Playing around with some methods to create a tangent to a circle of ellipse. Used to create cones in some jet figures.
Known radius r (or a, b) & distance q = |OQ|
% TANGENT to CIRCLE - known: r, q \usetikzlibrary{calc} \begin{tikzpicture} \def\r{1.5} % radius \def\q{4} % distance center-external point q = |OQ| \def\x{{\r^2/\q}} % Q x coordinate \def\y{{\r*sqrt(1-(\r/\q)^2}} % Q y coordinate \coordinate (O) at (0,0); % circle center O \coordinate (Q) at (\q,0); % external point Q \coordinate (P) at (\x,\y); % point of tangency, P \draw[->] (0,-1.3*\r) -- (0,1.5*\r); \draw[->] (-1.3*\r,0) -- (\q+0.4*\r,0); \draw[dashed] (\x,0) |- (0,\y); \draw[blue,thick] (O) circle(\r); \draw[green,thick] ($(Q)!-0.2!(P)$) -- ($(Q)!1.3!(P)$); \draw[green,thick] ($(O)!-0.3!(P)$) -- ($(O)!1.4!(P)$); \fill[red] (O) circle(0.05) node[below right] {O}; \fill[red] (Q) circle(0.05) node[below left] {Q}; \fill[red] (P) circle(0.05) node[above=3,right=4] {P}; \end{tikzpicture} % TANGENT to ELLIPSE - known: a, b, q \begin{tikzpicture} \def\a{1.5} % horizontal radius \def\b{1.0} % vertical radius \def\q{4} % distance center-external point q = |OQ| \def\x{{\a^2/\q}} % x coordinate P \def\y{{\b*sqrt(1-(\a/\q)^2}} % y coordinate P \coordinate (O) at (0,0); % circle center O \coordinate (Q) at (\q,0); % external point Q \coordinate (P) at (\x,\y); % point of tangency, P \draw[->] (0,-\b-0.3*\a) -- (0,\b+0.4*\a); \draw[->] (-1.3*\a,0) -- (\q+0.4*\a,0); \draw[dashed] (\x,0) |- (0,\y); \draw[blue,thick] (O) ellipse({\a} and {\b}); \draw[green,thick] ($(Q)!-0.2!(P)$) -- ($(Q)!1.3!(P)$); \draw[green,thick] ($(O)!-0.3!(P)$) -- ($(O)!1.4!(P)$); \fill[red] (O) circle(0.05) node[below right] {O}; \fill[red] (Q) circle(0.05) node[below left] {Q}; \fill[red] (P) circle(0.05) node[above=3,right=4] {P}; \end{tikzpicture}
Known radius r (or a, b) & angle α
% TANGENT to CIRCLE - known: r, alpha \usetikzlibrary{calc} \usetikzlibrary{math} % for \tikzmath \usetikzlibrary{angles,quotes} % for pic (angle labels) \begin{tikzpicture} \def\r{1.5} % radius \def\ang{25} % alpha angle \def\q{\r/sin(\ang)} % distance center-external point q = |OQ| \coordinate (O) at (0,0); % circle center O \coordinate (Q) at ({\q},0); % external point Q \coordinate (P) at (90-\ang:\r); % point of tangency, P \draw[->] (0,-1.3*\r) -- (0,1.5*\r); \draw[->] (-1.3*\r,0) -- ({\q+0.4*\r},0); \draw[dashed] ({\r*sin(\ang)},0) |- (0,{\r*cos(\ang)}); \draw[blue,thick] (O) circle(\r); \draw[green,thick] ($(Q)!-0.2!(P)$) -- ($(Q)!1.3!(P)$); \draw[green,thick] ($(O)!-0.3!(P)$) -- ($(O)!1.4!