If you haven't been to Wolfram Alpha yet, there is a lot of fun stuff for math geeks to do. I'm teaching polar coordinates to my trig students now, so I spent some time goofing around with the Polar Plot calculator. You could do the same thing with any graphing calculator, but Wolfram does it prettier and does it for free.
To make an ellipse, type in "polar plot r^2 = 6/(2 + sin(theta)^2)". (Don't include the quotation marks, obvs.) The 6 and the 2 in the fraction can be changed to adjust the size of the ellipse.
You can also draw a cardioid by typing "polar plot r = 1 + sin theta". That sharp point at the origin is called a cusp.
This cardioid with a loop is "polar plot r = .5 + sin theta". It can be interesting to see what changes you can make in a graph by changing the coefficients slightly.
You might also like "polar plot r = sin (3*theta)" or "polar plot r = sin (4*theta)", especially if you played with Spirograph as a kid.
1 hour ago