Code
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TeX
Below are some things I've drawn or coded in TeX.
The code below draws the slope field given by a function \(f(x,y)\) of two variables, once you've specified a few parameters. This example produces the slope field for \(f(x,y) = x^2 + y^3\). Be aware that this is unlikely to work with poorly-behaved functions, for instance when the slope is undefined.
You can download the TeX file here.
\documentclass { amsart}
\usepackage { tikz}
\begin { document}
\begin { center}
\begin { tikzpicture}
%% Set the window for x and y values.
\pgfmathsetmacro \xmin { -2} ;
\pgfmathsetmacro \ymin { -2} ;
\pgfmathsetmacro \xmax { 2} ;
\pgfmathsetmacro \ymax { 2} ;
%% Set the density of the slope field; a lower spacing gives a denser slope field.
\pgfmathsetmacro \xspacing { 0.2} ;
\pgfmathsetmacro \yspacing { 0.2} ;
%% Set the length of the tangent lines: all lines of the slope field are drawn with this length. This should be less than both spacing numbers above for a neat slope field.
\pgfmathsetmacro \length { 0.1} ;
%% Define the slope at (\x,\y) as a function f of \x and \y. Here, #1 is \x and #2 is \y. The function should be defined within \pgfmathparse{ }.
\pgfmathdeclarefunction { f} { 2} { \pgfmathparse { (#1)^2 + (#2)^3} } ;
%% This block generates the slope field.
%% Restrict drawing to the confines of the specified rectangle.
\clip (\xmin ,\ymin ) rectangle (\xmax ,\ymax );
%% Draw the axes.
\draw (\xmin ,0) -- (\xmax ,0);
\draw (0,\ymin ) -- (0,\ymax );
%% Some calculations (since we cannot do arithmetic inside a list).
\pgfmathsetmacro \xnext { \xmin +\xspacing } ;
\pgfmathsetmacro \ynext { \ymin +\yspacing } ;
%% Loop to draw the tangent lines.
\foreach \x in{ \xmin ,\xnext , ...,\xmax } { \foreach \y in{ \ymin ,\ynext , ...,\ymax } {
%% Calculate slope at (\x,\y).
\pgfmathsetmacro \slope { f(\x ,\y )} ;
\begin { scope}
%% Cut the drawing down to the circle of radius 0.5*\length centered at (\x,\y).
\clip (\x ,\y ) circle ({ 0.5*\length } );
%% Draw tangent lines of length at least \length, with midpoint (\x,\y).
\draw [ very thin] ({ \x - 0.5*\length } ,{ \y - 0.5*\slope *\length } ) -- ({ \x + 0.5*\length } ,{ \y + 0.5*\slope *\length } );
\end { scope}
}
}
\end { tikzpicture}
\end { center}
\end { document}
Using Euler's method, it is relatively easy to add a solution curve. The following amended code draws the slope field above along with a solution curve passing through the point \((0, 0.3)\).
You can download the TeX file here.
\documentclass { amsart}
\usepackage { tikz}
\begin { document}
\begin { center}
\begin { tikzpicture}
%% Set the window for x and y values.
\pgfmathsetmacro \xmin { -2} ;
\pgfmathsetmacro \ymin { -2} ;
\pgfmathsetmacro \xmax { 2} ;
\pgfmathsetmacro \ymax { 2} ;
%% Set the density of the slope field; a lower spacing gives a denser slope field.
\pgfmathsetmacro \xspacing { 0.2} ;
\pgfmathsetmacro \yspacing { 0.2} ;
%% Set the length of the tangent lines: all lines of the slope field are drawn with this length. This should be less than both spacing numbers above for a neat slope field.
\pgfmathsetmacro \length { 0.1} ;
%% Define the slope at (\x,\y) as a function f of \x and \y. Here, #1 is \x and #2 is \y. The function should be defined within \pgfmathparse{ }.
\pgfmathdeclarefunction { f} { 2} { \pgfmathparse { (#1)^2 + (#2)^3} } ;
%% Specify the step size to use in Euler's method.
\pgfmathsetmacro \stepsize { 0.005} ;
%% Specify the bounds of the interval on which to draw the solution curve, and the x co-ordinate of the initial point. Make sure that the pairwise differences \initialx - lower and \upper - \initialx are divisible by \stepsize.
