2)(C);
-\tzparabola"AA"(A)(B)(C)
-\foreach \x in {1,1.5,...,3.5}
- { \tzslopeat[thick,red]{AA}{\x}{2cm} }
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-
-\paragraph{Normal lines}
-With the option |[<rotate>]| \xem{immediately after} the last mandatory option |{<length>}|, you can draw a \iisw{normal line} to a curve by rotating the slope lines |90|\textdegree or |-90|\textdegree.
-
-\begin{tzcode}{.3}
-% \tzslopeat: slopes and normals
-\begin{tikzpicture}
-\tzhelplines*(5,4)
-\tzcoors*(0,0)(A)(2,2)(B){B}[b](3.5,1)(C)(5,3)(D);
-\tztos[thick]"AA"(A)[out=0,in=180]
- (B)[out=0,in=180]
- (C)[out=0,in=255]
- (D);
-\tzslopeat[red]{AA}{1}{3cm}
-\tzslopeat[blue]{AA}{2}{2cm}
-\tzslopeat{AA}{4}{2cm}{$f'(4)$}[r]
-% normal lines: roatate: [90] or [-90]
-\tzslopeat[blue]{AA}{1}{1cm}[90]
-\tzslopeat[red]{AA}{2}{2cm}[90]
-\tzslopeat[red]{AA}{4}{1cm}[-90]{normal}[b]
-\end{tikzpicture}
-\end{tzcode}
-
-
-
-
-
-\subsection{\protect\cmd{\tzslope}}
-\label{ss:tzslope}
-
-\icmd{\tzslope} is the same as |\tzslopeat|, except for one thing.
-|\tzslope| uses a coordinate instead of a value of $x$ to draw slope lines.
-So the mandatory arguments of |\tzslope| is |{<path>}|, |(<coor>)|, and |{<length>}|. The $y$ value of |(<x,y>)| is ignored.
-
-\begin{tzdef}{}
-% syntax
-\tzslope[<opt>]{<path>}(<x,y>)(<epsilon1>,<epsilon2>){<length>}[<angle>]
- {<text>}[<pos,opt>]<code.append>
-% defaults
- []{<m>}(<m>)(.01,.01){<m>}[0]{}[]<>
-\end{tzdef}
-
-
-
-\begin{tzcode}{.3}
-% \tzslope: slopes and normals
-\begin{tikzpicture}
-\tzhelplines*(5,4)
-\tzcoors*(0,0)(A)(2,2)(B){B}[b](3.5,1)(C)(5,3)(D);
-\tztos[thick]"AA"(A)[out=0,in=180]
- (B)[out=0,in=180]
- (C)[out=0,in=255]
- (D);
-\coordinate (B) at (2,2);
-\tzslope[red]{AA}(1,0){3cm}
-\tzgetxyval(B){\Bx}{\By}
-\tzslopeat[blue]{AA}{\Bx}{2cm}
-\tzslope{AA}(4,0){2cm}{$f'(4)$}[r]
-% normal lines: roatate: [90] or [-90]
-\tzslope[blue]{AA}(1,0){1cm}[90]
-\tzslope[red]{AA}(\Bx,0){2cm}[90]
-\tzslope[red]{AA}(4,0){1cm}[-90]{normal}[b]
-\end{tikzpicture}
-\end{tzcode}
-
-
-\begin{tzcode}{.3}
-% \tzslope: repeated
-\begin{tikzpicture}
-\tzhelplines(1,1)(5,4)
-\tzcoors*(0,1)(A)(2,3)(B)(4,2)(C);
-\tzparabola"AA"(A)(B)(C)
-\foreach \x in {1,1.5,...,3.5}
- { \tzslope[thick,red]{AA}(\x,0){2cm} }
-\end{tikzpicture}
-\end{tzcode}
-
-You can add \Tikz\ code with the last option |<code.append>|.
-
-\begin{tzcode}{.3}
-% \tztangent, \tzslope: extend/shorten lines
-\begin{tikzpicture}[scale=.5]
-\tzhelplines(9,8)
-\tzaxes(9,8)
-\tzplotcurve"AA"(1,7)(3,3)(8,1);
-\tzcoors(2,0)(K2)(4,0)(K4);
-\tzslope[red,tzextend={1cm}{2cm}]{AA}(2,0){1cm} %%
-\tztangent[blue]{AA}(K4)[1:7]{tangent}[red,b]
-\tzslope[->]{AA}(4,0){0.1pt}[90]
- <--([turn]0:2cm) node [r] {gradient} > %%
-\tzvXpoint*{AA}(K4)
-\tzticksx{1,7}
-\end{tikzpicture}
-\end{tzcode}
-
-