Watertight conversion of trimmed CAD surfaces to Clough-Tocher splines

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 C1C1 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.This research was supported by the Engineering and Physical Sciences Research Council through Grant EP/K503757/1.This is the final version. It was first published by Elsevier at http://www.sciencedirect.com/science/article/pii/S0167839615000795

Conversion of B-rep CAD models into globally G¹ triangular splines

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

Using Least Squares to Construct Improved Clough-Tocher Interpolant

In this thesis, a quartic Clough-Tocher interpolation scheme is introduced, and additional modifications, to adjust the macro-boundary and the order of continuity across domain triangles, are provided to improve both the mathematical and the visual quality of the resulting surface. Furthermore, a proof is given to show the convergence of the interpolation scheme under some specific constraints

On numerical quadrature for C1C^1 quadratic Powell-Sabin 6-split macro-triangles

The quadrature rule of Hammer and Stroud [16] for cubic polynomials has been shown to be exact for a larger space of functions, namely the $C^1$ cubic Clough-Tocher spline space over a macro-triangle if and only if the split-point is the barycentre of the macro-triangle [21]. We continue the study of quadrature rules for spline spaces over macro-triangles, now focusing on the case of $C^1$ quadratic Powell-Sabin 6-split macro-triangles. We show that the $3$-node Gaussian quadrature(s) for quadratics can be generalised to the $C^1$ quadratic Powell-Sabin 6-split spline space over a macro-triangle for a two-parameter family of inner split-points, not just the barycentre as in [21]. The choice of the inner split-point uniquely determines the positions of the edge split-points such that the whole spline space is integrated exactly by a corresponding polynomial quadrature. Consequently, the number of quadrature points needed to exactly integrate this special spline space reduces from twelve to three. For the inner split-point at the barycentre, we prove that the two 3-node quadratic polynomial quadratures of Hammer and Stroud exactly integrate also the $C^1$ quadratic Powell-Sabin spline space if and only if the edge split-points are at their respective edge midpoints. For other positions of the inner and edge split-points we provide numerical examples showing that three nodes suffice to integrate the space exactly, but a full classification and a closed-form solution in the generic case remain elusive

High-order adaptive methods for computing invariant manifolds of maps

The author presents efficient and accurate numerical methods for computing invariant manifolds of maps which arise in the study of dynamical systems. In order to decrease the number of points needed to compute a given curve/surface, he proposes using higher-order interpolation/approximation techniques from geometric modeling. He uses B´ezier curves/triangles, fundamental objects in curve/surface design, to create adaptive methods. The methods are based on tolerance conditions derived from properties of B´ezier curves/triangles. The author develops and tests the methods for an ordinary parametric curve; then he adapts these methods to invariant manifolds of planar maps. Next, he develops and tests the method for parametric surfaces and then he adapts this method to invariant manifolds of three-dimensional maps

Scattered data interpolation methods for electronic imaging systems: a survey

Numerous problems in electronic imaging systems involve the need to interpolate from irregularly spaced data. One example is the calibration of color input/output devices with respect to a common intermediate objective color space, such as XYZ or L*a*b*. In the present report we survey some of the most important methods of scattered data interpolation in two-dimensional and in three-dimensional spaces. We review both single-valued cases, where the underlying function has the form f:R2→R or f:R3→R, and multivalued cases, where the underlying function is f:R2→R2 or f:R3→R3. The main methods we review include linear triangular (or tetrahedral) interpolation, cubic triangular (Clough-Tocher) interpolation, triangle based blending interpolation, inverse distance weighted methods, radial basis function methods, and natural neighbor interpolation methods. We also review one method of scattered data fitting, as an illustration to the basic differences between scattered data interpolation and scattered data fittin

Triangular Bézier Surfaces with Approximate Continuity

When interpolating a data mesh using triangular Bézier patches, the requirement of C¹ or G¹ continuity imposes strict constraints on the control points of adjacent patches. However, fulfillment of these continuity constraints cannot guarantee that the resulting surfaces have good shape. This thesis presents an approach to constructing surfaces with approximate C¹/G¹ continuity, where a small amount of discontinuity is allowed between surface normals of adjacent patches. For all the schemes presented in this thesis, although the resulting surface has C¹/G¹ continuity at the data vertices, I only require approximate C¹/G¹ continuity along data triangle boundaries so as to lower the patch degree. For functional data, a cubic interpolating scheme with approximate C¹ continuity is presented. In this scheme, one cubic patch will be constructed for each data triangle and upper bounds are provided for the normal discontinuity across patch boundaries. For a triangular mesh of arbitrary topology, two interpolating parametric schemes are devised. For each data triangle, the first scheme performs a domain split and constructs three cubic micro-patches; the second scheme constructs one quintic patch for each data triangle. To reduce the normal discontinuity, neighboring patches across data triangle boundaries are adjusted to have identical normals at the middle point of the common boundary. The upper bounds for the normal discontinuity between two parametric patches are also derived for the resulting approximate G¹ surface. In most cases, the resulting surfaces with approximate continuity have the same level of visual smoothness and in some cases better shape quality

Investigation of smoothness-increasing accuracy-conserving filters for improving streamline integration through discontinuous fields

Journal ArticleStreamline integration of fields produced by computational fluid mechanics simulations is a commonly used tool for the investigation and analysis of fluid flow phenomena. Integration is often accomplished through the application of ordinary differential equation (ODE) integrators - integrators whose error characteristics are predicated on the smoothness of the field through which the streamline is being integrated, which is not available at the interelement level of finite volume and finite element data. Adaptive error control techniques are often used to ameliorate the challenge posed by interelement discontinuities

Multisided generalisations of Gregory patches

We propose two generalisations of Gregory patches to faces of any valency by using generalised barycentric coordinates in combination with two kinds of multisided Bézier patches. Our first construction builds on S-patches to generalise triangular Gregory patches. The local construction of Chiyokura and Kimura providing G1 continuity between adjoining Bézier patches is generalised so that the novel Gregory S-patches of any valency can be smoothly joined to one another. Our second construction makes a minor adjustment to the generalised Bézier patch structure to allow for cross-boundary derivatives to be defined independently per side. We show that the corresponding blending functions have the inherent ability to blend ribbon data much like the rational blending functions of Gregory patches. Both constructions take as input a polygonal mesh with vertex normals and provide G1 surfaces interpolating the input vertices and normals. Due to the full locality of the methods, they are well suited for geometric modelling as well as computer graphics applications relying on hardware tessellation