使用 Latex 绘制参数图

Question 1

为了好玩，pstricks方式如下：

\documentclass[svgnames, x11names, border=3pt]{standalone}%
\usepackage[utf8]{inputenc}
\usepackage{pst-plot, pst-math, pst-text, auto-pst-pdf}%

\begin{document}

 \psset{arrowinset=0.12, algebraic, plotstyle=curve, plotpoints=1000}%
    \def\paracurve{t*sin(t)|t*cos(t)}
    \def\tsup{\pscalculate{3*\psPiH}}
\begin{pspicture*}(-5.5,-5)(4,5)%
    \psset{labels=none}
    \psaxes[linecolor=LightSteelBlue3, ticksize =2pt -2pt, ]{->}(0,0)(-5.5,-5)(4,5)[$x$, -120][$y$, -135]
    \uput[dl](0,0){$O $}
     \uput[d](\psPiH, 0){$ \frac{\pi}{2}$} \uput[-120](-\tsup, 0){$ -\frac{3\pi}{2}$}
     \uput[120](0,\psPi){$\pi$} \uput[-120](0,-\psPi){$-\pi$} 
    \parametricplot[linewidth=1.2pt, linecolor=IndianRed]{-\tsup}{\tsup}{\paracurve}
    \psplotTangent[linecolor=Goldenrod1, linewidth=0.4pt]{\psPiH}{6}{\paracurve}
    \uput[l](0,2.467){$\bigl(\frac{\pi}{2}\bigr)^{\!\scriptscriptstyle2}$}
\end{pspicture*}

\end{document}

Answer

为了好玩，pstricks方式如下：

\documentclass[svgnames, x11names, border=3pt]{standalone}%
\usepackage[utf8]{inputenc}
\usepackage{pst-plot, pst-math, pst-text, auto-pst-pdf}%

\begin{document}

 \psset{arrowinset=0.12, algebraic, plotstyle=curve, plotpoints=1000}%
    \def\paracurve{t*sin(t)|t*cos(t)}
    \def\tsup{\pscalculate{3*\psPiH}}
\begin{pspicture*}(-5.5,-5)(4,5)%
    \psset{labels=none}
    \psaxes[linecolor=LightSteelBlue3, ticksize =2pt -2pt, ]{->}(0,0)(-5.5,-5)(4,5)[$x$, -120][$y$, -135]
    \uput[dl](0,0){$O $}
     \uput[d](\psPiH, 0){$ \frac{\pi}{2}$} \uput[-120](-\tsup, 0){$ -\frac{3\pi}{2}$}
     \uput[120](0,\psPi){$\pi$} \uput[-120](0,-\psPi){$-\pi$} 
    \parametricplot[linewidth=1.2pt, linecolor=IndianRed]{-\tsup}{\tsup}{\paracurve}
    \psplotTangent[linecolor=Goldenrod1, linewidth=0.4pt]{\psPiH}{6}{\paracurve}
    \uput[l](0,2.467){$\bigl(\frac{\pi}{2}\bigr)^{\!\scriptscriptstyle2}$}
\end{pspicture*}

\end{document}

Question 2

绘制情节非常简单：

\draw plot[variable=\t,domain=-3*pi/2:3*pi/2,smooth,samples=51] 
  ({\t*sin(\t)},{\t*cos(\t)});

其中\t是参数。这里我们使用trig format=rad切换到弧度。为了方便起见，我添加了一种样式tangent at，将切线附加到给定的 t 值处。分量仅由和的 t 导数给出x(t)。y(t)长度存储在中tangent length。要获得处长度为 2 的切线t=pi，例如，您可以编写

\draw[blue,tangent length=2,tangent at=pi];

完整代码：

\documentclass[tikz,border=3mm]{standalone}
\begin{document}
\begin{tikzpicture}[trig format=rad,tangent at/.style={insert path={
 [/utils/exec=\pgfmathsetmacro{\mylength}{%
 veclen(#1*cos(#1)+sin(#1),-1*#1*sin(#1)+cos(#1))/\pgfkeysvalueof{/tikz/tangent
 length}}]
 ({#1*sin(#1)},{#1*cos(#1)}) ++ 
 ({(#1*cos(#1)+sin(#1))/(2*\mylength)},{(-1*#1*sin(#1)+cos(#1))/(2*\mylength)})
 -- ++ ({(-2*#1*cos(#1)-2*sin(#1))/(2*\mylength)},
 {(2*#1*sin(#1)-2*cos(#1))/(2*\mylength)})}},
 tangent length/.initial=1]
 \draw plot[variable=\t,domain=-3*pi/2:3*pi/2,smooth,samples=51] 
  ({\t*sin(\t)},{\t*cos(\t)});
 \draw[tangent at=2];
 \draw[blue,tangent length=2,tangent at=pi];
 \draw[red,tangent length=3,tangent at=-pi/2];
\end{tikzpicture}
\end{document}

也可以使用来附加切线decorations.markings，但在您的情况下，曲线的参数化是已知的，因此最好通过分析来进行。

Answer