Graphing Polar Equations The command polarplot can be used to graph an equation in polar coordinates. The animatecurve command can be used to visualize the manner in which the graph is drawn. These commands can only be used after typing with(plots) restart:with(plots): Plot of r = 1 r1:=1; polarplot(r1,t=0..5*Pi,scaling=constrained,thickness=2); animatecurve([r1,t,t=0..Pi],coords=polar,thickness=2,frames=50,scaling=constrained);#Click on the plot and use the toolbar buttons to animate. Plot of r = NiNJJnRoZXRhRzYi r2:=t; polarplot(r2,t=0..20*Pi,scaling=constrained,thickness=2); animatecurve([r2,t,t=0..20*Pi],coords=polar,thickness=2,frames=50,scaling=constrained,numpoints=5000);#Click on the plot and use the toolbar buttons to animate. Plot of r = 2sinNiNJJnRoZXRhRzYi r:=2*sin(t): polarplot(r,t=0..Pi,scaling=constrained,thickness=2); animatecurve([r,t,t=0..Pi],coords=polar,thickness=2,frames=50,scaling=constrained);#Click on the plot and use the toolbar buttons to animate. Plot NiMvJSJyRywmIiIjIiIiKiZGJkYnLSUkc2luRzYjKiZGJkYnJSZ0aGV0YUdGJ0YnISIi restart:with(plots): f:=2-2*sin(2*t): polarplot(f,t=0..2*Pi,scaling=constrained,thickness=2); animatecurve([f,t,t=0..2*Pi],coords=polar,thickness=2,frames=50,scaling=constrained); The following set of commands creates a graph called a cardiod restart:with(plots): g:=1+sin(5*t); polarplot(g,t=0..2*Pi,scaling=constrained,thickness=2); animatecurve([g,t,t=0..2*Pi],coords=polar,thickness=2,frames=50,scaling=constrained); The following set of commands creates a graph called a four-leaved rose restart:with(plots): r:=cos(2*t): polarplot(r,t=0..2*Pi,scaling=constrained,thickness=2,color=blue); animatecurve([r,t,t=0..2*Pi],coords=polar,thickness=2,frames=50,scaling=constrained); int((cos(2*x))^2,x=0..2*Pi); The following family of graphs based on r = 1 + c sin(t) are called limacons after a French word for snail. Notice the shape of the graphs. Feel free to try other values of c to see what happens. restart:with(plots): a:=1+3*sin(t):polarplot(a,t=-arcsin(1/3)..(Pi+arcsin(1/3)),scaling=constrained,thickness=2,color=blue); The following commands were used to show how to calculate the area of the inside and the outside of a limacon seperately. You need to determine your limits of integration correctly. QyQtSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JCwkKiQsJiIiIkYtLUkkc2luR0YlNiNJInhHRigiIiQiIiMkIiImISIiL0YxOyIiISwkSSNQaUdGJkYzRi0= QyQ+SSJhRzYiLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQsJCokLCYiIiJGLy1JJHNpbkdGKDYjSSJ4R0YlIiIkIiIjJCIiJiEiIi9GMzssJC1JJ2FyY3NpbkdGKDYjI0YvRjRGOCwmSSNQaUdGKUYvRjxGL0Yv QyQ+SSJiRzYiLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQsJCokLCYiIiJGLy1JJHNpbkdGKDYjSSJ4R0YlIiIkIiIjJCIiJiEiIi9GMzssJkkjUGlHRilGLy1JJ2FyY3NpbkdGKDYjI0YvRjRGLywmRjxGNUY9RjhGLw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIitGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUTBtZWRpdW1tYXRoc3BhY2VGJy8lJ3JzcGFjZUdGTy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1GLDYlUSJiRidGL0YyLUY2NjBRIjtGJ0Y5RjsvRj9GMUZARkJGREZGRkhGSi9GTlEkMGVtRicvRlFRL3RoaWNrbWF0aHNwYWNlRidGUkZV Back to examples. b:=1+2*sin(t):polarplot(b,t=0..2*Pi,scaling=constrained,thickness=2,color=green); c:=1+.5*sin(t):polarplot(c,t=0..2*Pi,scaling=constrained,thickness=2,color=orange); d:=1-.5*sin(t):polarplot(d,t=0..2*Pi,scaling=constrained,thickness=2,color=red); e:=1-2*sin(t):polarplot(e,t=0..2*Pi,scaling=constrained,thickness=2,color=aquamarine); f:=1-3*sin(t):polarplot(f,t=0..2*Pi,scaling=constrained,thickness=2,color=violet); A lemniscate has the form NiMvKiQpJSJyRyIiIyIiIiomKSUiYUdGJ0YoLSUkY29zRzYjKiZGJ0YoJSJ0R0YoRig= Maple has a tough time with this plot, but the following will give you an idea of what the graph looks like. restart:with(plots): p:=(cos(2*t))^(1/2):q:=-(cos(2*t))^(1/2): polarplot({p,q},t=0..20*Pi,thickness=2,color=aquamarine); Here are a few other plots polarplot(sin(9*t/5),t=0..10*Pi,scaling=constrained, thickness=2); polarplot(sin(t)+(sin(5*t/2))^3,t=0..4*Pi,scaling=constrained, thickness=2); polarplot((sin(4*t))^2+cos(4*t),t=0..2*Pi,scaling=constrained, thickness=2); polarplot((5/sqrt(t)),t=0..6*Pi,scaling=constrained, thickness=2); polarplot(1/(16+4*cos(t)),t=0..2*Pi,scaling=constrained, thickness=2);