\pgfmathsetmacro \lower { -2} ;
\pgfmathsetmacro \initialx { 0} ;
\pgfmathsetmacro \upper { 1.4} ;
%% Specify the desired y-value at x=\initialx of the solution to be graphed.
\pgfmathsetmacro \initialy { 0.3} ;
%% This block generates the slope field.
%% Restrict drawing to the confines of the specified rectangle.
\clip (\xmin ,\ymin ) rectangle (\xmax ,\ymax );
%% Draw the axes.
\draw (\xmin ,0) -- (\xmax ,0);
\draw (0,\ymin ) -- (0,\ymax );
%% Some calculations (since we cannot do arithmetic inside a list).
\pgfmathsetmacro \xnext { \xmin +\xspacing } ;
\pgfmathsetmacro \ynext { \ymin +\yspacing } ;
%% Loop to draw the tangent lines.
\foreach \x in{ \xmin ,\xnext , ...,\xmax } { \foreach \y in{ \ymin ,\ynext , ...,\ymax } {
%% Calculate slope at (\x,\y).
\pgfmathsetmacro \slope { f(\x ,\y )} ;
\begin { scope}
%% Cut the drawing down to the circle of radius 0.5*\length centered at (\x,\y).
\clip (\x ,\y ) circle ({ 0.5*\length } );
%% Draw tangent lines of length at least \length, with midpoint (\x,\y).
\draw [ very thin] ({ \x - 0.5*\length } ,{ \y - 0.5*\slope *\length } ) -- ({ \x + 0.5*\length } ,{ \y + 0.5*\slope *\length } );
\end { scope}
}
}
%% This block generates the solution curve with the specificed parameters.
%% Some calculations again.
\pgfmathsetmacro \xnext { \initialx +\stepsize } ;
\pgfmathsetmacro \xprev { \initialx -\stepsize } ;
%% Iterate to draw the desired solution curve with Euler's method.
\foreach [ remember=\lasty as\y (initially\initialy )] \x in{ \initialx ,\xnext , ...,\upper } {
\draw (\x ,\y ) -- ({ \x +\stepsize } ,{ \y + f(\x ,\y )*\stepsize } );
\pgfmathsetmacro \lasty { \y + f(\x ,\y )*\stepsize } ;
}
\foreach [ remember=\lasty as\y (initially\initialy )] \x in{ \initialx ,\xprev , ...,\lower } {
\draw (\x ,\y ) -- ({ \x -\stepsize } ,{ \y - f(\x ,\y )*\stepsize } );
\pgfmathsetmacro \lasty { \y - f(\x ,\y )*\stepsize } ;
}
\end { tikzpicture}
\end { center}
\end { document}
Here is a torus. You can download the TeX file here.
\documentclass { amsart}
\usepackage { fullpage}
\usepackage { tikz}
\begin { document}
\begin { center}
\begin { tikzpicture}
\draw [ smooth] (0,0) to[ out=90,in=90] (3,0) to[ out=-90,in=-90] (0,0);
\draw [ smooth] (0.9,0) .. controls (1.3,-0.15) and (1.7,-0.15) .. (2.1,0);
\draw [ smooth] (1,-0.03) .. controls (1.2,0.2) and (1.8,0.2) .. (2,-0.03);
\end { tikzpicture}
\end { center}
\end { document}
Here is a surface of genus 2. You can download the TeX file here.
\documentclass { amsart}
\usepackage { fullpage}
\usepackage { tikz}
\begin { document}
\begin { center}
\begin { tikzpicture}
\draw [ smooth] (0,1) to[ out=30,in=150] (2,1) to[ out=-30,in=210] (3,1) to[ out=30,in=150] (5,1) to[ out=-30,in=30] (5,-1) to[ out=210,in=-30] (3,-1) to[ out=150,in=30] (2,-1) to[ out=210,in=-30] (0,-1) to[ out=150,in=-150] (0,1);
\draw [ smooth] (0.4,0.1) .. controls (0.8,-0.25) and (1.2,-0.25) .. (1.6,0.1);
\draw [ smooth] (0.5,0) .. controls (0.8,0.2) and (1.2,0.2) .. (1.5,0);
\draw [ smooth] (3.4,0.1) .. controls (3.8,-0.25) and (4.2,-0.25) .. (4.6,0.1);
\draw [ smooth] (3.5,0) .. controls (3.8,0.2) and (4.2,0.2) .. (4.5,0);
\end { tikzpicture}
\end { center}
\end { document}
Here is a pants decompositions for this surface. You can download the TeX file here.
