Approximation methods for the tensor product Gauss-Legendre surfaces

초록

This paper proposes a novel framework for approximating parametric surfaces using tensor product Gauss–Legendre (GL) patches. Although the Bézier patch is the standard surface model in Computer Aided Geometric Design (CAGD), its shape control can be limited for high-degree cases due to the large and oscillating control nets. To address this, we utilize the GL polynomial basis, which offers a more predictable relationship between control nets and surface geometry. Our approximation strategy follows a multi-step process: the corner and boundary control points are fixed first using the tangent data of the boundary curves of the reference surface at the GL nodes, followed by the determination of the interior control points. We present two computational methods for interior point optimization: a method using the tangent vectors at edge points and a method using the twist vectors at face points. We formulate these methods as overdetermined linear systems, and compute the least squares solutions using the QR decomposition method. Numerical experiments approximating complex surfaces, including unit spheres and Dupin cyclides, demonstrate that GL patches exhibit superior numerical stability even for high-degree cases. The results show that the proposed methods are both computationally efficient and highly accurate for the polynomial surface approximation. © 2026 Elsevier B.V.

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