Dupin Cyclides

It is worth searching the web for the fascinating properties of a particular type of cyclide named after Charles Dupin in his dissertation of 1803. They are currently used as tools in computer-aided geometric design (CAGD). Most striking is that their curvature lines are all circles, extended to infinity in the parabolic case.
Darboux studied the nonparabolic polynomial set to zero. It depends on three parameters , , and . It is usual to speak of ring cyclides for , horn cyclides for , and spindle cyclides for .

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