\documentclass { amsart}
\usepackage { fullpage}
\usepackage { tikz}
\begin { document}
\thispagestyle { empty}
\begin { center}
\begin { tikzpicture}
\draw [ smooth] (0,1) to[ out=30,in=150] (2,1) to[ out=-30,in=210] (3,1) to[ out=30,in=150] (5,1) to[ out=-30,in=30] (5,-1) to[ out=210,in=-30] (3,-1) to[ out=150,in=30] (2,-1) to[ out=210,in=-30] (0,-1) to[ out=150,in=-150] (0,1);
\draw [ smooth] (0.4,0.1) .. controls (0.8,-0.25) and (1.2,-0.25) .. (1.6,0.1);
\draw [ smooth] (0.5,0) .. controls (0.8,0.2) and (1.2,0.2) .. (1.5,0);
\draw [ smooth] (3.4,0.1) .. controls (3.8,-0.25) and (4.2,-0.25) .. (4.6,0.1);
\draw [ smooth] (3.5,0) .. controls (3.8,0.2) and (4.2,0.2) .. (4.5,0);
\draw (-0.5,0) arc(180:0:0.51 and 0.2);
\draw [ dashed] (-0.5,0) arc(180:0:0.51 and -0.2);
\draw (2.5,-0.85) arc(270:90:0.3 and 0.85);
\draw [ dashed] (2.5,-0.85) arc(270:450:0.3 and 0.85);
\draw (5.5,0) arc(180:0:-0.51 and 0.2);
\draw [ dashed] (5.5,0) arc(180:0:-0.51 and -0.2);
\end { tikzpicture}
\end { center}
\end { document}
Here is a surface of genus 3. You can download the TeX file here.
\documentclass { amsart}
\usepackage { fullpage}
\usepackage { tikz}
\begin { document}
\begin { center}
\begin { tikzpicture}
\draw [ smooth] (0,1) to[ out=30,in=150] (2,1) to[ out=-30,in=210] (3,1) to[ out=30,in=150] (5,1) to[ out=-30,in=210] (6,1) to[ out=30,in=150] (8,1) to[ out=-30,in=30] (8,-1) to[ out=210,in=-30] (6,-1) to[ out=150,in=30] (5,-1) to[ out=210,in=-30] (3,-1) to[ out=150,in=30] (2,-1) to[ out=210,in=-30] (0,-1) to[ out=150,in=-150] (0,1);
\draw [ smooth] (0.4,0.1) .. controls (0.8,-0.25) and (1.2,-0.25) .. (1.6,0.1);
\draw [ smooth] (0.5,0) .. controls (0.8,0.2) and (1.2,0.2) .. (1.5,0);
\draw [ smooth] (3.4,0.1) .. controls (3.8,-0.25) and (4.2,-0.25) .. (4.6,0.1);
\draw [ smooth] (3.5,0) .. controls (3.8,0.2) and (4.2,0.2) .. (4.5,0);
\draw [ smooth] (6.4,0.1) .. controls (6.8,-0.25) and (7.2,-0.25) .. (7.6,0.1);
\draw [ smooth] (6.5,0) .. controls (6.8,0.2) and (7.2,0.2) .. (7.5,0);
\end { tikzpicture}
\end { center}
\end { document}
Here is a pants decomposition for this surface. You can download the TeX file here.
