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如何制作椭圆的切线以及与其平行的线？

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如何制作椭圆的切线以及与其平行的线？

我已经按以下方式指定了一个椭圆和三个点：

\def\aa{2.5}
\def\bb{2}
\draw[thick] (0,0) ellipse [x radius=\aa,y radius=\bb];
\pgfmathsetmacro{\focus}{sqrt(\aa*\aa-\bb*\bb)}
\path coordinate (c) at (0,0)
coordinate (d) at (-\focus,0)
coordinate (r) at ($(0,0)+(36:{\aa} and {\bb})$);
\fill (c) circle (2pt)
(d) circle (2pt)
(r) circle (2pt);

现在我需要

  1. 过 (r) 的椭圆的切向量；
  2. 与该切线平行的通过 (c) 的线；
  3. 这最后一条线与连接 (r) 和 (d) 的线相交处的坐标。

我似乎找不到可以解决问题 1 的解决方案，从而无法解决问题 2 和 3。请问有什么建议吗？

答案1

以下是使用以下方法实现此目的的方法如何在 TikZ 中绘制路径上任意点的切线

\documentclass[border=5mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{calc, decorations.markings, intersections}
\begin{document}
\begin{tikzpicture}[
    tangent/.style={
        decoration={
            markings,% switch on markings
            mark=
                at position #1
                with
                {
                    \coordinate (tangent point-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (0pt,0pt);
                    \coordinate (tangent unit vector-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (1,0pt);
                    \coordinate (tangent orthogonal unit vector-\pgfkeysvalueof{/pgf/decoration/mark info/sequence number}) at (0pt,1);
                }
        },
        postaction=decorate
    },
    use tangent/.style={
        shift=(tangent point-#1),
        x=(tangent unit vector-#1),
        y=(tangent orthogonal unit vector-#1)
    },
    use tangent/.default=1
]

\def\aa{3.5}
\def\bb{2}
\pgfmathsetmacro{\focus}{sqrt(\aa*\aa-\bb*\bb)}

\draw[thick, tangent=0.07] (0,0) ellipse [x radius=\aa,y radius=\bb];

\path coordinate (c) at (0,0)
    coordinate (d) at (-\focus,0);
\fill (c) circle (2pt)
    (d) circle (2pt);

\fill (tangent point-1) circle [radius=2pt];
\draw [red, name path=rd] (tangent point-1) -- (d);
\draw [use tangent] (2,0) -- (-2,0);
\draw [use tangent, red, name path=parallel] (c) ++(2,0) -- +(-4,0);

\fill [red, name intersections={of={rd and parallel}}] (intersection-1) circle [radius=2pt];
\end{tikzpicture}

\end{document}

答案2

在此处输入图片描述

和Asymptote版本，ellipse.asy 以及通过 svg 翻译成 tikz

size(300);
void Dot(... pair[] p){ //  function takes a variable number of arguments
  for(int i=0;i<p.length;++i){
    fill(shift(p[i])*scale(0.06)*unitcircle,black);
    fill(shift(p[i])*scale(0.04)*unitcircle,white);
  }
}
real a=2.5, b=2, focus=sqrt(a*a-b*b);
pair c=(0,0), d=(-focus,0);
path el=ellipse(c,a,b);
path tline=rotate(36)*(c--(2a,0));
real tr=intersect(el,tline)[0];
pair r=point(el,tr);
pair tan_dir=dir(el,tr);
path tan_line=scale(b)*(-tan_dir--tan_dir);
pair w=intersectionpoint(d--r,shift(c)*tan_line);

pen linePen=darkblue+1.2pt;
pen elPen=red+1.5pt;

draw(el,elPen); draw(d--r,linePen);
draw(shift(r)*tan_line,linePen);
draw(shift(c)*tan_line,linePen);
Dot(c,d,r,w);
label("$C$",c,NE);label("$D$",d,NW);
label("$R$",r,NE);label("$W$",w,S);

运行asy ellipse.asy以获取ellipse.eps或asy -f pdf ellipse.asy获取ellipse.pdf。或者将其放在文档asy中的环境中LaTeX（参见texdoc asymptote）。

编辑：添加了一些评论。

一个用户定义的函数来绘制一串点，稍后用作Dot(c,d,r,w);：

void Dot(... pair[] p){ //  function takes a variable number of arguments
  for(int i=0;i<p.length;++i){
    fill(shift(p[i])*scale(0.06)*unitcircle,black);
    fill(shift(p[i])*scale(0.04)*unitcircle,white);
  }
}

该函数Dot是用... pair[] p构造定义的，这意味着它能够接受可变数量的参数，所有参数都将放置在一对数组（二维坐标）中p[]。

real a=2.5, b=2, focus=sqrt(a*a-b*b);

定义维度。

pair c=(0,0), d=(-focus,0);

通过坐标 定义点c和。dx,y

path el=ellipse(c,a,b);

