math - Split a cubic bezier curve at multiple points

Question

I'm writing an algorithm that will split a cubic bezier curve into multiple curves (up to 4). I have the t values for each point I want to split at from the beginning. I also have an algorithm already for splitting the curve once:

SubdivPoints subdivideBezier(Vector2 p0, Vector2 p1, Vector2 p2, Vector2 p3, float t)
{
    Vector2 p11 = (p1 - p0) * t + p0;
    Vector2 p21 = (p2 - p1) * t + p1;
    Vector2 p31 = (p3 - p2) * t + p2;

    Vector2 p12 = (p21 - p11) * t + p11;
    Vector2 p22 = (p31 - p21) * t + p21;

    Vector2 p13 = (p22 - p12) * t + p12;

    return SubdivPoints(p11, p12, p22, p31, p13);
}

My question is, is there a simple way to expand this for splitting multiple times? I imagine after each split I want to recalculate the t values; one thing I'm wondering is if simple arithmetic will work here. E.g. Let's say I have t values 0.2 and 0.6. I split the curve at t = 0.2, giving me two curves. The second curve covers t values 0.2 < t < 1 from the original. If I recalculate the second t value of 0.6 by division: (0.6 - 0.2) / (1 - 0.2) = 0.5, then divide the second curve at t = 0.75, will that work? Or do I need to recalculate it some other way?

Answer 1

Since your subdivideBezier() function did follow the De Casteljau algorithm, I am assuming that it works to subdivide a cubic Bezier curve at parameter t. So, yes, to continue subdividing at a different parameter (say t2), all you need to do is to figure out which subdivided curve t2 falls on, compute the new t2* against that curve and subdivide. In your example where you want to subdivide at t=0.2 and 0.6, the new parameter for 0.6 should be (0.6-0.2)/(1-0.2) = 0.5. So, you can simply subdivide the 2nd curve at t=0.5.