This is very fun and well done, great interactive illustrations, and I love the deep dive into curvature.
I have to wonder - if you want smooth joins, why not use uniform B-splines instead of Bézier curves? With B-splines you don’t have to set constraints or compute curvature, it’s built in. What are the reasons to prefer Bézier?
There’s an old 1985 tech report by Tony DeRose (formerly of Pixar Research) that categorized the different ways to smoothly join curves - "Geometric Continuity: A Parametrization Independent Measure of Continuity for Computer Aided Geometric Design.”
That paper has a simple constraint formula for matching curvature between cubic Béziers, one that has a couple degrees of freedom. It’s the “G2 (curvature continuity)” formula mentioned on Wikipedia here: https://en.wikipedia.org/wiki/Composite_B%C3%A9zier_curve#Sm.... It’d be interesting to know whether that somehow works out to be mathematically equivalent to this post’s technique or not.
I have to wonder - if you want smooth joins, why not use uniform B-splines instead of Bézier curves? With B-splines you don’t have to set constraints or compute curvature, it’s built in. What are the reasons to prefer Bézier?
There’s an old 1985 tech report by Tony DeRose (formerly of Pixar Research) that categorized the different ways to smoothly join curves - "Geometric Continuity: A Parametrization Independent Measure of Continuity for Computer Aided Geometric Design.”
That paper has a simple constraint formula for matching curvature between cubic Béziers, one that has a couple degrees of freedom. It’s the “G2 (curvature continuity)” formula mentioned on Wikipedia here: https://en.wikipedia.org/wiki/Composite_B%C3%A9zier_curve#Sm.... It’d be interesting to know whether that somehow works out to be mathematically equivalent to this post’s technique or not.