The computation of the envelope showed that indeed, the nephroid curve is part of the envelope of the family of reflected light rays, which explains how it ended up on the bottom of your coffee cup. But it also raises some questions: The first two questions start to make sense when we look closely at the graph surface F(t,x,y)=0 shown below, and think about how its singularities relate to the envelope. You can click on the picture to generate an MPEG movie (646 Kbytes).

If you haven't read about projections, profiles, and envelopes, take a look at it now. Given an algebraic surface like F(t,x,y)=0, there is also a notion of its singular set which you can also read about.

Question 8

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Vic Reiner <>
Frederick J. Wicklin <>
Last modified: Mon Apr 15 13:53:47 1996