Adaptive fitting with a cubic spline over planar hierarchical quadrilateral meshes

Pengxiao Wang; Chongjun Li; Pengxiao Wang; Chongjun Li

doi:10.3934/math.20251177

AIMS Mathematics

2025, Volume 10, Issue 11: 26767-26802. doi: 10.3934/math.20251177

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Research article

Adaptive fitting with a cubic spline over planar hierarchical quadrilateral meshes

Pengxiao Wang ¹,
Chongjun Li ^{2
,
,}

1.
School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing 211816, China
2.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Received: 13 August 2025 Revised: 30 October 2025 Accepted: 05 November 2025 Published: 18 November 2025
MSC : 65D07, 65D17

Automatic fitting techniques are required in many industrial applications, for example instrument calibration, data analysis, geometric modeling, and reverse engineering. In this paper, we present a surface construction algorithm for a cubic spline over planar hierarchical quadrilateral meshes. The surface was piecewisely constructed by interpolating the cubic spline surface of the 12 parameters at four vertices on each quadrilateral cell of the hierarchical quadrilateral mesh. For a given hierarchical quadrilateral mesh and geometric information (function values and two first-order partial derivatives) at the corresponding basis vertices of the hierarchical quadrilateral mesh, the surface can be constructed simply. Moreover, we give an adaptively refined surface algorithm for fitting scattered data points based on cubic spline surface construction. The numerical results show that the proposed adaptive algorithm is efficient in fitting scattered data points within a polygonal domain.
- scattered data fitting,
- the B-net method,
- hierarchical quadrilateral mesh,
- triangulated quadrangulation,
- surface reconstruction
Citation: Pengxiao Wang, Chongjun Li. Adaptive fitting with a cubic spline over planar hierarchical quadrilateral meshes[J]. AIMS Mathematics, 2025, 10(11): 26767-26802. doi: 10.3934/math.20251177

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Abstract

Automatic fitting techniques are required in many industrial applications, for example instrument calibration, data analysis, geometric modeling, and reverse engineering. In this paper, we present a surface construction algorithm for a cubic spline over planar hierarchical quadrilateral meshes. The surface was piecewisely constructed by interpolating the cubic spline surface of the 12 parameters at four vertices on each quadrilateral cell of the hierarchical quadrilateral mesh. For a given hierarchical quadrilateral mesh and geometric information (function values and two first-order partial derivatives) at the corresponding basis vertices of the hierarchical quadrilateral mesh, the surface can be constructed simply. Moreover, we give an adaptively refined surface algorithm for fitting scattered data points based on cubic spline surface construction. The numerical results show that the proposed adaptive algorithm is efficient in fitting scattered data points within a polygonal domain.

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