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楕円の接線とそれに平行な線を作成するにはどうすればよいでしょうか?

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楕円の接線とそれに平行な線を作成するにはどうすればよいでしょうか?

楕円と 3 つの点を次のように指定しました。

\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);

今必要なのは

  1. (r)を通る楕円の接線ベクトル
  2. この接線に平行な線（c）
  3. この最後の線と(r)と(d)を結ぶ線との交点の座標。

1. の解決策が見つからないのですが、2. と 3. を解決できる方法はありますか?

答え1

これを実現するには、次のアプローチを使用します。TikZ でパス上の任意の点の接線を描く方法

\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}

答え2

ここに画像の説明を入力してください

そしてAsymptoteバージョン、ellipse.asy svg経由でtikzへの翻訳も

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);

を実行するとがasy ellipse.asy取得され、を実行すると が取得されます。または、 をドキュメント内の 環境内に配置します( を参照)。ellipse.epsasy -f pdf ellipse.asyellipse.pdfasyLaTeXtexdoc asymptote

編集：コメントをいくつか追加しました。

後で使用するドットのリストを描画するユーザー定義関数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);
  }
}

この関数は構造Dotで定義されます... pair[] p。つまり、可変数の引数を受け入れることができ、それらはすべてペアの配列 (2D 座標) に配置されますp[]。

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

寸法を定義します。

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

点cと座標dを定義しますx,y。

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

後で使用する曲線（楕円のアウトライン）を定義します。

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

直線を36度回転した水平線として定義しますc--(2a,0)

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

tは、パス のパラメータである、いわゆる交差時間を定義します。これは、と楕円のアウトラインelの交差点に対応します。tlineel

pair r=point(el,tr);

交差点自体を定義します。

pair tan_dir=dir(el,tr);

点r（時刻tr）における接線方向を定義します。

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

原点を通る接線と平行な線を定義します。

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

d--r 直線と、原点（点c）を通る接線に平行な直線との間の交点を定義します。

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

線と楕円に使用するペン（色と幅）が定義されています

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

楕円と線を描くd--r

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

定義された関数を使用して、点rとを通る 2 本の平行線を描きます。 cDot

Dot(c,d,r,w);

4 つの点すべてにドットを描きますc,d,r,w。

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

最後に、(La)TeX文字列 (例"$C$") としてフォーマットされたラベルを、指定された位置 (例,c) に、指定された方向 (ed はNEポイントの北西を意味します) で描画します。

編集2: tikz翻訳を追加しました

[ここ][2]の回答に感謝しますHarish Kumar。私は[ソース][3]からインストールしたところです。ここでinkscape2tikz実行したところ、上記のコードから翻訳された解決策が記載されたドキュメントが見つかりました。asy -f svg ellipse.asy svg2tikz ellipse.svg > ellipse.texLaTeXtikzellipse.asy

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

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      (11.2680,-8.1720) .. controls (10.9080,-8.1720) and (10.5360,-8.1960) ..
      (10.1880,-8.1960) .. controls (10.1160,-8.1960) and (9.9720,-8.1960) ..
      (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}

グラフィックは問題ないように見えますが、ラベルが何らかの理由で消えてしまいました。

答え3

PSTricksと説明付き。

ここに画像の説明を入力してください

\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}

アニメーション

ここに画像の説明を入力してください

\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}

答え4

使用tzplot:

ここに画像の説明を入力してください

\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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