Bezier Curve always the same length

Question

I'm working on a game in HTML5 canvas.

I want is draw an S-shaped cubic bezier curve between two points, but I'm looking for a way to calculate the coordinates of the control points so that the curve itself is always the same length no matter how close those points are, until it reaches the point where the curve becomes a straight line.

Answer 1

This is solvable numerically. I assume you have a cubic bezier with 4 control points. at each step you have the first (P0) and last (P3) points, and you want to calculate P1 and P2 such that the total length is constant.

Adding this constraint removes one degree of freedom so we have 1 left (started with 4, determined the end points (-2) and the constant length is another -1). So you need to decide about that.

The bezier curve is a polynomial defined between 0 and 1, you need to integrate on the square root of the sum of elements (2d?). for a cubic bezier, this means a sqrt of a 6 degree polynomial, which wolfram doesn't know how to solve. But if you have all your other control points known (or known up to a dependency on some other constraint) you can have a save table of precalculated values for that constraint.