Martin Peternell

A Laguerre geometric approach to rational offsets

Laguerre geometry provides a simple approach to the design of rational curves and surfaces with rational offsets. These so-called PH curves and PN surfaces can be constructed from arbitrary rational curves or surfaces with help of a geometric transformation which describes a change between two models of Laguerre geometry. Closely related to that is their optical interpretation as anticaustics of arbitrary rational curves/surfaces for parallel illumination. A theorem on rational parametrizations for envelopes of natural quadrics leads to algorithms for the computation of rational parametrizations of surfaces; those include canal surfaces with rational spine curve and rational radius function, offsets of rational ruled surfaces or quadrics, and surfaces generated by peripheral milling with a cylindrical or conical cutter. Laguerre geometry is also useful for the construction of PN surfaces with rational principal curvature lines. New families of such principal PN surfaces are determined.

Computing rational parametrizations of canal surfaces

A canal surface is the envelope of a one-parameter set of spheres with radii r(t) and centers m(t). It is shown that any canal surface to a rational spine curve m(t) and a rational radius function r(t) possesses rational parametrizations. We derive algorithms for the computation of these parametrizations and put particular emphasis on low degree representations. c ∞ 1997 Academic Press Limited

Applications of Laguerre geometry in CAGD

Abstract We briefly introduce to the basics of Laguerre geometry and then show that this classical sphere geometry can be applied to solve various problems in geometric design. In the present part, we focus on applications of the cyclographic model of Laguerre geometry and the cyclographic map. It relates the medial axis and Voronoi curves/surfaces to special surface/surface intersection and the corresponding trimming procedures to hidden line removal. Rational canal surfaces are treated as cyclographic images of rational curves in R 4. This leads to a simple control structure for rational canal surfaces. Its practical use is demonstrated at hand of modeling techniques with Dupin cyclides.

An introduction to line geometry with applications

Abstract The article presents a brief tutorial on classical line geometry and investigates new aspects of line geometry which arise in connection with a computational treatment. These mainly concern approximation and interpolation problems in the set of lines or line segments in Euclidean three-space. In particular, we study the approximation of data lines by, in a certain sense, ‘linear’ families of lines. These sets are, for instance linear complexes and linear congruences. An application is the reconstruction of helical surfaces or surfaces of revolution from scattered data points. This is based on the fact that the normals of these surfaces lie in linear complexes; in particular, normals of surfaces of revolution intersect the axis of revolution. Approximation with linear complexes or congruences is also useful in detecting singular positions of serial or parallel robots. These are positions where the robot should be a rigid system but possesses an undesirable and unexpected instantaneous self motion.

On the computational geometry of ruled surfaces

Abstract This article presents a brief introduction to the classical geometry of ruled surfaces with emphasis on the Klein image and studies aspects which arise in connection with a computational treatment of these surfaces. As ruled surfaces are one parameter families of lines, one can apply curve theory and algorithms to the Klein image, when handling these surfaces. We study representations of rational ruled surfaces and get efficient algorithms for computation of planar intersections and contour outlines. Further, low degree boundary curves, useful for tensor product representations, are studied and illustrated at hand of several examples. Finally, we show how to compute efficiently low degree rational G 1 ruled surfaces.

Developable surface fitting to point clouds

Given a set of data points as measurements from a developable surface, the present paper investigates the recognition and reconstruction of these objects. We investigate the set of estimated tangent planes of the data points and show that classical Laguerre geometry is a useful tool for recognition, classification and reconstruction of developable surfaces. These surfaces can be generated as envelopes of a one-parameter family of tangent planes. Finally we give examples and discuss the problems especially arising from the interpretation of a surface as set of tangent planes.

Rational surfaces with linear normals and their convolutions with rational surfaces

It is shown that polynomial (or rational) parametric surfaces with a linear field of normal vectors are dual to graphs bivariate polynomials (or rational functions). We discuss the geometric properties of these surfaces. In particular, using the dual representation it is shown that the convolution with general rational surfaces yields again rational surfaces. Similar results hold in the case of curves.

Geometric Properties of Bisector Surfaces

This paper studies algebraic and geometric properties of curve?curve, curve?surface, and surface?surface bisectors. The computation is in general difficult since the bisector is determined by solving a system of nonlinear equations. Geometric considerations will help us to determine several distinguished curve and surface pairs which possess elementary computable bisectors. Emphasis is on low-degree rational curves and surfaces, since they are of particular interest in surface modeling.

Reconstruction of piecewise planar objects from point clouds

Abstract This article discusses the reverse engineering problem of reconstructing objects with planar faces. We will present the main geometric features of a modeling system which are the detection of planar faces and the generation of a cad model. The algorithms are applied to the problem of reconstruction of buildings from airborne laser scanner data.

Approximation in Line Space — Applications in Robot Kinematics and Surface Reconstruction

Combining classical line geometry with techniques from numerical approximation, we develop algorithms for approximation in line space. In particular, linear complexes, linear congruences and reguli are fitted to given sets of lines or line segments. The results are applied to computationally robust detection of special robot configurations and to reconstruction of fundamental surface shapes from scattered points.