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author | Norbert Preining <norbert@preining.info> | 2022-03-01 03:01:11 +0000 |
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committer | Norbert Preining <norbert@preining.info> | 2022-03-01 03:01:11 +0000 |
commit | 3e9a44ea155766214d8657ed79db2206592b6e8e (patch) | |
tree | 8bac160137045a4244d7a7462bb2ee51adc70e9c /graphics/pgf/contrib/tzplot/tzplot-doc-D-v2.0.tex | |
parent | 7a8d1925127ad121a8f591db931524f9b8eacb93 (diff) |
CTAN sync 202203010301
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diff --git a/graphics/pgf/contrib/tzplot/tzplot-doc-D-v2.0.tex b/graphics/pgf/contrib/tzplot/tzplot-doc-D-v2.0.tex new file mode 100644 index 0000000000..70148a3c9e --- /dev/null +++ b/graphics/pgf/contrib/tzplot/tzplot-doc-D-v2.0.tex @@ -0,0 +1,2850 @@ +%!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} + + |