The paper generalizes the classical C1 cubic Clough–Tocher spline space over a triangulation to C1 spaces of any degree higher that three. It shows that the considered spaces can be equipped with a ...basis consisting of non-negative locally supported functions forming a partition of unity and demonstrates the applicability of the basis in the context of the finite element method. The studied spaces have optimal approximation power and are defined by enforcing additional smoothness inside the triangles of the triangulation where the Clough–Tocher splitting is used. Locally, over each triangle of the triangulation, the splines are expressed in the Bernstein–Bézier form, which enables one to take the full advantage of the geometric properties and computational techniques that come with such a representation. Solving boundary problems with Galerkin discretization is thus relatively straightforward and is illustrated with several examples.
We consider a C1 cubic spline space defined over a triangulation with Powell–Sabin refinement. The space has some local C2 super-smoothness and can be seen as a close extension of the classical cubic ...Clough–Tocher spline space. In addition, we construct a suitable normalized B-spline representation for this spline space. The basis functions have a local support, they are nonnegative, and they form a partition of unity. We also show how to compute the Bézier control net of such a spline in a stable way.
•We consider a C1 cubic spline space defined over a Powell–Sabin refined triangulation.•The space has some local C2 super-smoothness and can be seen as a close extension of the classical cubic Clough–Tocher spline space.•We construct a suitable normalized B-spline representation for this spline space.•We also show how to compute the Bézier control net of such a spline in a stable way.
We present the construction of a suitable normalized B-spline representation for reduced cubic 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.
► Clough–Tocher splines. ► Normalized B-splines. ► Control triangles. ► Bezier control net.
A numerical integration rule for multivariate cubic polynomials over n-dimensional simplices was designed by Hammer and Stroud (1956). The quadrature rule requires n+2 quadrature points: the ...barycentre of the simplex and n+1 points that lie on the connecting lines between the barycentre and the vertices of the simplex. In the planar case, this particular rule belongs to a two-parameter family of quadrature rules that admit exact integration of bivariate polynomials of total degree three over triangles. We prove that this rule is exact for a larger space, namely the C1 cubic Clough–Tocher spline space over macro-triangles if and only if the split-point is the barycentre. This results in a factor of three reduction in the number of quadrature points needed to integrate the Clough–Tocher spline space exactly.
Existing techniques that convert B-rep (boundary representation) patches into Clough-Tocher splines guarantee watertight, that is C0, conversion results across B-rep edges. In contrast, our approach ...ensures global tangent-plane, that is G1, continuity of the converted B-rep CAD models. We achieve this by careful boundary curve and normal vector management, and by converting the input models into Shirman-Séquin macro-elements near their (trimmed) B-rep edges. We propose several different variants and compare them with respect to their locality, visual quality, and difference with the input B-rep CAD model. Although the same global G1 continuity can also be achieved by conversion techniques based on subdivision surfaces, our approach uses triangular splines and thus enjoys full compatibility with CAD.
•We present a globally G1 conversion method for CAD models into triangular splines.•A careful B-rep edge management is detailed.•Our method is based on Clough-Tocher and Shirman-Séquin macro-elements.
The boundary representations (B-reps) that are used to represent shape in Computer-Aided Design systems create unavoidable gaps at the face boundaries of a model. Although these inconsistencies can ...be kept below the scale that is important for visualisation and manufacture, they cause problems for many downstream tasks, making it difficult to use CAD models directly for simulation or advanced geometric analysis, for example. Motivated by this need for watertight models, we address the problem of converting B-rep models to a collection of cubic C1 Clough–Tocher splines. These splines allow a watertight join between B-rep faces, provide a homogeneous representation of shape, and also support local adaptivity.
We perform a comparative study of the most prominent Clough–Tocher constructions and include some novel variants. Our criteria include visual fairness, invariance to affine reparameterisations, polynomial precision and approximation error. The constructions are tested on both synthetic data and CAD models that have been triangulated. Our results show that no construction is optimal in every scenario, with surface quality depending heavily on the triangulation and parameterisation that are used.
•Watertight conversion of boundary representations to Clough–Tocher splines.•Comparative study of the most prominent Clough–Tocher constructions and some novel variants.•Comparison based on visual fairness, invariance to affine reparameterisations, polynomial precision and approximation error.
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