(P)$); \fill[red] (O) circle(0.05) node[below right] {O}; \fill[red] (Q) circle(0.05) node[below left] {Q}; \fill[red] (P) circle(0.05) node[above=3,right=4] {P}; \draw pic[<-,"$\alpha$"{above=1,left=0},draw=black,angle radius=28,angle eccentricity=1.0] {angle = P--Q--O}; \end{tikzpicture} % TANGENT to ELLIPSE - known: a, b, alpha \begin{tikzpicture} \def\a{1.5} % horizontal radius \def\b{1.0} % vertical radius \def\ang{25} % alpha angle \def\q{\a/sin(\ang)} % distance center-external point q = |OQ| \coordinate (O) at (0,0); % circle center O \coordinate (Q) at ({\q},0); % external point Q \coordinate (P) at (90-\ang:{\a} and {\b}); % point of tangency, P \draw[->] (0,-\b-0.3*\a) -- (0,\b+0.5*\a); \draw[->] (-1.3*\a,0) -- ({\q+0.4*\a},0); \draw[dashed] ({\a*sin(\ang)},0) |- (0,{\b*cos(\ang)}); \draw[blue,thick] (O) ellipse({\a} and {\b}); \draw[green,thick] ($(Q)!-0.2!(P)$) -- ($(Q)!1.3!(P)$); \draw[green,thick] ($(O)!-0.3!(P)$) -- ($(O)!1.4!(P)$); \fill[red] (O) circle(0.05) node[below right] {O}; \fill[red] (Q) circle(0.05) node[below left] {Q}; \fill[red] (P) circle(0.05) node[above=3,right=4] {P}; \draw pic[<-,"$\alpha$"{above=1,left=0},draw=black,angle radius=28,angle eccentricity=1.0] {angle = P--Q--O}; \end{tikzpicture}
Known vector & angle α
Cones given by vector and opening angle α
Full code
Edit and compile if you like:
% Author: Izaak Neutelings (May 2021) % Description: % Experimenting with finding the tangent to an ellipse % with the purpose of finding a nice way to draw a 3D cone. \documentclass[border=3pt,tikz]{standalone} \usepackage{amsmath} \usepackage{mathtools} % for \eqcolon \usepackage{physics} \usepackage{siunitx} \usepackage{xcolor} \usepackage{etoolbox} %ifthen \usepackage[outline]{contour} % glow around text \usetikzlibrary{calc} \usetikzlibrary{math} % for \tikzmath \usetikzlibrary{angles,quotes} % for pic (angle labels) \tikzset{>=latex} % for LaTeX arrow head \contourlength{1.6pt} \colorlet{myblue}{blue!70!black} \colorlet{mygreen}{green!40!black} \colorlet{myred}{red!65!black} \tikzstyle{vector}=[->,very thick,myblue,line cap=round] \tikzstyle{cone}=[thin,blue!50!black,fill=blue!50!black!30,fill opacity=0.8] \newcommand\rightAngle[4]{ \pgfmathanglebetweenpoints{\pgfpointanchor{#2}{center}}{\pgfpointanchor{#3}{center}} \coordinate (tmpRA) at ($(#2)+(\pgfmathresult+45:#4)$); %\draw[white,line width=0.6] ($(#2)!(tmpRA)!(#1)$) -- (tmpRA) -- ($(#2)!(tmpRA)!(#3)$); \draw[blue!40!black] ($(#2)!(tmpRA)!(#1)$) -- (tmpRA) -- ($(#2)!(tmpRA)!(#3)$); } \begin{document} % TANGENT to CIRCLE - known: r, q \begin{tikzpicture} \def\r{1.5} % radius \def\q{4} % distance center-external point q = |OQ| \def\x{{\r^2/\q}} % Q x coordinate \def\y{{\r*sqrt(1-(\r/\q)^2}} % Q y coordinate \coordinate (O) at (0,0); % circle center O \coordinate (Q) at (\q,0); % external point Q \coordinate (P) at (\x,\y); % point of tangency, P \draw[->] (0,-1.3*\r) -- (0,1.5*\r); \draw[->] (-1.3*\r,0) -- (\q+0.4*\r,0); \draw[dashed] (\x,0) |- (0,\y); \draw[myblue,thick] (O) circle(\r); \draw[mygreen,thick] ($(Q)!-0.2!(P)$) -- ($(Q)!1.3!(P)$); \draw[mygreen,thick] ($(O)!-0.3!(P)$) -- ($(O)!1.4!