Geometric Modelling

Frontmatter

Applications of a Fruitful Relation between a Dupin Cyclide and a Right Circular Cone

Abstract

A powerful way of handling a Dupin cyclide is presented. It is based on the concept of inversion, which yields a fruitful relation between the symmetric Dupin horn cyclide, from which all other Dupin cyclides may be obtained by offsetting, and a right circular cone. This relation has two important applications. First, it is used for constructing rational rectangular and triangular Bézier patches on the cyclide. Second, it allows to establish an approximative isometry between cyclide and cone patches, a useful result e.g. for scattered data interpolation techniques on Dupin cyclides.

G. Albrecht

Interrogative Visualization

Abstract

We present an environment for interrogative visualization of large datasets such as from the Visible Human project. Components of the environment have been developed to support rapid computation, rendering, and querying of large amounts of data. A cluster of graphics workstations connected by ATM to an Intel Paragon provides the computational and display power required to support visualization in an interactive sense. We discuss the three components of interrogative visualization and how they are addressed in our design.

C. L. Bajaj

Representation Conversions for Nef Polyhedra

Abstract

A Nef polyhedron is any set in ℝ^d which can be obtained by applying a finite number of Boolean set operations cpl and ∩ to finitely many (open) linear halfspaces. After resuming some fundamentals, it is shown in which sense several kinds of well-known polyhedra are special cases of Nef polyhedra. Then a number of representations of Nef polyhedra are presented and discussed, and algorithms for converting them into each other are given.

Hanspeter Bieri

The Shape Parametrisation of an Aircraft Engine Nacelle and Pylon

Abstract

The PDE Method generates a surface patch using a boundary-value approach. We describe how complicated surface shapes may be constructed by joining together a collection of PDE patches, either by joining adjacent patches at mutual boundaries or else by cutting a ‘hole’ in an existing surface, removing the patch of the surface which lies inside the hole, and using boundary conditions imposed at the edges of the hole to generate a second surface which meets the first at the hole boundaries. The particular object we use to illustrate the technique is an aircraft engine nacelle and pylon.

M. I. G. Bloor, M. J. Wilson

Multiresolution Analysis with Non-Nested Spaces

Abstract

Two multiresolution analyses (MRA) intended to be used in scientific visualization, and that are both based on a non-nested set of approximating spaces, are presented. The notion of approximated refinement is introduced to deal with non nested spaces. The first MRA scheme, referred to as BLaC (Blending of Linear and Constant) wavelets is based on a one parameter family of wavelet bases that realizes a blend between the Haar and the linear wavelet bases. The approximated refinement is applied in the last part to build a second MRA scheme for data defined on an arbitrary planar triangular mesh.

G. P. Bonneau

Optimal Degree Reduction of Free Form Curves

Abstract

Optimal degree reductions, i.e. best approximations of n-th degree Bezier curves by Bezier curves of degree n - 1, with respect to different norms are studied. It is shown that for any L_p-norm the Euclidean degree reduction where the norm is applied to the Euclidean distance function of two curves is identical to component-wise degree reduction. The Bezier points of the degree reductions are found to lie on parallel lines through the Bezier points of any Taylor expansion of degree n - 1 of the original curve. The Bezier points of the degree reduction are explicitly given p = 1 and p = 2.

G. Brunnett, T. Schreiber

Intersection Methods of Convergence

Abstract

This article describes a numerical method developed in the 60’s at Citroën in order to compute intersections between conical sections.

P. de Casteljau

Approximation of Curves by a Measure of Shape

Abstract

Based on the support function, a well-known tool in the theory of convex sets, a new measure of a “distance” between two planar parametric curves is introduced and its main properties are established. The measure is invariant under affine transformations and depends not only on the position vectors but also on the tangent vectors of the curves. An algorithm to compute the optimal Bézier approximant to a given curve is derived.

W. L. F. Degen

Geometric Design of Rational Bézier Line Congruences and Ruled Surfaces Using Line Geometry

Abstract

This paper presents a method for constructing rational Bézier line congruences and ruled surfaces suitable for Computer Aided Geometric Design based on line geometry. Directed lines in the Euclidean three-space are represented by vectors with three homogeneous components over the ring of dual numbers. A projective deCasteljau algorithm is presented for construction of rational Bézier line congruences and ruled surfaces. In the case of ruled surface patches an intrinsic representation of a ruled patch based on segmentation of the ruling about the striction curve of the surface is presented. This leads to a coordinate independent representation of such surface patches. An algorithm is presented that would allow geometric construction of the striction points on rulings of a ruled surface at each intermediate step in the deCasteljau’s algorithm. This provides for a geometric method for constructing a coordinate independent representation of a rational ruled surface patch which can be used for geometric design purposes.

