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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, \cmd{\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/$x_3$}[blue]
+ {2/$\sqrt{x}$,3/y,4/m,5,7/$k$}[red]
+\end{tikzpicture}
+\end{tzcode}
+
+\paragraph{Tick marks}
+To prints tick marks you need to specify |(<x-from:x-to>)| for $x$ ticks and/or |(<y-from:y-to>)| for $y$ ticks. (The default is |(0pt:0pt)|.)
+
+\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]
+ (-5pt:10pt){1,...,7,8/$\alpha$}
+ (0pt:3cm) {2,...,6,7/$\beta$}
+\end{tikzpicture}
+\end{tzcode}
+
+
+\paragraph{Shift}
+The position of tick labels does not depend on the length of the tick marks.
+To change the position, use 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(*)}}
+\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}
+
+
+
+
+%%==================================
+\chapter{Plot Functions}
+\label{c:functions}
+
+
+%%------------------------------------------------------------
+\section{\protect\cmd{\tzfn}: Plot 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=200]<>""{<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{2*\x+1}[1:5] % works like:
+ \draw [samples=200,domain=1:5] plot (\x,{2*\x+1});
+\end{tztikz}
+
+
+\subsection{Define and name functions}
+
+To use |\tzfn| you need to express a function as a function of |\x|.
+You can add text (as a function name) 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)
+\def\Dx{7-2/3*\x}
+\tzfn{\Dx}[0:7]{$D$}[r]
+\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>"| 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=200,,domain=1:5,name path=mypath] plot (\x,{\Fx});
+\end{tztikz}
+
+\remark
+Suppose that the function's expression |<fn of \x>| consists \xem{only of a macro name}, say |\Fx|. Then
+\begin{itemize}
+\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}.)
+\begin{tztikz}{}
+\tzfn\Fx[1:5] % works like:
+ \draw [samples=200,domain=1:5,name path=Fx] plot (\x,{\Fx});
+\end{tztikz}
+\end{itemize}
+
+\begin{tzcode}{.3}
+% \tzfn: name path: intersection point
+\begin{tikzpicture}[scale=.5]
+\tzhelplines(8,8)
+\tzaxes(8,8)
+\def\Dx{7-2/3*\x}
+\def\Sx{1+\x}
+\tzfn"Dx"\Dx[0:7]{$D$}[r] % name path = Dx
+\tzfn \Sx[0:7]{$S$}[blue,r] % name path = Sx
+\tzXpoint*{Dx}{Sx}(E){E}
+\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 empty shift option |<>| is not allowed.
+
+\begin{tzcode}{.3}
+% \tzfn: shift
+\begin{tikzpicture}[scale=.6]
+\tzhelplines(8,8)
+\def\Dx{7-2/3*\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]{$D'$}[right]
+\tzfn[dashed]<1,-1>"supplyA"\Sx[0:7]{$S'$}[right]
+\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 created by |\tzfn| from the end of the graph, 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 with \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]
+% < to [bend right] ++(2,-2) node [b] {End!} >
+\end{tikzpicture}
+\end{tzcode}
+
+
+\paragraph{\texttt{\bs tzfnAtBegin}}
+You can use \icmd{\tzfnAtBegin} (immediately) \xem{before each} |\tzfn| to insert \Tikz\ code before 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{ (0,0) -- }
+\tzfn[->]\Fx[-2:2*pi]{$\sin\,x+3$}[blue,r]
+\end{tikzpicture}
+\end{tzcode}
+
+
+%%------------------------------------------------------------
+\section{\protect\cmd{\tzLFn}: Plot linear functions}
+\label{s:tzLFn}
+
+
+\icmd{\tzLFn} draws a linear function: $y=ax+b$.
+
+\begin{itemize}
+\item
+When you know two coordinates on a line,
+|\tzLFn(<coor1>)(<coor2>)| draws the line.
+\item
+When you know one coordinate and the slope of a line,
+|\tzLFn(<coor1>){<slope>}| draws the line.
+\item
+If you specify all the three arguments |(<coor1>)(<coor2>){<slope>}|, then the slope is ignored.
+\end{itemize}
+
+\begin{tzdef}{}
+% syntax
+\tzLFn[<opt>]<shift coor>"<path name>"
+ (<coor1>)(<coor2>){<slope>}[<domain>]{<text>}[<node opt>]<code.append>
+% defaults
+[]<>""(<m>)(){1}[<m>]{}[]<>
+\end{tzdef}
+
+|\tzLFn| accepts two mandatory arguments: |(<coor1>)| and |[<domain>]|.
+\begin{itemize}
+\item The 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}
+
+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 the slope |.5|, over $0\leq x\leq 4$.
+
+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>"| immediately before the mandatory coordinate.
+
+\begin{tzcode}{.3}
+% \tzLFn
+\begin{tikzpicture}
+\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 empty shift option |<>| is not allowed.)
+You can also expand the path of |\tzLFn| by writing \Tikz\ code in the last optional argument |<code.append>|.
+
+\begin{tzcode}{.3}
+% \tzLFn: shift, expanding path
+\begin{tikzpicture}
+\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}
+
+
+%%------------------------------------------------------------
+\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 expand 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}
+
+
+
+\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}
+
+
+
+%%------------------------------------------------------------
+\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 expand 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.)
