Question

I’m looking for an algorithm to insert a new control point on a Bézier curve, without deforming.

Does anybody know a library or reference for Bézier algorithms (insertion, optimize, de Casteljau …)?

Answer 1

This is called the “knot insertion problem”. For Bézier curves, the de Casteljau algorithm will give you the right answer. Here is the simple algorithm for a degree 3 Bézier.

Say you want to insert a knot at a fraction t of the parameter space inside the Bézier curve defined by P0, P1, P2, P3. Here’s what you do:

P0_1 = (1-t)*P0 + t*P1
P1_2 = (1-t)*P1 + t*P2
P2_3 = (1-t)*P2 + t*P3

P01_12 = (1-t)*P0_1 + t*P1_2
P12_23 = (1-t)*P1_2 + t*P2_3

P0112_1223 = (1-t)*P01_12 + t*P12_23

Then your first Bézier will be defined by: P_0, P0_1, P01_12, P0112_1223; your second Bézier is defined by: P0112_1223, P12_23, P2_3, P3.

The geometrical interpretation is simple: you split each segment of the Bézier polygon at fraction t, then connect these split points in a new polygon and iterate. When you’re left with 1 point, this point lies on the curve and the previous/next split points form the previous/next Bézier polygon. The same algorithm also works for higher degree Bézier curves.

Now it can get trickier if you want to insert the control point not at a specific value of t but at a specific location in space. Personally, what I would do here is simply a binary search for a value of t that falls close to the desired split point… But if performance is critical, you can probably find a faster analytic solution.