Splines in 5 Minutes: Part 2 -- Catmull-Rom and Natural Cubic Splines

Equivalent to a 50 minute university lecture on Catmull-Rom and Natural Cubic Splines. Part 2 of 3 on splines.
Part 1: • Splines in 5 minutes: ...
Part 3: • Splines in 5 minutes: ...
0:00 - intro
0:16 - higher order polynomial fitting
0:49 - piece-wise cubic curves
1:32 - knots and C1 continuity
2:03 - Catmull-Rom splines
2:37 - Catmull-Rom fitting = convolution with a cubic kernel
3:00 - natural cubic spline
5:16 - C2 continuity
5:39 - natural cubic spline minimizes the squared second derivative
6:16 - local control
Graphics in 5 minutes is a series of cartoon-style videos that teach computer graphics in 10x less time. You can take the equivalent of a University level computer graphics course in just over two hours. The playlist is here:
• Graphics in 5 minutes ...
See here for more information: g5m.cs.washington.edu/

[@ireoluwaTH]

This is gold!!!

[@UzunKamis]

Thank you very much for this very explanatory video. It is the first time we are studying this with my project students. We would like to make some comments about what we get from the explanations. Please correct us if we are wrong. For NCS, we use f''[x_{k+1}]=g''[x_{k+1}] condition in the middle points together with f''[x_{1}]==0 and g''[x_{n}]==0 at the end points. While, for CRS, we use f'[x_{k+1}]=g'[x_{k+1}]=(g[x_{k+2}]-f[x_{k}])/(x_{k+2}-x_{k}) together with some derivative constraints (such as f'[x_{1}]=0 and g'[x_{n}]=-1) at the end points.

There is also monotonic cubic interpolation which is commonly used and you might be interested in it if you're watching this: en.wikipedia.org/wiki/Monotone_cubic_interpolation

[@JustAnotherAlchemist]

I was going to mention cubic Hermite interpolation, a form of monotone interpolation. Gotta preface it's use with the fact that the "steps along X" must be equal distant, aka uniform in interval. In other words, CHI doesn't work if X is allowed to be arbitrary real numbers on the number line.
I explored both cubic Hermite and Catmull-Rom splines for their use in 8-bit microcontrolers several years ago. I'd link to the stack exchange question, but just googling "cubic spline 8 bit" gets you to it almost the first hit.