I have a following situation:
Say I have a rational cubic Bezier curve $B_0$. I then project it into 4D, to make it non-rational, and then use De Casteljau's algorithm to subdivide it into two segments at point $u = 0.5$. Now I have two rational Cubic Bezier curves $B_1$ and $B_2$. My question is, can we say that $B_0 (0.75) = B_2(0.5)$? Meaning that, after subdivision of curves, is there any connection between the parametrization of the original curve and the two resulting curves?
I think it is true that $B_0(0.75) = B_2(0.5)$.
To see this, assume the original rational curve is 2D. Then it’s easy to visualize the “projection” of this curve into a polynomial curve in 3D. Then run the de Casteljau algorithm to split this 3D polynomial curve into two. This will produce two curves whose parameterization matches the unsplit one. Projection back to 2D will preserve this.
If that argument is not convincing, just compute $B_0(0.75)$ and $B_2(0.5)$ using the de Casteljau algorithm. I’m pretty sure you’ll get the same answers.