+
+
+\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}
+
+
+
+
+
+%%==================================
+\chapter{Intersections}
+\label{c:intersections}
+
+%%------------------------------------------------------------
+\section{\protect\cmd{\tzXpoint(*)}: Intersection points}
+\label{s:intersections}
+
+\icmd{\tzXpoint} finds 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}
+
+
+\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}[0] % <angle>
+\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 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}[-45]
+\end{tikzpicture}
+\end{tzcode}
+
+
+%%------------------------------------------------------------
+\subsection{\protect\cmd{\tzvXpoint(*)}}
+\label{ss:tzvXpoint}
+
+\icmd{\tzvXpoint} accepts |{<path>}| and |(<coor>)| as mandatory arguments to find 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}[-45]
+\end{tikzpicture}
+\end{tzcode}
+
+
+
+%%------------------------------------------------------------
+\section{Horizontal intersection points}
+\label{s:tzhXpointat}
+
+\subsection{\protect\cmd{\tzhXpointat(*)}}
+\label{ss:tzhXpointat}
+
+\icmd{\tzhXpointat} determines 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}[135]
+\end{tikzpicture}
+\end{tzcode}
+
+%%------------------------------------------------------------
+\subsection{\protect\cmd{\tzhXpoint(*)}}
+\label{ss:tzhXpoint}
+
+\icmd{\tzhXpoint} accepts |{<path>}| and |(<coor>)| as mandatory arguments to find 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}[135]
+\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 secant line of a curve, on the |behind| layer by default.
+You need to specify a path name and two values of $x$.
+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 should be of the form |[<from num:to num>]|.
+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}
+
+You can optionally specify the |[<domain>]|.
+
+\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|.
+
+\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 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}
+
+You can move the secant line and extend the path.
+
+\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 tangent line to a curve at a specified value of $x$.
+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}
+|\tztangentat| takes three mandatory arguments: |{<path>}|, |{<x-val>}|, and |[<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|.
+
+\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}
+The slope of the tangent line is approximate, so 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>)| 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 tangent line.
+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}
+
+
+|\tztangent| accepts three mandatory arguments: |{<path>}|, |(<coor>)|, and |<domain>|.
+
+\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}
+
+
+%%==================================
+\chapter{Miscellany}
+\label{c:misc}
+
+%%------------------------------------------------------------
+\section{\protect\cmd{\tzbrace(')}}
+\label{s:tzbrace}
+
+\icmd{\tzbrace} takes two coordinates as mandatory arguments to draw a \xem{calligraphic brace} connecting them.
+
+\begin{tzdef}{}
+% syntax: minimum
+\tzbrace(<coor>)(<coor>)
+% syntax: medium
+\tzbrace(<coor>)(<coor>){<text>}[<node opt>]
+% syntax: full
+\tzbrace[<draw opt>]{<raise>}[<decoration opt>]<shift coor>
+ (<coor>)(<coor>){<text>}[<node opt>]
+% defaults
+ []{5pt}[amplitude=5pt]<>(<m>)(<m>){}[]
+\end{tzdef}
+
+The |raise| value of a brace is |5pt| by default and the value can be changed by the first curly brace optional argument |{<raise>}|.
+
+The |amplitude| of a brace is |5pt| by default. You can control the amplitude by writing the option |amplitude=<dim>| in the second bracket option |[<decoration opt>]|.
+
+\begin{tztikz}{}
+\tzbrace[thick](0,0)(3,1) % works like:
+ \draw [thick,decorate,decoration={calligraphic brace, amplitude=5pt, raise=5pt}]
+ (0,0) to (3,1);
+\end{tztikz}
+
+The \iisw{wap version} \icmd{\tzbrace'} swaps the coordinates.
+So it prints a mirror image of |\tzbrace|.
+For example, |\tzbrace'(0,0)(3,1)| is equivalent to |\tzbrace(3,1)(0,0)|.
+
+\begin{tzcode}{.3}
+% \tzbrace(')
+\begin{tikzpicture}[sloped]
+\tzhelplines(4,2)
+\tzline(0,0)(3,1)
+\tzbrace(0,0)(3,1){AAA}[above=10pt,blue]
+\tzbrace'[red](0,0)(3,1){BBB}[below=10pt]
+\end{tikzpicture}
+\end{tzcode}
+
+You can change the style of the decorating brace by the second bracket optional argument |[<decoration opt>]|.
+
+The color of the calligraphic brace can be changed by the option |pen colour| in the list of |[<draw option>]|.
+
+\begin{tzcode}{.3}
+% \tzbrace('): decoration options
+\begin{tikzpicture}[sloped]
+\tzhelplines(4,3)
+\tzline(0,0)(3,1)
+\tzbrace [very thick,pen colour=blue]
+ [amplitude=10pt]
+ (0,0)(3,1){AAA}[a=15pt]
+\tzbrace'[red,very thick]{10pt}
+ [brace,amplitude=10pt]
+ (0,0)(3,1){BBB}[b=20pt]
+\end{tikzpicture}
+\end{tzcode}
+
+You can also move a brace by specifying the option |<shift coor>| immediately before the the first mandatory coordinate.
+The empty shift option |<>| is not allowed.
+
+\begin{tzcode}{.3}
+% \tzbrace('): shift
+\begin{tikzpicture}[sloped]
+\tzhelplines(4,3)
+\tzline(0,0)(3,1)
+\tzbrace [very thick,pen colour=blue][amplitude=10pt]
+ <.5,.5>(0,0)(3,1){AAA}[a=15pt]
+\tzbrace'[red,very thick]{10pt}[brace,amplitude=10pt]
+ (0,0)(3,1){BBB}[b=20pt]
+\end{tikzpicture}
+\end{tzcode}
+
+
+
+