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Positive NURBS weights do not imply an admissible patch

Statement

On $[0, 1]^{2}$ , take degree-one tensor Bernstein functions in each coordinate, all four weights equal to one, and control net $P_{00} = (0, 0), P_{10} = (1, 0), P_{01} = (0, 1), P_{11} = (1, - 1) .$ The resulting positive-weight bilinear NURBS map is $F (ξ, η) = (ξ, η (1 - 2 ξ))$ . Its Jacobian vanishes at $ξ = 1 / 2$ , changes sign across that line, and the map is not injective; hence positive weights alone do not give an admissible patch.

Facts & Assumptions

Given: The degree-one tensor Bernstein basis, the four control points and unit weights in the Statement.

[F1]

Tensor-product NURBS patch maps and rational spline spaces defines $F$ as the rational weighted sum of its control points and requires global injectivity and positive cell Jacobian for admissibility.

[F2]

Positive weights give a bounded partition-of-unity NURBS basis: unit weights yield denominator one and nonnegative basis functions summing to one.

Proof

1.1F1F2givenalgebra

The degree-one Bernstein functions are $1 - ξ, ξ$ and $1 - η, η$ . With all weights one, [F2] gives $W = 1$ , so the tensor rational basis is the ordinary bilinear basis. Its first component is $ξ$ and its second component is $(1 - ξ) η + ξ (- η) = η (1 - 2 ξ)$ .

2.1F1step 1.1algebra∎

Differentiation gives $D F = (\begin{array}{cc} 1 & 0 \\ - 2 η & 1 - 2 ξ \end{array}), \det D F = 1 - 2 ξ .$ Thus the Jacobian is zero on $ξ = 1 / 2$ , positive for $ξ < 1 / 2$ , and negative for $ξ > 1 / 2$ . Moreover, $F (1 / 2, 0) = F (1 / 2, 1) = (1 / 2, 0)$ with distinct parameter points. These violate two separate admissibility requirements in [F1] despite all weights being positive.

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Sources

Stefan Takacs, Numerical Analysis of Isogeometric Methods, complete JKU lecture notes, 2018/19 (standard reference, not scraped)
Annalisa Buffa, Gregor Gantner, Carlotta Giannelli, Dirk Praetorius, Rafael Vázquez, Mathematical Foundations of Adaptive Isogeometric Analysis, Archives of Computational Methods in Engineering 29 (2022) (standard reference, not scraped)