The mathematical beauty of hyperbezier curves

I love bezier curves. In community-college, it was the first time I ever encountered a subject that made me want to go do more research on my own. One of my core memories is toiling away for multiple nights when the rest of the house was asleep on my 2015 Macbook Pro, writing janky p5.js code and pressing refresh on the browser page over and over until suddenly, I started see real, beautiful curves blossom from my control points.

I just went back to dig up some old sources[0], and I can't believe this post is almost a decade old now. This guy's explainers and animations were leagues beyond any other resource I could find through Google searc at the time.

[0] https://jamie-wong.com/post/bezier-curves/

One big aspect that made Bezier popular was local support, i.e. moving control points resulted in expected changes in the same direction. It's interesting that this is the result of a search for 'a curve family better suited for interactive design than cubic Beziers.' I think it's better for achieving curve quality in the sense of curvature changes, but not so much in interactivity.

Related: Last year I had the pleasure to work with Bezier curves to make a very specific UI for an app I was working on.

I wrote down a bunch of posts on how I achieved the UI primarly to explain the process to my future self[0]

As a mobile developer it was so much fun to do something other than building a CRUD app.

[0]https://www.krtkush.com/computer-graphics-basics-with-compos...

Very cool demo, The curve does sort of go screwball when the control points are close together.

After playing with it some more. Not close together, but more side loaded. On reflection while many of these forms would be awkward to find in a drawing program, I think this is the source of the superior spiral the author likes.

One unintuitive thing I noticed about these curves, at least from the demo on the page, is that they have both areas where their behavior quickly 'snaps' from one configuration to another, and areas where they saturate such that changes to the control handles no longer produce any movement. Standard Bezier curves don't have either of these; I wonder if that's essentially coupled to the higher expressivity they offer?