\documentclass { amsart}
\usepackage { fullpage}
\usepackage { tikz}
\begin { document}
\begin { center}
\begin { tikzpicture}
\draw [ smooth] (0,1) to[ out=30,in=150] (2,1) to[ out=-30,in=210] (3,1) to[ out=30,in=150] (5,1) to[ out=-30,in=210] (6,1) to[ out=30,in=150] (8,1) to[ out=-30,in=30] (8,-1) to[ out=210,in=-30] (6,-1) to[ out=150,in=30] (5,-1) to[ out=210,in=-30] (3,-1) to[ out=150,in=30] (2,-1) to[ out=210,in=-30] (0,-1) to[ out=150,in=-150] (0,1);
\draw [ smooth] (0.4,0.1) .. controls (0.8,-0.25) and (1.2,-0.25) .. (1.6,0.1);
\draw [ smooth] (0.5,0) .. controls (0.8,0.2) and (1.2,0.2) .. (1.5,0);
\draw [ smooth] (3.4,0.1) .. controls (3.8,-0.25) and (4.2,-0.25) .. (4.6,0.1);
\draw [ smooth] (3.5,0) .. controls (3.8,0.2) and (4.2,0.2) .. (4.5,0);
\draw [ smooth] (6.4,0.1) .. controls (6.8,-0.25) and (7.2,-0.25) .. (7.6,0.1);
\draw [ smooth] (6.5,0) .. controls (6.8,0.2) and (7.2,0.2) .. (7.5,0);
\draw (-0.5,0) arc(180:0:0.51 and 0.2);
\draw [ dashed] (-0.5,0) arc(180:0:0.51 and -0.2);
\draw (2.5,-0.85) arc(270:90:0.3 and 0.85);
\draw [ dashed] (2.5,-0.85) arc(270:450:0.3 and 0.85);
\draw (4.0,0.15) arc(270:90:0.3 and 1.14/2);
\draw [ dashed] (4.0,0.15) arc(270:450:0.3 and 1.14/2);
\draw (4.0,-0.15) arc(270:90:0.3 and -1.14/2);
\draw [ dashed] (4.0,-0.15) arc(270:450:0.3 and -1.14/2);
\draw (5.5,-0.85) arc(270:90:0.3 and 0.85);
\draw [ dashed] (5.5,-0.85) arc(270:450:0.3 and 0.85);
\draw (8.5,0) arc(180:0:-0.51 and 0.2);
\draw [ dashed] (8.5,0) arc(180:0:-0.51 and -0.2);
\end { tikzpicture}
\end { center}
\end { document}
Here is a surface of genus n. You can download the TeX file here.
\documentclass { amsart}
\usepackage { fullpage}
\usepackage { tikz}
\begin { document}
\thispagestyle { empty}
\begin { center}
\begin { tikzpicture}
\draw [ smooth] (5.5,-0.85) to[ out=180,in=30] (5,-1) to[ out=210,in=-30] (3,-1) to[ out=150,in=30] (2,-1) to[ out=210,in=-30] (0,-1) to[ out=150,in=-150] (0,1) to[ out=30,in=150] (2,1) to[ out=-30,in=210] (3,1) to[ out=30,in=150] (5,1) to[ out=-30,in=180] (5.5,0.85);
\draw [ smooth] (0.4,0.1) .. controls (0.8,-0.25) and (1.2,-0.25) .. (1.6,0.1);
\draw [ smooth] (0.5,0) .. controls (0.8,0.2) and (1.2,0.2) .. (1.5,0);
\draw [ smooth] (3.4,0.1) .. controls (3.8,-0.25) and (4.2,-0.25) .. (4.6,0.1);
\draw [ smooth] (3.5,0) .. controls (3.8,0.2) and (4.2,0.2) .. (4.5,0);
\node at (6.5,0){ $.\ ;\ ; .\ ;\ ; .$} ;
\draw [ smooth] (7.5,0.85) to[ out=0,in=210] (8,1) to[ out=30,in=150] (10,1) to[ out=-30,in=210] (11,1) to[ out=30,in=150] (13,1) to[ out=-30,in=30] (13,-1) to[ out=210,in=-30] (11,-1) to[ out=150,in=30] (10,-1) to[ out=210,in=-30] (8,-1) to[ out=150,in=0] (7.5,-0.85);
\draw [ smooth] (8.4,0.1) .. controls (8.8,-0.25) and (9.2,-0.25) .. (9.6,0.1);
\draw [ smooth] (8.5,0) .. controls (8.8,0.2) and (9.2,0.2) .. (9.5,0);
\draw [ smooth] (11.4,0.1) .. controls (11.8,-0.25) and (12.2,-0.25) .. (12.6,0.1);
\draw [ smooth] (11.5,0) .. controls (11.8,0.2) and (12.2,0.2) .. (12.5,0);
\draw (5.5,-0.85) arc(270:90:0.3 and 0.85);
\draw (5.5,-0.85) arc(270:450:0.3 and 0.85);
\draw (7.5,-0.85) arc(270:90:0.3 and 0.85);
\draw [ dashed] (7.5,-0.85) arc(270:450:0.3 and 0.85);
\end { tikzpicture}
\end { center}
\end { document}
Here is a pants decomposition for this surface. You can download the TeX file here.
