Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

Conchoid surfaces of spheres

The conchoid of a surface $F$ with respect to given fixed point $O$ is roughly speaking the surface obtained by increasing the radius function with respect to $O$ by a constant. This paper studies {\it conchoid surfaces of spheres} and shows that these surfaces admit rational parameterizations. Explicit parameterizations of these surfaces are constructed using the relations to pencils of quadrics in $\R^3$ and $\R^4$. Moreover we point to remarkable geometric properties of these surfaces and their construction

Generalized plane offsets and rational parameterizations

In the first part of the paper a planar generalization of offset curves is introduced and some properties are derived. In particular, it is seen that these curves exhibit good regularity properties and a study on self-intersection avoidance is performed. The representation of a rational curve as the envelope of its tangent lines, following the approach of Pottmann, is revisited to give the explicit expression of all rational generalized offsets. Other famous shapes, such as constant width curves, bicycle tire-tracks curves and Zindler curves are related to these generalized offsets. This gives rise to the second part of the paper, where the particular case of rational parameterizations by a support function is considered and explicit families of rational constant width curves, rational bicycle tire-track curves and rational Zindler curves are generated and some examples are shown

Exact Symbolic-Numeric Computation of Planar Algebraic Curves

We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional bivariate system, and an algorithm denoted GeoTop to analyze a single algebraic curve. Compared to existing approaches based on elimination techniques, we considerably improve the corresponding lifting steps in both subroutines. As a result, generic position of the input system is never assumed, and thus our algorithm never demands for any change of coordinates. In addition, we significantly limit the types of involved exact operations, that is, we only use resultant and gcd computations as purely symbolic operations. The latter results are achieved by combining techniques from different fields such as (modular) symbolic computation, numerical analysis and algebraic geometry. We have implemented our algorithms as prototypical contributions to the C++-project CGAL. They exploit graphics hardware to expedite the symbolic computations. We have also compared our implementation with the current reference implementations, that is, LGP and Maple's Isolate for polynomial system solving, and CGAL's bivariate algebraic kernel for analyses and arrangement computations of algebraic curves. For various series of challenging instances, our exhaustive experiments show that the new implementations outperform the existing ones.Comment: 46 pages, 4 figures, submitted to Special Issue of TCS on SNC 2011. arXiv admin note: substantial text overlap with arXiv:1010.1386 and arXiv:1103.469

Choosing roots of polynomials with symmetries smoothly

The roots of a smooth curve of hyperbolic polynomials may not in general be parameterized smoothly, even not $C^{1,\alpha}$ for any $\alpha > 0$. A sufficient condition for the existence of a smooth parameterization is that no two of the increasingly ordered continuous roots meet of infinite order. We give refined sufficient conditions for smooth solvability if the polynomials have certain symmetries. In general a $C^{3n}$ curve of hyperbolic polynomials of degree $n$ admits twice differentiable parameterizations of its roots. If the polynomials have certain symmetries we are able to weaken the assumptions in that statement.Comment: 19 pages, 2 figures, LaTe

Fibers of rational maps and elimination matrices: an application oriented approach

28 pages. Dedicated to David Eisenbud on the occasion of his seventy-fifth birthday.International audienceParameterized algebraic curves and surfaces are widely used in geometric modeling and their manipulation is an important task in the processing of geometric models. In particular, the determination of the intersection loci between points, pieces of parameterized algebraic curves and pieces of algebraic surfaces is a key problem in this context. In this paper, we survey recent methods based on syzygies and blowup algebras for computing the image and the finite fibers of a curve or surface parameterization, more generally of a rational map. Conceptually, the main idea is to use elimination matrices, mainly built from syzygies, as representations of rational maps and to extract geometric informations from them. The construction and main properties of these matrices are first reviewed and then illustrated through several settings, each of them highlighting a particular feature of this approach that combines tools from commutative algebra, algebraic geometric and elimination theory

Polygonization of Multi-Component Non-Manifold Implicit Surfaces through A Symbolic-Numerical Continuation Algorithm

In computer graphics, most algorithms for sampling implicit surfaces use a 2-points numerical method. If the surface-describing function evaluates positive at the first point and negative at the second one, we can say that the surface is located somewhere between them. Surfaces detected this way are called sign-variant implicit surfaces. However, 2-points numerical methods may fail to detect and sample the surface because the functions of many implicit surfaces evaluate either positive or negative everywhere around them. These surfaces are here called sign-invariant implicit surfaces. In this paper, instead of using a 2-points numerical method, we use a 1-point numerical method to guarantee that our algorithm detects and samples both sign-variant and sign-invariant surface components or branches correctly. This algorithm follows a continuation approach to tessellate implicit surfaces, so that it applies symbolic factorization to decompose the function expression into symbolic components, sampling then each symbolic function component separately. This ensures that our algorithm detects, samples, and triangulates most components of implicit surfaces

Representation and application of spline-based finite elements

Isogeometric analysis, as a generalization of the finite element method, employs spline methods to achieve the same representation for both geometric modeling and analysis purpose. Being one of possible tool in application to the isogeometric analysis, blending techniques provide strict locality and smoothness between elements. Motivated by these features, this thesis is devoted to the design and implementation of this alternative type of finite elements. This thesis combines topics in geometry, computer science and engineering. The research is mainly focused on the algorithmic aspects of the usage of the spline-based finite elements in the context of developing generalized methods for solving different model problems. The ability for conversion between different representations is significant for the modeling purpose. Methods for conversion between local and global representations are presented