Q. J. Ge, B. Ravani

Variational Design and Parameter Optimized Surface Fitting

Abstract

Computer Aided Geometric Design has emerged from the need for freeform surfaces in CAD/CAM technologies; it has become a major research topic in computer science with direct applications for all engineering sciences. The main task of the so-called variational design is to determine smooth curves or surfaces, which minimizes certain functionals and fulfill special constraints. First we interpret one widely used functional in the sense of physics. This approach naturally extends to a physical-based modification of surfaces. As an optimization step we include the parametrization as an additional parameter in the variational design process.

H. Hagen, A. Nawotki

Shape Improvement of Surfaces

Abstract

An automatic and local fairing algorithm for bicubic B-spline surfaces is proposed. A local fairness criterion selects the knot, where the spline surface has to be faired. A fairing step is then applied, which locally modifies the control net by a constrained least-squares approximation. It consists of increasing locally the smoothness of the surface from C ² to C ³. Some extensions of this method are also presented, which show how to build further methods by the same basic fairing principle.

S. Hahmann

A Philosophy for Smooth Contour Reconstruction

Abstract

We propose an approach to the reconstruction of point samples into smooth models based upon the generation of isocurves, with concentration on contour data. We show how to build an isocurve through any point and how to choose which isocurves are computed. The isocurves are used to build a tensor product surface for each component of the model isomorphic to a cylinder.

J. K. Johnstone, K. R. Sloan

The Design of Physically Accurate Fluid Flow

Abstract

We first review work in which a bivariate parametric polynomial is derived that maps one curve onto another and deforms regions about the curves conformally. The curves are defined in Bézier form. In this paper the approximation is extended to B-Spline curves that generate near-conformal maps. It is then applied to approximating and animating ideal fluid flows. The B-spline curve underpins the design approach. With it an interface is developed to design fluid flow applications which also incorporates potential field theory. We give parameter maps that set up the flow from the confomally designed regions. Examples of animated fluid flow include designed channel flows with obstacles.

Z. Kadi, A. Rockwood

A Quadratic-Programming Method for Removing Shape-Failures from Tensor-Product B-Spline Surfaces

Abstract

We first study the effect caused, on the local shape of a tensor-product B-spline surface, by the movement of a subset of its control net. We then propose two (2) discrete approaches for removing shape failures from such surfaces, without altering them more than is needed. The second approach is a simple Quadratic-Programming method, that is suitable for restoring the shape of almost shape-preserving tensor-product B-spline surfaces. The performance of this method is tested and discussed for three industrial surfaces.

P. D. Kaklis, G. D. Koras

Fill-in Regions

Abstract

We approach the problem of three-dimensional shape preserving interpolation by defining fill-in regions that are regions in space which contain all interpolating curves with a desired shape. These regions are defined for special types of curves called coils, which are non-strict versions of curves of geometric order three in three dimensions. We establish that the fill-in regions are a set of tetrahedra and show how they must be restricted to form interpolating curves.

C. Labenski, B. Piper

Smoothing Polyhedra Using Trimmed Bicubic Patches

Abstract

Several efficient constructions of smooth surfaces generate three-sided patches. To demonstrate that these constructions are compatible with existing software based on tensor-product patches, the particular scheme in [3] is expressed in terms of linearly-trimmed bicubic patches. Explicit formulas relating the coefficients of the patches to the vertices of an arbitrary input polyhedron are given. Four of these patches can be grouped together into a NURBS surface.

J. Peters

A G 2-Subdivision Algorithm

Abstract

In this paper we present a method to optimize the smoothness order of subdivision algorithms generating surfaces of arbitrary topology. In particular we construct a subdivision algorithm with some negative weights producing G ²-surfaces. These surfaces are piecewise bicubic and are flat at their extraordinary points. The underlying ideas can also be used to improve the smoothness order of subdivision algorithms for surfaces of higher degree or triangular nets.

H. Prautzsch, G. Umlauf

Octtrees Meet Splines

Abstract

In this paper, the smooth isosurface generation from a set of voxels representing either a volumetric model or a discretization of a given solid is addressed. It is assumed that a certain isosurface value has been already given and that the set of voxels stabbing this value has been extracted from the volumetric model. The goal is to obtain an smooth, C ², piecewise algebraic surface representing the isosurface. The proposed approach adds one dimension to the problem and works by considering a scalar function with domain coinciding with the octtree (or voxels) universe. An iterative local filtering operator is derived and discussed through several examples.

A. Vinacua, I. Navazo, P. Brunet

Degree Reduction of B-Spline Curves

Abstract

This work deals with the problem of degree reducing a B-spline curve without intermediate conversion to Bézier form. Degree elevation for B-spline curves may easily be described in blossom form. We also use blossoms for the degree reduction problem, and set up least squares problems for finding the B-spline coefficients directly. We also give an application to curve fairing.

H. J. Wolters, G. Wu, G. Farin

Backmatter