Watertight conversion of trimmed CAD surfaces to Clough–Tocher splines

Abstract

The boundary representations (B-reps) that are used to represent shape in ComputerAided 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 C 1 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.

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

“…The dimension of S 1 2 (T ) is 9, which can be determined, for instance, using the concept of minimal determining sets [24]. A discussion on choosing the split-point S can be found in [22,32]. Although various options exist, the split-point is typically placed at the barycentre of T .…”

“…The positions of the quadrature points t i are determined by t i , the barycentric coordinate on the micro-edge SV i , i.e., 1,1 , p 1,0,1 , p 1,1,0 ) are shown. The error values, see (22), between the exact integrals and the results of the quadrature rule (red/green bullets; Table 1, top) are displayed.…”

Gaussian quadrature for C1 cubic Clough–Tocher macro-triangles

“…It is known that dim(S 1 3 (T )) = 12. A discussion on choosing the split point S can be found in [20,28]. Although various options exist, the split point is typically placed at the barycentre of T .…”

“…For later use, we now recall C 0 , C 1 , and C 2 continuity conditions between Bézier cubic triangles [20,23,29]. Let p i be a cubic Bézier triangle defined on T i , i = 0, 1, 2.…”

A survey on 3D CAD model quality assurance and testing tools

“…First, each shape requires its own particular algorithm to simulate its behavior, and second, this information is unavailable in models generated from point clouds [30] or sets of scanned images [31]. In other words, there is a need to convert the trimmed surfaces of B-reps into a form that removes the gaps between adjacent faces and allows a homogeneous representation of shape [32]. In these cases, the best solution involves discretizing the model into a mesh of smaller elementary mesh cells of mesh elements (usually triangles) that approximates a geometric domain [33].…”