\documentclass { amsart}
\usepackage { fullpage}
\usepackage { tikz}
\begin { document}
\begin { center}
\begin { tikzpicture}
\draw [ smooth] (5.5,-0.85) to[ out=180,in=30] (5,-1) to[ out=210,in=-30] (3,-1) to[ out=150,in=30] (2,-1) to[ out=210,in=-30] (0,-1) to[ out=150,in=-150] (0,1) to[ out=30,in=150] (2,1) to[ out=-30,in=210] (3,1) to[ out=30,in=150] (5,1) to[ out=-30,in=180] (5.5,0.85);
\draw [ smooth] (0.4,0.1) .. controls (0.8,-0.25) and (1.2,-0.25) .. (1.6,0.1);
\draw [ smooth] (0.5,0) .. controls (0.8,0.2) and (1.2,0.2) .. (1.5,0);
\draw [ smooth] (3.4,0.1) .. controls (3.8,-0.25) and (4.2,-0.25) .. (4.6,0.1);
\draw [ smooth] (3.5,0) .. controls (3.8,0.2) and (4.2,0.2) .. (4.5,0);
\node at (6.5,0){ $.\ ;\ ; .\ ;\ ; .$} ;
\draw [ smooth] (7.5,0.85) to[ out=0,in=210] (8,1) to[ out=30,in=150] (10,1) to[ out=-30,in=210] (11,1) to[ out=30,in=150] (13,1) to[ out=-30,in=30] (13,-1) to[ out=210,in=-30] (11,-1) to[ out=150,in=30] (10,-1) to[ out=210,in=-30] (8,-1) to[ out=150,in=0] (7.5,-0.85);
\draw [ smooth] (8.4,0.1) .. controls (8.8,-0.25) and (9.2,-0.25) .. (9.6,0.1);
\draw [ smooth] (8.5,0) .. controls (8.8,0.2) and (9.2,0.2) .. (9.5,0);
\draw [ smooth] (11.4,0.1) .. controls (11.8,-0.25) and (12.2,-0.25) .. (12.6,0.1);
\draw [ smooth] (11.5,0) .. controls (11.8,0.2) and (12.2,0.2) .. (12.5,0);
\draw (-0.5,0) arc(180:0:0.51 and 0.2);
\draw [ dashed] (-0.5,0) arc(180:0:0.51 and -0.2);
\draw (2.5,-0.85) arc(270:90:0.3 and 0.85);
\draw [ dashed] (2.5,-0.85) arc(270:450:0.3 and 0.85);
\draw (4.0,0.15) arc(270:90:0.3 and 1.14/2);
\draw [ dashed] (4.0,0.15) arc(270:450:0.3 and 1.14/2);
\draw (4.0,-0.15) arc(270:90:0.3 and -1.14/2);
\draw [ dashed] (4.0,-0.15) arc(270:450:0.3 and -1.14/2);
\draw (5.5,-0.85) arc(270:90:0.3 and 0.85);
\draw (5.5,-0.85) arc(270:450:0.3 and 0.85);
\draw (7.5,-0.85) arc(270:90:0.3 and 0.85);
\draw [ dashed] (7.5,-0.85) arc(270:450:0.3 and 0.85);
\draw (9.0,0.15) arc(270:90:0.3 and 1.14/2);
\draw [ dashed] (9.0,0.15) arc(270:450:0.3 and 1.14/2);
\draw (9.0,-0.15) arc(270:90:0.3 and -1.14/2);
\draw [ dashed] (9.0,-0.15) arc(270:450:0.3 and -1.14/2);
\draw (10.5,-0.85) arc(270:90:0.3 and 0.85);
\draw [ dashed] (10.5,-0.85) arc(270:450:0.3 and 0.85);
\draw (13.5,0) arc(180:0:-0.51 and 0.2);
\draw [ dashed] (13.5,0) arc(180:0:-0.51 and -0.2);
\end { tikzpicture}
\end { center}
\end { document}
