Chapter Fourteen

How Many Different Rational Parametric Cubic Curves Are There? Part I, Inflection Points

July–August 1999

u14-01-9781558608603

In my never-ending quest to build intuition about the relationship between algebra and geometry, I have recently turned my attention once again to cubic curves. The basic question is this: what sorts of shapes can a given symbolic expression generate? In Chapter 4, “How Many Different Cubic Curves Are There?” and Chapter 6, “Cubic Curve Update,” of Jim Blinn's Corner: Dirty Pixels I asked this question about algebraic cubic curves of the form

Ax3+3Bx2y+3Cxy2+Dy3+3Ex2w+6Fxyw+3Gy2w?+3Hxw2+3fyw2+Kw3=0

I am now going to ask the same ...

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