 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-66/6/a/solution

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

Choose the knot origin so the span most influenced by

P_{i}

[i, i + 1]

t \in [0, 1]

. The three active uniform quadratic B-spline basis functions give

C_{i} (t) = \frac{1}{2} (1 - t)^{2} P_{i - 1} + (\frac{1}{2} + t - t^{2}) P_{i} + \frac{1}{2} t^{2} P_{i + 1} .

The central coefficient reaches its maximum at

t = 1/2

, so this is indeed the span centered on the strongest influence of

P_{i}

. Express it in the quadratic Bernstein basis:

C_{i} (t) = (1 - t)^{2} B_{0} + 2 t (1 - t) B_{1} + t^{2} B_{2} .

Matching constant, linear and quadratic coefficients gives the quadratic Bézier curve controls

B_{0} = \frac{P _{i - 1} + P _{i}}{2}, B_{1} = P_{i}, B_{2} = \frac{P _{i} + P _{i + 1}}{2} .

For example,

C_{i}^{'} (0) = 2 (B_{1} - B_{0}) = P_{i} - P_{i - 1}

and

C_{i}^{'} (1) = 2 (B_{2} - B_{1}) = P_{i + 1} - P_{i}

, checking the endpoint derivatives and the

C^{1}

joins of neighboring spans.

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