How do I make tangents to ellipses and lines parallel to these?
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}
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)}}]
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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
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(-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}
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width=0.960pt] (-45.1687,-0.0000) -- (55.7293,-40.4897);
\end{scope}
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\end{scope}
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\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}
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(1.8068,0.9978) .. (1.8068,0.0000) -- cycle;
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(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.
With PSTricks and explanation.
\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
\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}