What Math Equations Look Like in 3-D

In a parametric curve, a derivative of zero may indicate a cusp, which I feel is where the comparison is coming from.

I only read your comment and article but didn't see the video, so this comment may be off.

No, they are the same thing. What this sentence is referring to is the vanishing of the Jacobian determinant [1] (which is defined using the derivatives of the defining equations).

A simple example is the equations y^3 - x^2 = 0. This is a "cusp" (use wolfram alpha to see what it looks like) and has a singularity at the origin.

The jacobian is the matrix:

[ -2x, 3y^2 ]

This has rank 1 unless x and y are zero in which case it has rank zero. The fact that the rank is less than 1 indicates a singularity.

[1]: http://en.wikipedia.org/wiki/Singularity_(mathematics)#Algeb...