@whdawn: Not 100% sure what you're asking, but if you're wondering where we get the values u0 and u1, you can imagine those values are specified "by the user." For instance, if someone is drawing a cubic Bezier curve in a 2D illustration program, then the endpoints p0, p1 are determined by where they click, and the tangents u0, u1 are determined by where they drag the cursor after clicking. Or, you might have data that comes from a physical simulation, in which case p0, p1 correspond to configurations, and u0, u1 correspond to velocities.

How to get p'(0) and p'(1) when computing?

@whdawn: Not 100% sure what you're asking, but if you're wondering where we get the values u0 and u1, you can imagine those values are specified "by the user." For instance, if someone is drawing a cubic Bezier curve in a 2D illustration program, then the endpoints p0, p1 are determined by where they click, and the tangents u0, u1 are determined by where they drag the cursor after clicking. Or, you might have data that comes from a physical simulation, in which case p0, p1 correspond to configurations, and u0, u1 correspond to velocities.

Oh, I see. Thanks Keenan!