Josef Hoschek

Intrinsic parametrization for approximation

Abstract In parametric approximation an ordered set of points P i or P ik is given and the points are parametrized by parameter values ti for a curve approximation and by parameter values (ui, vk) for a surface approximation. The crucial point is to find intrinsic parameter values which lead to an optimal approximation curve or surface. In the present paper an iterative (Newton-like) approach of intrinsic parametrization is proposed.

An algebraic approach to curves and surfaces on the sphere and on other quadrics

Abstract An explicit representation for any irreducible rational Bezier curve and Bezier surface patch on the unit sphere is given. The extension to general quadrics (ellipsoids, hyperboloids, paraboloids) is outlined. The construction is based on an algebraic result concerning Pythagorean quadruples in polynomial rings and can be additionally interpreted as a generalized stereographic projection onto the sphere. This projection is shown to be the composition of a hyperbolic projection (a special net projection) with a stereographic projection. The investigation of its properties leads to new results for the biquadratic Bezier patch on the sphere. Further attention is payed to the interpolation of a given point set with a spherical rational curve. The results are extended to rational B-spline curves and tensor product B-spline surfaces.

Spline approximation of offset curves

Abstract This paper deals with the approximation of (regular) offset curves (of a given spline curve of degree n ) by a spline curve of arbitrarily chosen degree m . The approximating spline curves are determined by geometric continuity conditions and by parameter optimization for minimizing the range of the approximation error.

Approximate conversion of spline curves

In German car body industries (VDA) different manufacturers and their subcontractors have different geometric modeling systems for curve and surface representations. For exchanging data between the different geometric modeling systems conversions of curve and surface representation are required in order to compensate differences in the types of polynomial bases, maximum polynomial degrees and mesh sizes. Conversion means reducing the degree of a spline curve (and splitting more than one segment) or elevating the degree of more than one spline segment (and merging to one segment). For this purpose a set of methods was developed by DANNENBERG and NOWACKI [ 2]. They have extended a conversion method introduced by HOLZLE [ 8] for plane curves to surfaces by interpreting a surface as a net of curves. The extension of the algorithm to surfaces is implemented in the VDA-Software, but often a great number of new patches is obtained. So the question arises how to develop a method which yields to a more economical patch number. In the present paper a new conversion method for spline curves is introduced, which works yery effectively for plane spline curves. The method can be extended to approximate conversion of spline surfaces and to approximation of offset curves and offset surfaces by spline curves and spline surfaces (s. [6,7]).

Offset curves in the plane

Abstract For applications such as the generation of ornamental patterns for the numerical control of sewing machines in the textile industry or in the shoe industry or the numerical control of milling machines in the car body industry, offset curves must be generated from curves Di given by a designer. During the generation process further problems arise, for example finding the intersection points of neighbouring branches of the offset curves or deleting undesirable portions of the offset curves with cusps or with self-intersection points. In this paper methods are developed for attacking this problems.

GC 1 continuity conditions between adjacent rectangular and triangular Bézier surface patches

Abstract Necessary and sufficient conditions for geometric C1 continuity (GC1) that cover all the four combinations between rectangular and triangular Bezier surface patches are presented. Further, some more practical sufficient conditions are developed. The GC1 conditions is several special cases might be useful for CAD surface modelling.

Scattered Data Interpolation

Bei Anwendungen z.B. in der Geologie, Meteorologie, Kartographie, aber auch beim Digitalisieren von Modell Oberflachen, konnen unregelmasig verteilte Daten (scattered data) auftreten, die durch eine Flache interpoliert oder approximiert werden sollen.1) Das zu losende Interpolationsproblem lautet also: Gegeben sind N Abszissen xi = (xi,yi) ∈ ℝ2, i=1(1)N, mit zugehorigen Ordinaten (z.B. Meswerten) zi, gesucht ist eine Funktion f(x) = f (x,y) derart, das zi = (xi,yi) gilt. Das Approximationsproblem kann als (weighted oder auch als moving) least square Problem I(f) = Σ ωi (x) (f(xi, yi) - zi)2 → Min. behandelt werden, oder, was in jungster Zeit immer haufiger geschieht, als smoothing Problem I(f) = Σ ωi(x)(f(xi,yi) - zi)2 + λ J(f) → Min., mit Glattungsparameter λ und “physikalischem Term” J(f), z.B. der Biegeenergie einer eingespannten, elastischen dunnen Platte, etc. Wobei bei der scattered data Interpolation bzw. Approximation jedoch, im Gegensatz zur Aufgabenstellung der vorausgehenden Kapitel, keine speziellen Forderungen an die Datenpunkte (xi, yi, zi), insbesondere in Bezug auf Verteilungsanordnung und -dichte, gestellt werden. Wir wollen uns hier auf das Interpolationsproblem beschranken. Das Approximationsproblem wurde bereits in den Kap. 2.3, 4.4 und 6.2.5 angesprochen. Weiterhin sei auch verwiesen auf [Die 81], [Farw 86], [Fol 87b], [Fra 87], [Hay 74], [Hu 86], [Mcla 74, 76], [Mcm 87], [Lan 79], [LAN 86], [Schm 79, 83, 85], [Schu 76], sowie auf die Literaturliste [Fra 87a]; zum Smoothing s.a. Kap. 13.

Optimal approximate conversion of spline curves and spline approximation of offset curves

Abstract The paper introduces an effective method for approximate spline conversion. The method uses mainly parameter transformations and nonlinear optimization techniques. Geometric continuity conditions are used as parameter invariant spline conditions. For geometric continuity of order 1, 2, 3, 4, algorithms are introduced for approximate reducing of the polynomial degree of a given spline segment (and splitting into as few spline segments as possible) or elevating the polynomial degree (and merging as many spline segments as possible). The method is extended to spline approximation of offset curves (and splitting into as few new spline segements as possible).

Optimal approximate conversion of spline surfaces

Abstract The paper introduces an effective method for approximate conversion of spline surfaces. The method uses mainly geometric continuity conditions, parameter transformations and nonlinear optimization techniques. Degree reduction to bicubic and biquintic surfaces is introduced, curvature oriented segmentation is developed. The method can be extended to merging spline surfaces and to spline approximation of offset surfaces.

Rational patches on quadric surfaces

Abstract The paper discusses rational curve segments and surface patches on quadric surfaces. Detailed constructions of rational Bezier patches from given boundaries on a unit sphere and on a hyperbolic paraboloid are presented based on a generalization of the stereographic projection. The method is applied to interpolation with rational curves on quadrics. The results are extended to rational B-spline representations by discussion of products of B-spline functions. Finally, the generalization of the constructions to arbitrary nondegenerated quadric surfaces is outlined.