see this is where I can simply ask them to pay the educational license fee for the censored version of the video
I like it! Have you had any interest from educators? Random tangent, but I’m glad you joined the discussion! Maybe you’re sick of curves by now, and I realize you said you left out many topics, but I was curious if you’d considered the quadratic B-spline or any other quadratic curves? I feel like we always talk about linear and cubic and hardly anyone considers quadratics. There are good reasons for that, but I like…
no educators have reached out yet! As for quadratics, I haven't found much use of them, except in cases where you really need an exact solution for arc length or area. Usually the cubics give you a ton more control for not that much extra cost, they kinda strike that nice balance! And yes, the quadratic B-spline is C1, just like the catrom
ah gosh, a collab would be cool but I think you overestimate my knowledge in this area! I have no formal education or even an eighth of the vocabulary let alone mathematical background to contribute much ;-; I've been reading your paper on the spiro splines and a lot of it goes way over my head! I'd have to research every step along the way
Oh, I wouldn't worry. What would make a collab video cool is not dazzling people with mathematical BS, but rather visually showing the geometric and physical principles behind the splines, especially their relationship with bending elastic strips. If you want to have a chat to brainstorm (even if nothing comes of it), my contact info should be pretty easy to find.
I’d definitely up my Patreon contributions to see the results of such collaboration.
I like it! Have you had any interest from educators? Random tangent, but I’m glad you joined the discussion! Maybe you’re sick of curves by now, and I realize you said you left out many topics, but I was curious if you’d considered the quadratic B-spline or any other quadratic curves? I feel like we always talk about linear and cubic and hardly anyone considers quadratics. There are good reasons for that, but I like…
no educators have reached out yet! As for quadratics, I haven't found much use of them, except in cases where you really need an exact solution for arc length or area. Usually the cubics give you a ton more control for not that much extra cost, they kinda strike that nice balance! And yes, the quadratic B-spline is C1, just like the catrom
That’s fair, compelling uses of quadratics are legitimately hard to find, and usually it’s not significantly more costly to go to cubics. Here are few more reasons I’ve collected: the quadratic is easier to explain and understand, useful from a pedagogical perspective; quadratics, unlike cubics, have an analytic rotation minimizing moving frame (a normal and binormal that minimize twist along the curve); quadratics can be analytically sampled in steps of equal change in tangent angle (very handy for drawing, path rendering, tessellating for GPUs); and quadratics are much easier to ray trace (using a swept-circle or swept-sphere formulation). Ray tracing a swept-sphere cubic is solving an order 10 polynomial, while the quadratic is order 6 - but you can reduce the order to 4 with some constraints, which suddenly allows it to be evaluated analytically (in theory), while the cubic is permanently stuck in iterative numeric methods. One of the reasons I liked investigating the quadratic B-spline is because it suddenly makes clear that the order of a B-spline is nothing more than a smoothing factor. Quadratic and cubic and quartic are all the same curve, more or less, just successively smoother. Maybe that’s unsurprising, but from a practical point of view, it can allow you to reduce or increase the order without changing the high level behavior of your curves. Anyway sorry to geek out, I was just curious - I’ve done some work in ray tracing of hair, which is what led me down this path. I did not use quadratics when I worked in games or films, but looking back on it, I think there might have been good reasons to consider it especially on lower powered hardware, maybe for simulation and procedural animation more than hand-animated stuff…