A normalized basis for reduced Clough–Tocher splines

Abstract

We present the construction of a suitable normalized B-spline representation for reduced Clough-Tocher splines. The basis functions have a local support, they are nonnegative, and they form a partition of unity. Geometrically, the problem can be interpreted as the determination of a set of triangles that must contain a specific set of points. This leads to a natural definition of tangent control triangles. We also consider a stable computation of the Bézier control net of the spline surface.Keywords : Clough-Tocher splines, normalized B-splines, control points, control triangles, Bézier control net MSC : Primary : 65D07, Secondary : 65D17, 68U07A normalized basis for reduced Clough-Tocher splines Hendrik SpeleersDepartment of Computer Science, Katholieke Universiteit Leuven Celestijnenlaan 200A, B-3001 Leuven, Belgium AbstractWe present the construction of a suitable normalized B-spline representation for reduced Clough-Tocher splines. The basis functions have a local support, they are nonnegative, and they form a partition of unity. Geometrically, the problem can be interpreted as the determination of a set of triangles that must contain a specific set of points. This leads to a natural definition of tangent control triangles. We also consider a stable computation of the Bézier control net of the spline surface.

Isogeometric analysis on triangulations

“…These include the Clough-Tocher split [24,25] where one triangle is subdivided into three micro-triangles with polynomials of degree d = 3r for continuity r-odd and d = 3r + 1 for r-even, the Powell-Sabin split [26] with polynomials of degree d = ⌊ 9r+1 4 for r-odd, and of degree d = ⌊ 9r+4 4 for r-even, and numerous others [21]. Convenient B-spline-like bases for PS-6 split [27], recently for PS-12 split [28], normalized Clough-Tocher split [29], Quintic PSsplines [30], and a family of PS splines [31] have been proposed. Fig.…”

On numerical quadrature for C1 quadratic Powell–Sabin 6-split macro-triangles

“…We now recall C 0 , C 1 , and C 2 continuity conditions between quadratic Bézier triangles [23,34]. Let p i be a quadratic Bézier triangle defined on T i , i = 1, .…”

“…More information on continuity conditions can be found e.g. in [1,14,23,34]. Figure 5: Two Gaussian quadrature rules for quadratic polynomials [37].…”

A new B-spline representation for cubic splines over Powell–Sabin triangulations

“…Section 5 discusses some strategies to reduce the number of degrees of freedom in the proposed spline space. In particular, we detail the relation with the reduced CT3-splines developed by Speleers (2010b). Finally, in Section 6 we end with some concluding remarks.…”

“…Later on, they were also applied in the area of scattered data interpolation (see, e.g., Farin, 1985;Kashyap, 1996;Mann, 1999). A normalized B-spline basis has been constructed by Speleers (2010b) for a certain subspace of the CT3-spline space. Yet, it is still an open question whether or not it is possible to construct a normalized B-spline basis for the full CT3-spline space.…”