Thomas J. Peters

Polyhedral perturbations that preserve topological form

Abstract The idea, that we are willing to accept variation in an object but that we insist it should retain its original topological form, has powerful intuitive appeal, and the concept appears in many applied fields. Some of the most important of these are tolerancing and metrology, solid modeling, engineering design, finite element analysis, surface reconstruction, computer graphics, path planning in robotics, fairing procedures, image analysis, and medical imaging. In this paper we focus on the field of tolerancing and metrology. The requirement that two objects or sets should have the same topological form requires a precise definition. We specify “same topological form” to mean that there exists a “space homeomorphism” from R 3 onto R 3 that carries a nominal object S onto another design object. In general, establishing the existence of such space homeomorphisms can be considerably more difficult than demonstrating classical topological equivalence by a homeomorphism. In the special case when the boundary of S is a polyhedral two-sphere in R 3 , one of the authors has previously given a simple sufficient condition for the existence of a space homeomorphism mapping S onto another design object. This paper presents an analogous sufficient condition for the case when S is a finite polyhedron in R 3 . The result relies upon a triangulation of the boundary and upon a dependent parameter that specifies the maximum size of permissible perturbations of the vertices of the polyhedron.

Encoding mechanical design features for recognition via neural nets

Within mechanical computer-aided design (CAD), pattern-recognition techniques are fundamental to feature recognition. The use of neural net software as the pattern-recognition element depends upon encoding schemes which extract critical information from candidate geometric subsets. The trained memory can then determine if a particular candidate geometric subset corresponds to a feature of interest. Successful experiments with particular encoding schemes over a restricted class of features will be presented. Neural nets were chosen with the long-term view toward a feature-recognition architecture where the end-user could customize the domain of features that can be recognized. The training of the neural net memory would be achieved through a graceful graphics interface. Extensive programming and knowledge bases would be avoided. This envisioned architecture will be presented to provide a context for the encoding schemes.

The role of topology in engineering design research

Aspects of the mathematical specialty of topology appear within several seemingly distinct areas of engineering design and engineering design theory. Indeed, the expression “topology of a design” is often used informally. In this article a primary intent is to demonstrate the diversity of applications of topology within engineering design. A complementary goal is to introduce the engineering design community to topology as a rich, formal, well-established mathematical discipline that may be of value for wider study. Upon reviewing some of these topological applications, it appears that topology holds promise as a basis for formalizing engineering design theory. This article considers topology as a basis for unifying design abstractions. The potential benefit may be the realization of commonalities between design aspects previously considered separately, where each now has its own attendant specialized, expensive analyses.

EQUIVALENCE OF TOPOLOGICAL FORM FOR CURVILINEAR GEOMETRIC OBJECTS

Given a curvilinear geometric object in R3, made up of properly-joined parametric patches defined in terms of control points, it is of interest to know under what conditions the object will retain its original topological form when the control points are perturbed. For example, the patches might be triangular BΘzier surface patches, and the geometric object may represent the boundary of a solid in a solid-modeling application. In this paper we give sufficient conditions guaranteeing that topological form is preserved by an ambient isotopy. The main conditions to be satisfied are that the original object should be continuously perturbed in a way that introduces no self-intersections of any patch, and such that the patches remain properly joined. The patches need only have C0 continuity along the boundaries joining adjacent patches. The results apply directly to most surface modeling schemes, and they are of interest in several areas of application.

Topological Neighborhoods for Spline Curves: Practice & Theory

The unresolved subtleties of floating point computations in geometric modeling become considerably more difficult in animations and scientific visualizations. Some emerging solutions based upon topological considerations for curves will be presented. A novel geometric seeding algorithm for Newtons method was used in experiments to determine feasible support for these visualization applications.

Ambient isotopic approximations for surface reconstruction and interval solids

Given a nonsingular compact 2-manifold <i>F</i> without boundary, we present methods for establishing a family of surfaces which can approximate <i>F</i> so that each approximant is ambient isotopic to <i>F</i>. The current state of the art in surface reconstruction is that both theory and practice are limited to generating a piecewise linear (PL) approximation. The methods presented here offer broader theoretical guidance for a rich class of ambient isotopic approximations. They are also used to establish sufficient conditions for an interval solid to be ambient isotopic to the solid it is approximating.The methods are based on <i>global</i> theoretical considerations and are compared to existing <i>local</i> methods. Practical implications of these methods are also presented. For the global case, a differential surface analysis is performed to find a positive number <i>ρ</i> so that the offsets <i>F_o(± ρ)</i> of <i>F</i> at distances <i>± ρ</i> are nonsingular. In doing so, a normal tubular neighborhood, <i>F(ρ)</i>, of <i>F</i> is constructed. Then, each approximant of <i>F</i> lies inside <i>F(ρ)</i>. Comparisons between these global and local constraints are given.

Preserving computational topology by subdivision of quadratic and cubic Bézier curves

Non-self-intersection is both a topological and a geometric property. It is known that non-self-intersecting regular Bézier curves have non-self-intersecting control polygons, after sufficiently many uniform subdivisions. Here a sufficient condition is given within ℝ3 for a non-self-intersecting, regular C2 cubic Bézier curve to be ambient isotopic to its control polygon formed after sufficiently many subdivisions. The benefit of using the control polygon as an approximant for scientific visualization is presented in this paper.

Isotopic approximations and interval solids

Given a nonsingular compact two-manifold F without boundary, we present methods for establishing a family of surfaces which can approximate F so that each approximant is ambient isotopic to F: The methods presented here offer broad theoretical guidance for a rich class of ambient isotopic approximations, for applications in graphics, animation and surface reconstruction. They are also used to establish sufficient conditions for an interval solid to be ambient isotopic to the solid it is approximating. Furthermore, the normals of the approximant are compared to the normals of the original surface, as these approximating normals play prominent roles in many graphics algorithms. The methods are based on global theoretical considerations and are compared to existing local methods. Practical implications of these methods are also presented. For the global case, a differential surface analysis is performed to find a positive number r so that the offsets Foð^rÞ of F at distances ^r are nonsingular. In doing so, a normal tubular neighborhood, FðrÞ; of F is constructed. Then, each approximant of F lies inside FðrÞ: Comparisons between these global and local constraints are given. q 2004 Elsevier Ltd. All rights reserved.

Topological properties that model feature-based representation conversions within concurrent engineering

One of the fundamental axioms of concurrent engineering is that undertaking functional design without foreseeing the manufacturing process leads to production delays and increased costs. This widely accepted concurrent engineering principle is given a formal basis by development of a mathematical model for the conversion of a feature-based design representation to a manufacturing representation. Within the domain of thin-walled components, it is shown that the conversion to tooling cost representations can result in a discontinuous function when the sets of design and manufacturing representations have been formulated as topological spaces. This discontinuity formally reflects the folklore that a small design change can significantly increase product cost. The mathematical sophistication required within this model is suggestive of why manufacturability evaluations can be quite difficult.

Modeling time and topology for animation and visualization with examples on parametric geometry

The art of animation relies upon modeling objects that change over time. A sequence of static images is displayed to produce an illusion of motion. Even for simple cases, a careful analysis exposes that formal topological guarantees are often lacking. This absence of rigor can result in subtle, but significant, topological flaws. A new modeling approach is proposed to integrate topological rigor with a continuous model of time. Examples will be given for Bezier curves, while indicating extensions to a richer class of parametric curves and surfaces. Applications to scientific visualization for molecular modeling are discussed. Prototype animations are available for viewing over the web.