Understanding Quadratic Bézier Curves (opens in new tab)

Thanks, I liked this better than the originally posted article, mostly because it explains higher order beziers.

Polynomial splines are pretty shitty compared to rational splines for animation and modeling, but they both have the problem of non-constant speed. It gets much worse with higher degree curves. You fix that by letting the animator separately specify a scalar speed curve so he can control where things speed up and down. To make this curve be natural to manipulate, it must be specified relative to an arc length parameterization. You can sometimes profitably combine the arc length reparameterization curve and the pre-integrated user-specified speed remapping into a single remapping curve that's baked out to disk, so the run-time curve evaluator only has to deal with a single level of parameter remapping on top of the raw spline.

This is just a fact of life with splines. They have non-uniform speed in their raw form and you have to deal with it. It's also why quaternion slerping being constant angular speed isn't a real advantage over lerp-and-normalize because they both trace out the same path and often you'll want to overlay a speed curve anyway (and lerp-and-normalize's speed is only barely non-uniform). Evaluate-and-normalize is the way to go for quaternions. It works great for multi-way skeletal animation blending and higher-order rotation interpolation (just do the spline evaluation in R^4 and project back onto the unit sphere). Casey has an old email that goes into some of this: http://mollyrocket.com/911.txt

Yes, if you want a constant speed you need to re-parametrize by arclength, which is straightforward for quadratic (or, more generally, polynomial) curves as there is a simple closed expression for their length.