Elmar Schömer

A Computational Basis for Conic Arcs and Boolean Operations on Conic Polygons

We give an exact geometry kernel for conic arcs, algorithms for exact computation with low-degree algebraic numbers, and an algorithm for computing the arrangement of conic arcs that immediately leads to a realization of regularized boolean operations on conic polygons. A conic polygon, or polygon for short, is anything that can be obtained from linear or conic halfspaces (= the set of points where a linear or quadratic function is non-negative) by regularized boolean operations. The algorithm and its implementation are complete (they can handle all cases), exact (they give the mathematically correct result), and efficient (they can handle inputs with several hundred primitives).

A constraint-based approach to rigid body dynamics for virtual reality applications

The GALILEO-system is a developmental state-of-the-art rigid body simulation tool with a strong bias to the simulation of unilateral contacts for virtual reality applications. On the one hand the system is aimed at closing the gap between the ‘paradigms of impulse-based simulation’ and of ‘constraint-based simulation’. On the other hand the chosen simulation techniques enable a balancing of the trade-off between the real-time demands of virtual environments (i.e. 15-25 visualizations per second) and the degree of physical correctness of the simulation. The focus of this paper lies on the constraint-based simulation approach to threedimensional multibody systems including a scalable friction model. This is only one of the two main components of the GALILEO-software-module. A nonlinear complementarity problem (NCP) describes the equations of motion, the contact conditions of the objects and the Coulomb friction model. Further on we show, as an interesting evaluation example from the field of ‘classical mechanics’, the first rigid body simulation of the tippe-top.

An exact and efficient approach for computing a cell in an arrangement of quadrics

We present an approach for the exact and efficient computation of a cell in an arrangement of quadric surfaces. All calculations are based on exact rational algebraic methods and provide the correct mathematical results in all, even degenerate, cases. By projection, the spatial problem is reduced to the one of computing planar arrangements of algebraic curves. We succeed in locating all event points in these arrangements, including tangential intersections and singular points. By introducing an additional curve, which we call the Jacobi curve, we are able to find non-singular tangential intersections. We show that the coordinates of the singular points in our special projected planar arrangements are roots of quadratic polynomials. The coefficients of these polynomials are usually rational and contain at most a single square root. A prototypical implementation indicates that our approach leads to good performance in practice.

Efficient collision detection for moving polyhedra

In this paper we consider the following problem: given two general polyhedra of complexity n, one of which is moving translationally or rotating about a fixed axis, determine the first collision (if any) between them. We present an algorithm with running time O(n8/5+’) for the case of translational movements and running time qn5/3+f f ) or rotational movements, where c is an arbitrary positive constant. This is the first known algorithm with sub-quadratic running time.

An exact, complete and efficient implementation for computing planar maps of quadric intersection curves

We present the first exact, complete and efficient implementation that computes for a given set P=p1,...,pn of quadric surfaces the planar map induced by all intersection curves p1∩ pi, 2 ≤ i ≤ n, running on the surface of p1. The vertices in this graph are the singular and x-extreme points of the curves as well as all intersection points of pairs of curves. Two vertices are connected by an edge if the underlying points are connected by a branch of one of the curves. Our work is based on and extends ideas developed in [20] and [9].Our implementation is complete in the sense that it can handle all kind of inputs including all degenerate ones where intersection curves have singularities or pairs of curves intersect with high multiplicity. It is exact in that it always computes the mathematical correct result. It is efficient measured in running times.

Efficient distance computation for quadratic curves and surfaces

Virtual prototyping and assembly planning require physically based simulation techniques. In this setting the relevant objects are mostly mechanical parts, designed in CAD-programs. When exported to the prototyping and planning systems, curved parts are approximated by large polygonal models, thus confronting the simulation algorithms with high complexity Algorithms for collision detection in particular are a bottleneck of efficiency and suffer from accuracy and robustness problems. To overcome these problems, our algorithm directly operates on the original CAD-data. This approach reduces the input complexity and avoids accuracy problems due to approximation errors. We present an efficient algorithm for computing the distance between patches of quadratic surfaces trimmed by quadratic curves. The distance calculation problem is reduced to the problem of solving univariate polynomials of a degree of at most 24. Moreover, we identify an important subclass for which the degree of the polynomials is bounded by 8.

Computing a 3-dimensional cell in an arrangement of quadrics: exactly and actually!

We present two approaches to the problem of calculating a cell in a 3- dimensional arrangement of quadrics. The first approach solves the problem using rational arithmetic. It works with reductions to planar arrangements of algebraic curves. Degenerate situations such as tangential intersections and self-intersections of curves are intrinsic to the planar arrangements we obtain. The coordinates of the intersection points are given by the roots of univariate polynomials. We succeed in locating all intersection points either by extended local box hit counting arguments or by globally characterizing them with simple square root expressions. The latter is realized by a clever factorization of the univariate polynomials. Only the combination of these two results facilitates a practical and implementable algorithm. The second approach operates directly in 3-space by applying classical solid modeling techniques. Whereas the first approach guarantees a correct solution in every case the second one may fail in some degenerate situations. But with the help of verified floating point arithmetic it can detect these critical cases and is faster if the quadrics are in general position.

Interactive simulation of one-dimensional flexible parts

Computer simulations play an ever growing role for the development of automotive products. Assembly simulation, as well as many other processes, are used systematically even before the first physical prototype of a vehicle is built in order to check whether particular components can be assembled easily or whether another part is in the way. Usually, this kind of simulation is limited to rigid bodies. However, a vehicle contains a multitude of flexible parts of various types: cables, hoses, carpets, seat surfaces, insulations, weatherstrips... Since most of the problems using these simulations concern one-dimensional components and since an intuitive tool for cable routing is still needed, we have chosen to concentrate on this category, which includes cables, hoses and wiring harnesses.This paper presents an interactive, real-time, numerically stable and physically accurate simulation tool for one-dimensional components. The modeling of bending and torsion follows the Cosserat model and is implemented with a generalized spring-mass system with a mixed coordinate system which features usual space coordinates for the positions of the points and quaternions for the orientation of the segments joining them. This structure allows us to formulate the springs based on the coordinates that are most appropriate for each type of interaction and leads to a banded system that is then solved iteratively with an energy minimizing algorithm.

EXACUS: efficient and exact algorithms for curves and surfaces

We present the first release of the Exacus C++ libraries. We aim for systematic support of non-linear geometry in software libraries. Our goals are efficiency, correctness, completeness, clarity of the design, modularity, flexibility, and ease of use. We present the generic design and structure of the libraries, which currently compute arrangements of curves and curve segments of low algebraic degree, and boolean operations on polygons bounded by such segments.

Complete, exact, and efficient computations with cubic curves

The Bentley-Ottmann sweep-line method can be used to compute thearrangement of planar curves provided a number of geometricprimitives operating on the curves are available. We discuss themathematics of the primitives for planar algebraic curves of degreethree or less and derive efficient realizations. As a result, weobtain a complete, exact, and efficient algorithm for computingarrangements of cubic curves. Conics and cubic splines are specialcases of cubic curves. The algorithm is complete in that it handles all possibledegeneracies including singularities. It is exact in that itprovides the mathematically correct result. It is efficient in thatit can handle hundreds of curves with a quarter million of segmentsin the final arrangement.