The Beauty of Bézier Curves

Bezier curves. Cubic bezier curves (generally the most common type) are represented by cubic polynomials x(t) and y(t) (or just one polynomial P(t) where the coefficients are vectors). The first derivative gives you the tangent and normal. The second derivative gives you the direction the normal will gradually become as t increases. The third derivative gives you the change in this change (the fourth derivative is zero). Curvature is also computed from the first and second derivatives: K = det(P', P'')/||P'||^3 Other kinds of bezier curves are similar. In quadratic ones the first and second derivatives are normal and normal change but the third derivative is zero. In higher-polynomial bezier curves the third derivative changes over time as the fourth is non-zero, and so on. Source is a really interesting video on bezier curves which I highly recommend: [this video] .

It is not the case that cubic Beziers are 3D. There’s a good chance that the font you’re using to read this (especially if you print it out) defines the outline of (2D) letters with a sequence of cubic Beziers. Lots of 2D computer graphics uses cubic Beziers because they ~balance the needs of having lots of DOF for expressivity, while not being so complicated that they’re a pain to evaluate. And the control points that define a quadratic Bezier could be in any dimension, including 3, or even 1, which is good for plotting the basis functions. Where you are shapes how you make sense of things. As you learn to see Bézier curves from different viewpoints, the particular physical viewpoint you have now will coexist with others. It’s always that way learning math. That feeling of being disoriented is something to treasure, btw; you won’t always have it. My hunch is that this video [this video] comes from someone who’s path through learning Bézier curves was non-traditional, but they do a beautiful job of explaining and illustrating things. It showed me that I have more to learn too.

A wonderful example of this is: The Continuity of Splines [link] and The Beauty of Bézier Curves [this video] by Freya Holmér [link] While not animated, I find: _Euclid's Elements_ (Joyce's Java Version) very helpful [link] Since my current project has pretty much devolved to putting a UI on various trigonometric functions and allowing the user to program them, I'd be very glad of other resources in this space (currently working through Hilbert's _Geometry and the Imagination_ and the book which matches [link] and just finished the _Make:Geometry/Trigonometry/Calculus_ series, and have had recommended Curves and Surfaces for CAGD: A Practical Guide by Gerald Farin)