Evan C. Sherbrooke

Computation of the solutions of nonlinear polynomial systems

Abstract: A fundamental problem in computer aided design is the efficient computation of all roots of a system of nonlinear polynomial equations inn variables which lie within ann-dimensional @?. We present two techniques designed to solve such problems, which rely on representation of polynomials in the multivariate Bernstein basis and subdivision. In order to isolate all of the roots within the given domain, each method uses a different scheme for constructing a series of bounding @?es; the first method projects control polyhedra onto a set of coordinate planes, and the second employs linear optimization. We also examine in detail the local convergence properties of the two methods, proving that the former is quadratically convergent forn=1 and linearly convergent forn 1, while the latter is quadratically convergent for alln. Worst-case complexity analysis, as well as analysis of actual running times are performed.

An algorithm for the medial axis transform of 3D polyhedral solids

The medial axis transform (MAT) is a representation of an object which has been shown to be useful in design, interrogation, animation, finite element mesh generation, performance analysis, manufacturing simulation, path planning and tolerance specification. In this paper, an algorithm for determining the MAT is developed for general 3D polyhedral solids of arbitrary genus without cavities, with nonconvex vertices and edges. The algorithm is based on a classification scheme which relates different pieces of the medial axis (MA) to one another, even in the presence of degenerate MA points. Vertices of the MA are connected to one another by tracing along adjacent edges, and finally the faces of the axis are found by traversing closed loops of vertices and edges. Representation of the MA and its associated radius function is addressed, and pseudocode for the algorithm is given along with recommended optimizations. A connectivity theorem is proven to show the completeness of the algorithm. Complexity estimates and stability analysis for the algorithms are presented. Finally, examples illustrate the computational properties of the algorithm for convex and nonconvex 3D polyhedral solids with polyhedral holes.

Differential and topological properties of medial axis transforms

Abstract The medial axis transform is a representation of an object which has been shown to be useful in design, interrogation, animation, finite element mesh generation, performance analysis, manufacturing simulation, path planning, and tolerance specification. In this paper, the theory of the medial axis transform for 3-D objects is developed. For objects with piecewise C 2 boundaries, relationships between the curvature of the boundary and the position of the medial axis are developed. For n -dimensional submanifolds of R n with boundaries which are piecewise C 2 and completely G 1 , a deformation retract is set up between each object and its medial axis, which demonstrates that if the object is path connected, then so is its medial axis. Finally, it is proven that path connected polyhedral solids without cavities have path connected medial axes.

Computation of the Medial Axis Transform of 3-D polyhedra

The Medial Axis Transform is important in path planning, analysis of growth, determination of symmetries, and tinite element mesh generation, In [his paper, an algorithm forcomputing the Medial Axis Tranformof a 3-D polyhedral solid is presented, along with adiscussion of its complexity and stability. lle algorithm is based on a classification scheme for points on the Medial Axis which is also discussed. The algorithm provides the continuous representation of the Medial Axis andassociated radius function. Examples of the algorithm are also presented to illustrate the method.

Computation of stationary points of distance functions

This paper presents an algorithm for computation of the stationary points of the squared distance functions between two point sets. One point set consists of a single space point, a rational B-spline curve, or a rational B-spline surface. The problem is reformulated in terms of solution of n polynomial equations with n variables expressed in the tensor product Bernstein basis. The solution method is based on subdivision relying on the convex hull property of the n-dimensional Bernstein basis and minimization techniques. We also cover classification of the stationary points of these distance functions, and include a method for tracing curves of stationary points in case the solution set is not zerodimensional. The distance computation problem is shown to be equivalent to the geometrically intuitive problem of computing collinear normal points. Finally, examples illustrate the applicability of the method

Efficient and reliable methods for rounded-interval arithmetic

We present an efficient and reliable method for computing the unit-in-the-last-place (ulp) of a double-precision floating-point number, taking advantage of the standard binary representation for floatingpoint numbers defined by IEEE Std 754-1985. The ulp is necessary to perform software rounding for robust rounded-interval arithmetic (RIA) operations. Hardware rounding, using two of the standard rounding modes defined by IEEE-754, may be more efficient. RIA has been used to produce robust software systems for the solution of systems of nonlinear equations, interrogation of geometric and differential properties of curves and surfaces, curve and surface intersections, and solid modeling.

Solving nonlinear polynomial systems in the barycentric Bernstein basis

We present a method for solving arbitrary systems of N nonlinear polynomials in n variables over an n-dimensional simplicial domain based on polynomial representation in the barycentric Bernstein basis and subdivision. The roots are approximated to arbitrary precision by iteratively constructing a series of smaller bounding simplices. We use geometric subdivision to isolate multiple roots within a simplex. An algorithm implementing this method in rounded interval arithmetic is described and analyzed. We find that when the total order of polynomials is close to the maximum order of each variable, an iteration of this solver algorithm is asymptotically more efficient than the corresponding step in a similar algorithm which relies on polynomial representation in the tensor product Bernstein basis. We also discuss various implementation issues and identify topics for further study.

Computation of Singularities for Engineering Design

The computation of singularities or critical points of polynomial and other more complex vector fields in a finite subdomain of the n-dimensional Euclidean space is the underlying fundamental process behind several important engineering and scientific problems. These include, for example, design, analysis, scientific visualization, and manufacture of complex objects in a computer environment. This paper starts with a review of extant solution techniques and focuses on recent research by the Design Laboratory in this general area. Specifically, we summarize the algorithmic techniques we have developed on computation of solutions of systems of non-linear polynomial equations and other more complex equations involving irrational functions. Such equations arise in shape interrogation problems including intersections of sculptured objects, symmetry transforms, distance function computations, visualization of rational and offset or parallel surfaces, stationary point computations of maps of physical properties, and in detailed analysis of differential geometry properties of complex free-form surfaces. Examples illustrate our techniques and their applications.