Check out Greg Egan's new page on caustics!   The blue curve here is an ellipse. When we shine light on it, the reflected rays bunch up along a curve called a catacaustic.  The ellipse is described by a quadratic equation, but its catacaustic is described by a polynomial equation of degree 6.  This is an example of a general pattern discussed by Egan: the catacaustic of a curve of degree d has degree at most 3d(d-1).  The math is serious, but the pictures will delight you!

http://www.gregegan.net/SCIENCE/Catacaustics/Catacaustics.html

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