Chapter Fourteen

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

## July–August 1999

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

$\begin{array}{c}A{x}^{3}+3B{x}^{2}y+3Cx{y}^{2}+D{y}^{3}\\ +3E{x}^{2}w+6Fxyw+3G{y}^{2}w\\ \text{?}+3Hx{w}^{2}+3fy{w}^{2}\\ +K{w}^{3}=0\end{array}$

I am now going to ask the same ...