I have a tensor-product Bézier patch and I want to subidivide this adding a curve inside the patch, which creates two rectangular subpatches. I found that the following statement holds: "if we subdivide a Bézier patch into two patches two opposite edges are divided in the same ratio because of the tensor-product property of a Bézier patch". I don't understand why the ratio must be the same. Thanks.
2026-03-27 08:16:55.1774599415
Subdividing a Bézier patch
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You do have Bézier patch, and now you want to divide it into subpatches you do use de Casteljau algorithm, which can divide Bézier curve into two at arbitrary point. But now you do divide it into two subpatches at some point (parametric value), so they create the same surface as did one you have splitted. If you split somwhere at t = 0.4 (with best parametrization from 0-1), one patch is now from 0 to 0.4 and the second from 0.4 to 1.
As far as I understand your question, there is no obvious algorithm that splits Bezier surfaces by arbitrary curve - Bézier surface interpolates corner control points, so you have to split 4 curves at one direction at the same point (4 curves for bicubic patch), otherwise you change the surface and yours split is no longer like the original.
Subdivision is supposed to give you two subpatches with total partitioning of the original surface.