Why can't we do this with quadratic Bezier curves? It seems like only the degrees of freedom increases, while the # of constraints stays the same, meaning DOF > # of constraints so we can find a satisfying assignment of control points.
The DoF decreases, because we are moving from cubic Bezier (3rd order curve made by weighting 4 basis functions) to quadratic (2nd order curve made by weighting 3 basis functions)
What does u refer to in this slide?
u is simply a parameter that varies from 0 at the first control point to 1 at the second control point. For animations, it may be time. However, for a spline represented based on control points in space, it does not have a specific meaning. In particular, changes in u do not match changes in arclength of the curve, so we cannot even use it directly to break up the curve into identical arclength segments.