定义一条稍后要使用的曲线（椭圆轮廓）；

path tline=rotate(36)*(c--(2a,0));

将直线定义为旋转 36 度的水平线c--(2a,0)

real tr=intersect(el,tline)[0];

定义所谓的交叉时间，t即路径的参数，它对应于与椭圆轮廓el的交点。tlineel

pair r=point(el,tr);

定义交点本身。

pair tan_dir=dir(el,tr);

r定义点（时间）处的切线方向tr。

path tan_line=scale(b)*(-tan_dir--tan_dir);

定义一条通过原点与切线平行的线。

pair w=intersectionpoint(d--r,shift(c)*tan_line);

定义感兴趣的交点，位于该线d--r 与通过原点（点c）的切线平行的线之间。

pen linePen=darkblue+1.2pt;
pen elPen=red+1.5pt;

定义用于线条和椭圆的笔（颜色和宽度）

draw(el,elPen); draw(d--r,linePen);

画椭圆和直线d--r

draw(shift(r)*tan_line,linePen);
draw(shift(c)*tan_line,linePen);

使用定义的函数通过点r和绘制两条平行线。 cDot

Dot(c,d,r,w);

在所有四个点上画出奇特的点c,d,r,w。

label("$C$",c,NE);label("$D$",d,NW);
label("$R$",r,NE);label("$W$",w,S);

最后，在指定位置（例如）(La)TeX绘制标签，格式化为字符串（例如），按照指定的方向（ed表示该点的西北方向）。"$C$",cNE

编辑2：添加了 tikz 翻译

感谢Harish Kumar[此处][2]的回答，我刚刚inkscape2tikz从[来源][3]安装了它，运行之后，asy -f svg ellipse.asy svg2tikz ellipse.svg > ellipse.tex它是一个LaTeX包含解决方案的文档tikz，是从ellipse.asy上面显示的代码翻译而来的：

\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage{tikz}

\begin{document}
\definecolor{cff0000}{RGB}{255,0,0}
\definecolor{c000040}{RGB}{0,0,64}
\definecolor{cffffff}{RGB}{255,255,255}


\begin{tikzpicture}[y=0.80pt,x=0.80pt,yscale=-1, inner sep=0pt, outer sep=0pt]
\begin{scope}[cm={{0.996,0.0,0.0,0.996,(0.0,0.0)}}]
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      (9.9720,-7.9680) .. controls (9.9720,-7.8600) and (10.0680,-7.8480) ..
      (10.1400,-7.8480) .. controls (10.3800,-7.8360) and (10.7640,-7.7640) ..
      (10.7640,-7.3920) .. controls (10.7640,-7.2360) and (10.7160,-7.1520) ..
      (10.5960,-6.9480) -- (7.3200,-1.2120) -- (6.8880,-7.4640) .. controls
      (6.8880,-7.6080) and (7.0200,-7.8360) .. (7.6920,-7.8480) .. controls
      (7.8480,-7.8480) and (7.9680,-7.8480) .. (7.9680,-8.0760) .. controls
      (7.9680,-8.1960) and (7.8480,-8.1960) .. (7.7880,-8.1960) .. controls
      (7.3680,-8.1960) and (6.9240,-8.1720) .. (6.4920,-8.1720) -- (5.8680,-8.1720)
      .. controls (5.6880,-8.1720) and (5.4720,-8.1960) .. (5.2920,-8.1960) ..
      controls (5.2200,-8.1960) and (5.0760,-8.1960) .. (5.0760,-7.9680) .. controls
      (5.0760,-7.8480) and (5.1600,-7.8480) .. (5.3640,-7.8480) .. controls
      (5.9160,-7.8480) and (5.9160,-7.8360) .. (5.9640,-7.1040) -- (6.0000,-6.6720)
      -- (2.8920,-1.2120) -- (2.4480,-7.4040) .. controls (2.4480,-7.5360) and
      (2.4480,-7.8360) .. (3.2640,-7.8480) .. controls (3.3960,-7.8480) and
      (3.5280,-7.8480) .. (3.5280,-8.0640) .. controls (3.5280,-8.1960) and
      (3.4200,-8.1960) .. (3.3480,-8.1960) .. controls (2.9280,-8.1960) and
      (2.4840,-8.1720) .. (2.0520,-8.1720) -- (1.4280,-8.1720) .. controls
      (1.2480,-8.1720) and (1.0320,-8.1960) .. (0.8520,-8.1960) .. controls
      (0.7800,-8.1960) and (0.6360,-8.1960) .. (0.6360,-7.9680) .. controls
      (0.6360,-7.8480) and (0.7320,-7.8480) .. (0.9000,-7.8480) .. controls
      (1.4640,-7.8480) and (1.4760,-7.7760) .. (1.5000,-7.3920) -- (2.0280,-0.0240)
      .. controls (2.0400,0.1800) and (2.0520,0.2520) .. (2.1960,0.2520) .. controls
      (2.3160,0.2520) and (2.3400,0.2040) .. (2.4480,0.0240) -- (6.0240,-6.2280) --
      (6.4680,-0.0240) .. controls (6.4800,0.1800) and (6.4920,0.2520) ..
      (6.6360,0.2520) .. controls (6.7560,0.2520) and (6.7920,0.1920) ..
      (6.8880,0.0240) -- (10.8360,-6.8640) -- cycle;
  \end{scope}
\end{scope}

