10

I have specified an ellipse and three points in the following way:

\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);

Now I need

  1. a tangent vector to the ellipse through (r);
  2. a line parallel to this tangent through (c);
  3. a coordinate at the intersection of this last line with the line joining (r) and (d).

I cannot seem to find a solution to 1. that allows me to solve 2. and 3. Any suggestions, please?

4

4 Answers 4

12

Here's a way of accomplishing this using the approach from How to draw tangent line of an arbitrary point on a path in TikZ

\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}
5
  • @jake: do you mind explaining the last \draw command? I cannot figure out why that ends up going through (c)? Replacing (c) with (0,0) does not give the correct answer.
    – daleif
    Apr 10, 2013 at 13:42
  • @jake, I know, but I do not understand why it ends up at (c) instead of (tangent point-1)
    – daleif
    Apr 10, 2013 at 14:04
  • @daleif: We are talking about \draw [use tangent, red, name path=parallel] (c) ++(2,0) -- +(-4,0);, right? The line goes through (c) because nodes do not get transformed by the coordinate transformation (so while (0,0) is at a different place if we use use tangent, (c) is still at the same place).
    – Jake
    Apr 10, 2013 at 14:08
  • @jake that is it, takes a bit you get ones head twisted through that one.
    – daleif
    Apr 10, 2013 at 14:10
  • 1
    Very interesting, I discover with your answer the /pgf/decoration/mark info/sequence number and how to use it Apr 10, 2013 at 17:08
8

enter image description here

And Asymptote version, ellipse.asy along with a translation to tikz via svg

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);

run asy ellipse.asy to get ellipse.eps or asy -f pdf ellipse.asy to get ellipse.pdf. Or put it inside the asy environment in a LaTeX document (see texdoc asymptote).

Edit: Some comments added.

A user defined function to draw a list of dots, to be used later as 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);
  }
}

The function Dot is defined with ... pair[] p construction, that means it is able to accept a variable number of arguments, all of them will be placed in an array of pairs (2D coordinates) p[].

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

defines dimensions.

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

defines points c and d by x,y coordinates.

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

defines a curve (ellipse outline) to be used later;

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

defines a straight line as a rotated by 36 degrees horizontal line c--(2a,0)

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

defines a so-called intersection time, a parameter t for the path el, which corresponds to the point of intersection of tline with the ellipse outline el.

pair r=point(el,tr);

define the point of intersection itself.

pair tan_dir=dir(el,tr);

defines a tangent direction at the point r (at time tr).

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

defines a line through the origin parallel to the tangent.

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

defines an intersection point of interest, between the line d--r and a line parallel to the tangent through the origin (point c).

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

defined are pens (color and width) to be used for lines and ellipse

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

draw the ellipse and the line d--r

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

draw two parallel lines through points r and c, using defined function Dot.

Dot(c,d,r,w);

draw fancy dots at all four points c,d,r,w.

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

and finally, draw labels, formatted as a (La)TeX string (e.g. "$C$") at specified position (e.g. ,c), oriented as specified (e.d. NE means to the North-West of the point).

Edit2: tikz translation added

Thanks to Harish Kumar for his answer [here][2], I've just installed inkscape2tikz from [source][3] and after running asy -f svg ellipse.asy svg2tikz ellipse.svg > ellipse.tex here it is a LaTeX document with tikz solution, translated from the ellipse.asy code shown above:

