Excellent, if lengthy, presentation. The narration is good and the graphics are exceptional, clear, unfussy. Highly recommended watching for anyone having to deal with the math of splines for the first time.
At about 12:00 she says non-uniform splines are much more complicated and this video won't discuss them. True, but the non-uniform knot intervals give a lot more power over parametric continuity (C), which is introduced at 15:30 and finally defined by name at 26:05. It's only at that point (no pun intended) that geometric continuity (G) is introduced. At 36:20 there's an implied hint of that power when geometric continuity is finally defined. Zebra striping to assess continuity at 28:30. (At 31:46 there's a clever bit of promotion for geometric algebra while discussing computation of curvature.) Definition of "Class A" surfaces as G2 continuous at 35:25. "Class A" is a term which is huge in industrial design, but might never be mentioned in a math course. Catmull-Rom splines discussed at 48:50, as a special case of cardinal splines, with really lovely and consistent local control, leading directly to B-splines, the first guaranteed C2 continuous splines discussed, at 53:40. Good summary and comparison of various spline types around 1:00:00.
One consideration that isn't mentioned is that applying textures (decals) to spline surfaces is generally going to be much happier with parametric (C) continuity, rather than just geometric (G) continuity, although reflections and shading are content with G2.