\end{tikzpicture}
\end{document}

图形看起来很好，但标签不知为何消失了。

答案3

附有PSTricks及解釋。

在此处输入图片描述

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-eucl,pst-plot}

\edef\A{2}% semi-major
\edef\B{1}% semi-minor
\edef\Cx{3}% center abscissa
\edef\Cy{3}% center ordinate 

% parametric representation of an ellipse
\edef\X(#1){\A*cos(#1)+\Cx}
\edef\Y(#1){\B*sin(#1)+\Cy}

% the left focus point in RPN notation
% [-sqrt(A^2-B^2)+Cx,Cy]
\edef\F{!\A\space 2 exp \B\space 2 exp sub sqrt neg \Cx\space add 
\Cy }

\psset{algebraic}

\begin{document}
\begin{pspicture}[showgrid](6,6)
    \psparametricplot{0}{Pi 2 mul}{\X(t)|\Y(t)}% plot the ellipse from 0 to 2*pi
    \curvepnode{Pi 4 div}{\X(t)|\Y(t)}{P}% define the point P through which the tangent line passes
    % \curvepnode also produces a unit tangent vector named Ptang
    %----------------------------------------------------------------------------------------------
    \pnode(\Cx,\Cy){C}% define the center
    \pnode(\F){F}% define the focus
    %----------------------------------------------------------------------------------------------
    \nodexn{-2(Ptang)+(C)}{S}% vector S = -2 Ptang + C
    \nodexn{2(Ptang)+(C)}{T}% vector T = 2 Ptang + C
    %-----------------------------------------------------------------------------------------------
    \psline[linecolor=red](S)(T)% draw the line passing through C and parallel to the unit tangent vector
    \psxline[linecolor=green](P){(S)-(C)}{(T)-(C)}% draw a line from vector P + S - C to P + T - C
    \pcline[nodesep=-1,linecolor=blue](F)(P)% drawn a line from F to P
    \pstInterLL[PointName=none]{F}{P}{S}{T}{I}% find the intersection point I between line FP and ST
    \psdots(P)(C)(F)% draw the points P, C, F
\end{pspicture}
\end{document}

动画片

在此处输入图片描述

\documentclass[pstricks,border=12pt]{standalone}
\usepackage{pst-eucl,pst-plot}
\usepackage[nomessages]{fp}

\def\X(#1){2*cos(#1)+3}
\def\Y(#1){sin(#1)+3}
\FPset\N{20}
\FPeval\Step{round(2*pi/N:2)}

\psset{algebraic,unit=0.5}

\begin{document}
\multido{\n=0.00+\Step}{\N}{%
\begin{pspicture*}[showgrid=false](6,6)
    \psparametricplot{0}{Pi 2 mul}{\X(t)|\Y(t)}
    \curvepnode{\n}{\X(t)|\Y(t)}{P}
    \pnode(3,3){Q}
    \pnode(!3 sqrt neg 3 add 3){F}
    \nodexn{-3(Ptang)+(Q)}{A}
    \nodexn{3(Ptang)+(Q)}{B}
    \psline[linecolor=red](A)(B)
    \psxline[linecolor=green](P){(A)-(Q)}{(B)-(Q)}
    \pcline[nodesep=-2,linecolor=blue](F)(P)
    \pstInterLL[PointName=none]{F}{P}{A}{B}{I}  
    \psdots(P)(Q)(F)
\end{pspicture*}}
\end{document}

答案4

使用tzplot：

在此处输入图片描述

\documentclass[border=1mm]{standalone} 

\usepackage{tzplot}

\begin{document}

\begin{tikzpicture}[scale=1.4]
\def\aa{2.5}
\def\bb{2}
\tzellipse[thick]"AA"(0,0)(\aa cm and \bb cm)
\pgfmathsetmacro{\focus}{sqrt(\aa*\aa-\bb*\bb)}
\tzcoors*(0,0)(c){c}(-\focus,0)(d){d}($(0,0)+(36:{\aa} and {\bb})$)(r){r};(4pt)
% 1. tangent at (r)
\tztangent{AA}(r)[1:3]
% 2. parallel line through (c)
\tzgetxyval($(r)-(c)$){\rcx}{\rcy}
\tztangent[blue]<-\rcx,-\rcy>"BB"{AA}(r)[1:3]{shifted}[b]
% 3. intersection
\tzline"RD"(r)(d)
\tzXpoint*[red]{RD}{BB}(X){X}(4pt)
\end{tikzpicture}

\end{document}

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