\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)}}]
  \begin{scope}[cm={{1.0,0.0,0.0,1.0,(207.183,174.51)}}]
    \path[draw=cff0000,line join=round,line cap=round,miter limit=10.04,line
      width=1.200pt] (75.2812,0.0000) .. controls (75.2812,-33.2613) and
      (41.5767,-60.2250) .. (0.0000,-60.2250) .. controls (-41.5767,-60.2250) and
      (-75.2812,-33.2613) .. (-75.2812,-0.0000) .. controls (-75.2812,33.2613) and
      (-41.5767,60.2250) .. (0.0000,60.2250) .. controls (41.5767,60.2250) and
      (75.2812,33.2613) .. (75.2812,0.0000) -- cycle;
  \end{scope}
  \begin{scope}[cm={{1.0,0.0,0.0,1.0,(207.183,174.51)}}]
    \path[draw=c000040,line join=round,line cap=round,miter limit=10.04,line
      width=0.960pt] (-45.1687,-0.0000) -- (55.7293,-40.4897);
  \end{scope}
  \begin{scope}[cm={{1.0,0.0,0.0,1.0,(207.183,174.51)}}]
    \path[draw=c000040,line join=round,line cap=round,miter limit=10.04,line
      width=0.960pt] (100.9330,-0.6942) -- (10.5257,-80.2853);
  \end{scope}
  \begin{scope}[cm={{1.0,0.0,0.0,1.0,(207.183,174.51)}}]
    \path[draw=c000040,line join=round,line cap=round,miter limit=10.04,line
      width=0.960pt] (45.2036,39.7955) -- (-45.2036,-39.7955);
  \end{scope}
  \begin{scope}[cm={{1.0,0.0,0.0,1.0,(207.183,174.51)}}]
    \path[fill=black] (1.8068,0.0000) .. controls (1.8068,-0.9978) and
      (0.9978,-1.8068) .. (0.0000,-1.8068) .. controls (-0.9978,-1.8068) and
      (-1.8068,-0.9978) .. (-1.8068,-0.0000) .. controls (-1.8068,0.9978) and
      (-0.9978,1.8068) .. (0.0000,1.8068) .. controls (0.9978,1.8068) and
      (1.8068,0.9978) .. (1.8068,0.0000) -- cycle;
  \end{scope}
  \begin{scope}[cm={{1.0,0.0,0.0,1.0,(207.183,174.51)}}]
    \path[fill=cffffff] (1.2045,0.0000) .. controls (1.2045,-0.6652) and
      (0.6652,-1.2045) .. (0.0000,-1.2045) .. controls (-0.6652,-1.2045) and
      (-1.2045,-0.6652) .. (-1.2045,-0.0000) .. controls (-1.2045,0.6652) and
      (-0.6652,1.2045) .. (0.0000,1.2045) .. controls (0.6652,1.2045) and
      (1.2045,0.6652) .. (1.2045,0.0000) -- cycle;
  \end{scope}
  \begin{scope}[cm={{1.0,0.0,0.0,1.0,(207.183,174.51)}}]
    \path[fill=black] (-43.3620,-0.0000) .. controls (-43.3620,-0.9978) and
      (-44.1709,-1.8068) .. (-45.1687,-1.8068) .. controls (-46.1666,-1.8068) and
      (-46.9755,-0.9978) .. (-46.9755,-0.0000) .. controls (-46.9755,0.9978) and
      (-46.1666,1.8068) .. (-45.1687,1.8068) .. controls (-44.1709,1.8068) and
      (-43.3620,0.9978) .. (-43.3620,-0.0000) -- cycle;
  \end{scope}
  \begin{scope}[cm={{1.0,0.0,0.0,1.0,(207.183,174.51)}}]
    \path[fill=cffffff] (-43.9642,-0.0000) .. controls (-43.9642,-0.6652) and
      (-44.5035,-1.2045) .. (-45.1687,-1.2045) .. controls (-45.8340,-1.2045) and
      (-46.3732,-0.6652) .. (-46.3732,-0.0000) .. controls (-46.3732,0.6652) and
      (-45.8340,1.2045) .. (-45.1687,1.2045) .. controls (-44.5035,1.2045) and
      (-43.9642,0.6652) .. (-43.9642,-0.0000) -- cycle;
  \end{scope}
  \begin{scope}[cm={{1.0,0.0,0.0,1.0,(207.183,174.51)}}]
    \path[fill=black] (57.5361,-40.4897) .. controls (57.5361,-41.4876) and
      (56.7272,-42.2965) .. (55.7293,-42.2965) .. controls (54.7315,-42.2965) and
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      (57.5361,-39.4919) .. (57.5361,-40.4897) -- cycle;
  \end{scope}
  \begin{scope}[cm={{1.0,0.0,0.0,1.0,(207.183,174.51)}}]
    \path[fill=cffffff] (56.9338,-40.4897) .. controls (56.9338,-41.1550) and
      (56.3946,-41.6942) .. (55.7293,-41.6942) .. controls (55.0641,-41.6942) and
      (54.5248,-41.1550) .. (54.5248,-40.4897) .. controls (54.5248,-39.8245) and
      (55.0641,-39.2852) .. (55.7293,-39.2852) .. controls (56.3946,-39.2852) and
      (56.9338,-39.8245) .. (56.9338,-40.4897) -- cycle;
  \end{scope}
  \begin{scope}[cm={{1.0,0.0,0.0,1.0,(207.183,174.51)}}]
    \path[fill=black] (-12.3358,-12.4506) .. controls (-12.3358,-13.4484) and
      (-13.1447,-14.2573) .. (-14.1426,-14.2573) .. controls (-15.1404,-14.2573) and
      (-15.9493,-13.4484) .. (-15.9493,-12.4506) .. controls (-15.9493,-11.4528) and
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      (-12.3358,-11.4528) .. (-12.3358,-12.4506) -- cycle;
  \end{scope}
  \begin{scope}[cm={{1.0,0.0,0.0,1.0,(207.183,174.51)}}]
    \path[fill=cffffff] (-12.9381,-12.4506) .. controls (-12.9381,-13.1158) and
      (-13.4774,-13.6551) .. (-14.1426,-13.6551) .. controls (-14.8078,-13.6551) and
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      (-12.9381,-11.7854) .. (-12.9381,-12.4506) -- cycle;
  \end{scope}
  \begin{scope}[shift={(209.733,171.904)}]
    \path (8.9640,-8.3400) .. controls (8.9640,-8.4480) and (8.8800,-8.4480) ..
      (8.8560,-8.4480) .. controls (8.8320,-8.4480) and (8.7840,-8.4480) ..
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      (8.1600,-5.0640) .. (8.2080,-5.2800) -- (8.9640,-8.3400) -- cycle;
  \end{scope}
  \begin{scope}[shift={(149.391,171.904)}]
    \path (1.8840,-0.8880) .. controls (1.7760,-0.4680) and (1.7520,-0.3480) ..
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      (3.5520,-7.5960) and (3.5520,-7.5240) .. (3.5040,-7.3440) -- (1.8840,-0.8880)
      -- cycle(4.4160,-7.3800) .. controls (4.5240,-7.8240) and (4.5720,-7.8480) ..
      (5.0400,-7.8480) -- (6.3600,-7.8480) .. controls (7.4880,-7.8480) and
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      (2.7120,-0.5760) and (2.7360,-0.6480) .. (2.7600,-0.7560) -- (4.4160,-7.3800)
      -- cycle;
  \end{scope}
  \begin{scope}[shift={(265.462,131.415)}]
    \path (4.4160,-7.3800) .. controls (4.5240,-7.8240) and (4.5720,-7.8480) ..
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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}

The graphics looks fine, but the labels vanished somehow.

0
5

With PSTricks and explanation.

enter image description here

\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}

Animation

enter image description here

\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}
0
2

Using tzplot:

enter image description here

\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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