 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-66/4/ii/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

If each univariate basis sums to one, distribute the finite sums, or use local finiteness when the basis is infinite:

j \sum k \sum b_{jk} (u, v) = (j \sum ϕ_{j} (u)) (k \sum ψ_{k} (v)) = 1.

Thus the tensor-product surface basis is a partition of unity. In particular, translating every control point by a fixed vector translates the parametric surface by precisely that vector, and constant control data produce a constant parametric surface.

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