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A quadratic Bernstein B-spline basis

Example

Take degree $p = 2$ and clamped knot vector $(0, 0, 0, 1, 1, 1)$ . On $[0, 1]$ its three B-splines are $(1 - x)^{2}$ , $2 x (1 - x)$ , and $x^{2}$ . They form a nonnegative partition of unity and hence a basis of $P_{2}$ .

Facts & Assumptions

Given: The degree-two clamped knot vector $(0, 0, 0, 1, 1, 1)$ .

[F1]

Degree-zero B-splines are knot-interval indicators and higher degrees follow the stated recurrence (B-spline spaces and knot conventions).

[F2]

For a clamped knot vector, the degree- $p$ B-splines form a basis and reproduce every polynomial of degree at most $p$ (B-spline basis properties and polynomial reproduction).

[A1]

The degree is $p = 2$ , the domain is $[0, 1]$ , and the knot vector is $(0, 0, 0, 1, 1, 1)$ .

Verification

Given: $p = 2$ , knots $(0, 0, 0, 1, 1, 1)$ , and $x \in [0, 1]$ .

1.1A1F1algebra

The only nonzero degree-zero spline on $(0, 1)$ is $B_{3, 0} = 1$ . Applying the recurrence [F1] first at degree one gives $B_{2, 1} = 1 - x$ and $B_{3, 1} = x$ . Applying it again gives $B_{1, 2} = (1 - x)^{2}$ , $B_{2, 2} = x (1 - x) + x (1 - x) = 2 x (1 - x)$ , and $B_{3, 2} = x^{2}$ . Each is nonnegative on $[0, 1]$ and their sum is $(1 - x)^{2} + 2 x (1 - x) + x^{2} = 1$ .

2.1A1F2step 1.1algebra∎

For $q (x) = a + b x + c x^{2}$ , choose coefficients $a$ , $a + b / 2$ , and $a + b + c$ . Their B-spline combination is $a (1 - x)^{2} + 2 (a + b / 2) x (1 - x) + (a + b + c) x^{2} = a + b x + c x^{2} .$ Thus every quadratic is reproduced; uniqueness of the coefficient vector follows from the basis assertion [F2].

Depends on

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Sources

Tom Lyche, Carla Manni, and Hendrik Speleers, B-Splines and Spline Approximation, §1.1.1 (standard reference, not scraped)