(P)$); \rightAngle{Q}{P}{O}{0.40} \fill[myred] (O) circle(0.05) node[below right] {O}; \fill[myred] (Q) circle(0.05) node[below left] {Q}; \fill[myred] (P) circle(0.05) node[above=3,right=4] {P}; \end{tikzpicture} % TANGENT to ELLIPSE - known: a, b, q \begin{tikzpicture} \def\a{1.5} % horizontal radius \def\b{1.0} % vertical radius \def\q{4} % distance center-external point q = |OQ| \def\x{{\a^2/\q}} % x coordinate P \def\y{{\b*sqrt(1-(\a/\q)^2}} % y coordinate P \coordinate (O) at (0,0); % circle center O \coordinate (Q) at (\q,0); % external point Q \coordinate (P) at (\x,\y); % point of tangency, P \draw[->] (0,-\b-0.3*\a) -- (0,\b+0.4*\a); \draw[->] (-1.3*\a,0) -- (\q+0.4*\a,0); \draw[dashed] (\x,0) |- (0,\y); \draw[myblue,thick] (O) ellipse({\a} and {\b}); \draw[mygreen,thick] ($(Q)!-0.2!(P)$) -- ($(Q)!1.3!(P)$); \draw[mygreen,thick] ($(O)!-0.3!(P)$) -- ($(O)!1.4!(P)$); \fill[myred] (O) circle(0.05) node[below right] {O}; \fill[myred] (Q) circle(0.05) node[below left] {Q}; \fill[myred] (P) circle(0.05) node[above=3,right=4] {P}; \end{tikzpicture} % EQUATIONS, known: r (or a, b), q \begin{tikzpicture}[scale=1] \node[align=left] at (0,0) { Circle with radius $r$:\\[2mm]$\quad \begin{aligned} \abs{\mathrm{OQ}} &\eqqcolon q \\ \abs{\mathrm{PQ}} &= \sqrt{q^2-r^2} \\ \sin{\alpha} &= \frac{r}{q} \\ \mathrm{P} &= \left( \frac{r^2}{q}, r\sqrt{1-\frac{r^2}{q^2}} \right) \\ \end{aligned}$\\[6mm] Ellipse with horizontal radius $a$ and vertical radius $b$:\\[2mm]$\quad \begin{aligned} \abs{\mathrm{OQ}} &\eqqcolon q \\ \abs{\mathrm{PQ}} &= \sqrt{\left(q^2+b^2-a^2\right)\left(1-\frac{a^2}{q^2}\right) } \\ \sin{\alpha} &= \frac{a}{q} \\ \mathrm{P} &= \left( \frac{a^2}{q}, b\sqrt{1-\frac{a^2}{q^2}} \right) \\ \end{aligned}$}; \end{tikzpicture} % TANGENT to CIRCLE - known: r, alpha \begin{tikzpicture} \def\r{1.5} % radius \def\ang{25} % alpha angle \def\q{\r/sin(\ang)} % distance center-external point q = |OQ| \coordinate (O) at (0,0); % circle center O \coordinate (Q) at ({\q},0); % external point Q \coordinate (P) at (90-\ang:\r); % point of tangency, P \draw[->] (0,-1.3*\r) -- (0,1.5*\r); \draw[->] (-1.3*\r,0) -- ({\q+0.4*\r},0); \draw[dashed] ({\r*sin(\ang)},0) |- (0,{\r*cos(\ang)}); \draw[myblue,thick] (O) circle(\r); \draw[mygreen,thick] ($(Q)!-0.2!(P)$) -- ($(Q)!1.3!(P)$); \draw[mygreen,thick] ($(O)!-0.3!(P)$) -- ($(O)!1.4!(P)$); \rightAngle{Q}{P}{O}{0.40} \fill[myred] (O) circle(0.05) node[below right] {O}; \fill[myred] (Q) circle(0.05) node[below left] {Q}; \fill[myred] (P) circle(0.05) node[above=3,right=4] {P}; \draw pic[<-,"$\alpha$"{above=1,left=0},draw=black,angle radius=28,angle eccentricity=1.0] {angle = P--Q--O}; \end{tikzpicture} % TANGENT to ELLIPSE - known: a, b, alpha \foreach \b in {1.0,2.0}{ % vertical radius \foreach \ang in {25}{ % alpha angle \begin{tikzpicture} \def\a{1.5} % horizontal radius %\def\b{1.0} % vertical radius %\def\ang{25} % alpha angle \def\q{\a/sin(\ang)} % distance center-external point q = |OQ| \coordinate (O) at (0,0); % circle center O \coordinate (Q) at ({\q},0); % external point Q \coordinate (P) at (90-\ang:{\a} and {\b}); % point of tangency, P \draw[->] (0,-\b-0.3*\a) -- (0,\b+0.5*\a); \draw[->] (-1.3*\a,0) -- ({\q+0.4*\a},0); \draw[dashed] ({\a*sin(\ang)},0) |- (0,{\b*cos(\ang)}); \draw[myblue,thick] (O) ellipse({\a} and {\b}); \draw[mygreen,thick] ($(Q)!-0.2!(P)$) -- ($(Q)!1.3!(P)$); \draw[mygreen,thick] ($(O)!-0.3!(P)$) -- ($(O)!1.4!(P)$); \fill[myred] (O) circle(0.05) node[below right] {O}; \fill[myred] (Q) circle(0.05) node[below left] {Q}; \fill[myred] (P) circle(0.05) node[above=3,right=4] {P}; \draw pic[<-,"$\alpha$"{above=1,left=0},draw=black,angle radius=28,angle eccentricity=1.0] {angle = P--Q--O}; \node[right,align=left] at ({0.85*\q},\b+0.6) {$\begin{aligned} a &= \a\\ b &= \b\\ \alpha &= \SI{\ang}{\degree} \end{aligned}$}; \end{tikzpicture} }} % EQUATIONS, known: r (or a, b), alpha \begin{tikzpicture}[scale=1] \node[align=left] at (0,0) { Circle with radius $r$:\\[2mm]$\quad \begin{aligned} \abs{\mathrm{OQ}} &= \frac{r}{\sin\alpha} \\ \abs{\mathrm{PQ}} &= r\abs{\cot\alpha} \\ \mathrm{P} &= (r;90-\alpha) = (r\sin\alpha,r\cos\alpha) \\ \end{aligned}$\\[6mm] Ellipse with horizontal radius $a$ and vertical radius $b$:\\[2mm]$\quad \begin{aligned} \abs{\mathrm{OQ}} &= \frac{a}{\sin\alpha} \\ \abs{\mathrm{PQ}} &= \sqrt{\frac{a^2}{\sin^2\alpha}+b^2-a^2}\abs{\cos\alpha} \\ \mathrm{P} &= (a\sin\alpha,b\cos\alpha) \\ \end{aligned}$}; \end{tikzpicture} % TANGENT to CIRCLE - known: vector, alpha \begin{tikzpicture} \def\ang{20} % alpha angle \coordinate (O) at (0,0); % circle center O \coordinate (Q) at (4,1); % external point Q \tikzmath{ coordinate \C; \C = (O)-(Q); \r = veclen(\Cx,\Cy)*sin(\ang); } \pgfmathanglebetweenpoints{\pgfpointanchor{Q}{center}}{\pgfpointanchor{O}{center}} \edef\vecang{\pgfmathresult} \coordinate (P) at ($(O)+({\vecang-(90+\ang)}:\r pt)$); % point of tangency, P \draw[myblue,thick] (O) circle(\r pt); \draw[mygreen,thick] ($(Q)!-0.2!(P)$) -- ($(Q)!1.3!(P)$); \draw[mygreen,thick] ($(O)!-0.3!(P)$) -- ($(O)!1.4!(P)$); \draw ($(O)!-0.5!(Q)$) -- ($(O)!1.3!(Q)$); \draw[vector] (Q) -- (O); \rightAngle{Q}{P}{O}{0.40} \fill[myred] (O) circle(0.05) node[below right] {O}; \fill[myred] (Q) circle(0.05) node[below left] {Q}; \fill[myred] (P) circle(0.05) node[above right=1] {P}; \draw pic[<-,"$\alpha$"{above=1,left=0},draw=black,angle radius=28,angle eccentricity=1.0] {angle = P--Q--O}; \end{tikzpicture} % TANGENT to ELLIPSE - known: vector, angle, a/b \begin{tikzpicture} \def\ang{20} % alpha angle \def\e{0.4} % x scale \coordinate (O) at (0,0); % circle center O \coordinate (Q) at (4,1); % external point Q \pgfmathanglebetweenpoints{\pgfpointanchor{Q}{center}}{\pgfpointanchor{O}{center}} \edef\vecang{\pgfmathresult} % angle of vector QO \tikzmath{ coordinate \C; \C = (O)-(Q); \x = veclen(\Cx,\Cy)*\e*sin(\ang)^2; % x coordinate P \y = tan(\ang)*(veclen(\Cx,\Cy)-\x); % y coordinate P \a = veclen(\Cx,\Cy)*sqrt(\e)*sin(\ang); % vertical radius \b = veclen(\Cx,\Cy)*tan(\ang)*sqrt(1-\e*sin(\ang)^2); % horizontal radius } \coordinate (P) at ($(O)+(\vecang-180:\x pt)+(\vecang-90:\y pt)$); % point of tangency, P \draw[myblue,thick,rotate=\vecang] (O) ellipse({\a pt} and {\b pt}); \draw[mygreen,thick] ($(Q)!-0.2!(P)$) -- ($(Q)!1.3!(P)$); \draw[mygreen,thick] ($(O)!-0.3!(P)$) -- ($(O)!1.4!(P)$); \draw ($(O)!-0.5!(Q)$) -- ($(O)!1.3!(Q)$); \draw[vector] (Q) -- (O); \fill[myred] (O) circle(0.05) node[below right] {O}; \fill[myred] (Q) circle(0.05) node[below left] {Q}; \fill[myred] (P) circle(0.05) node[above right=1] {P}; \draw pic[<-,"$\alpha$"{above=1,left=0},draw=black,angle radius=28,angle eccentricity=1.0] {angle = P--Q--O}; \end{tikzpicture} % CONE \foreach \ang in {10,20}{ % alpha angle \foreach \e in {0.05,0.5}{ % x scale \begin{tikzpicture} %\def\ang{10} % alpha angle %\def\e{0.3} % x scale \coordinate (O) at (0,0); \coordinate (V) at (5,2); \pgfmathanglebetweenpoints{\pgfpointanchor{O}{center}}{\pgfpointanchor{V}{center}} \edef\vecang{\pgfmathresult} % angle of vector OV \tikzmath{ coordinate \C; \C = (O)-(V); \x = veclen(\Cx,\Cy)*\e*sin(\ang)^2; % x coordinate P \y = tan(\ang)*(veclen(\Cx,\Cy)-\x); % y coordinate P \a = veclen(\Cx,\Cy)*sqrt(\e)*sin(\ang); % vertical radius \b = veclen(\Cx,\Cy)*tan(\ang)*sqrt(1-\e*sin(\ang)^2); % horizontal radius \angb = acos(sqrt(\e)*sin(\ang)); % angle of P in ellipse } \coordinate (V') at ($(V)+(\vecang:1.15*\a pt + 10 pt)$); %($(O)!{1/cos(\ang)^2}!(V)$); \coordinate (R) at ($(V)+(\vecang-180:\x pt)+(\vecang-90:\y pt)$); % tangency \coordinate (L) at ($(V)+(\vecang-180:\x pt)+(\vecang+90:\y pt)$); % tangency %\coordinate (R') at ($(V)+(\vecang-90:\b pt)$); % tangency %\coordinate (L') at ($(V)+(\vecang+90:\b pt)$); % tangency \draw[cone,rotate=\vecang] (V) ellipse({\a pt} and {\b pt}); \draw[vector] (O) -- (V'); \draw[cone,rotate=\vecang] (L) arc(180-\angb:180+\angb:{\a pt} and {\b pt}) -- (O) -- cycle; \draw[mygreen,thick] (O) -- (L); \draw[mygreen,thick] (O) -- (R); \fill[myred] (O) circle(0.05) node[below left] {O}; \fill[myred] (V) circle(0.05) node[below right] {V}; \fill[myred] (V') circle(0.05) node[below right] {V$'$}; \fill[myred] (L) circle(0.05) node[above left=-1] {L}; \fill[myred] (R) circle(0.05) node[below=0] {R}; %\fill[myred] (L') circle(0.05) node[above right=-1] {L$'$}; %\fill[myred] (R') circle(0.05) node[below right] {R$'$}; \node[right,align=left] at (-0.2,2.5) {$\begin{aligned} \alpha &= \SI{\ang}{\degree} \\ \theta &= \angb \\ e &= \e \end{aligned}$}; \end{tikzpicture} }} \end{document}
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