Solid modeling is rapidly emerging as a central area of research and de- velopment in such diverse applications as engineering and product design, computer-aided manufacturing, electronic prototyping, o -line robot pro- gramming, and motion planning. All these applications require represent- ing the shapes of solid physical objects, and such representations and basic operations on them can be provided by solid modeling. As a eld, solid modeling spans several disciplines, including mathemat- ics, computer science, and engineering. In consequence, it is a broad subject that must accommodate a diversity of viewpoints and has to meet a diversity of goals. Sometimes, this diversity of goals can lead to con icting demands. Current thinking on the subject views the proper resolution of these con- icts to be application-dependent. That is, it is no longer thought realistic to envision a comprehensive solid-modeling system that satis es the needs of all potential users. Rather, as it is argued, we should concentrate on con- structing a software environment in which many tools for geometric and solid computation are available and can be combined with ease as appropriate for the speci c application under consideration.

Geometric and Solid Modeling

A polyhedron can model irregularly shaped objects such as asteroids, comet nuclei, and small planetary satellites. With minor effort, such a model can incorporate important surface features such as large craters. Here we develop closed-form expressions for the exterior gravitational potential and acceleration components due to a constant-density polyhedron. An equipotential surface of Phobos is illustrated.

The gravitational potential of a homogeneous polyhedron or don't cut corners

The volume, moments of inertia, and similar properties of solids are defined by triple (volumetric) integrals over subsets of three-dimensional Euclidean space. The automatic computation of such integral properties for geometrically complex solids is important in CAD/CAM, robotics, and other fields and raises interesting mathematical and computational problems that have received little attention from numerical analysts and computer scientists. This paper summarizes the known methods for calculating integral properties of solids that may be geometrically complex and identifies some significant gaps in our current knowledge.

Algorithms for computing the volume and other integral properties of solids. I. known methods and open issues

We construct efficient quadratures for the integration of polynomials over irregular convex polygons and polyhedrons based on moment fitting equations. The quadrature construction scheme involves the integration of monomial basis functions, which is performed using homogeneous quadratures with minimal number of integration points, and the solution of a small linear system of equations. The construction of homogeneous quadratures is based on Lasserre's method for the integration of homogeneous functions over convex polytopes. We also construct quadratures for the integration of discontinuous functions without the need to partition the domain into triangles or tetrahedrons. Several examples in two and three dimensions are presented that demonstrate the accuracy and versatility of the proposed method.

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Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons

An introduction to solid modelling

A computational scheme for determining global geometric properties of solid object models is presented. The method operates directly on the boundary representation of the model. The scheme is tested on a number of models produced by an experimental modelling system. Primitive objects combined for the tests are all represented in terms of parametric bicubic patches.

Computation of global geometric properties of solid objects

A representation for parametric cubic curves and surfaces is presented which incorporates the polygonal approach popularized by the Bezier and B-spline schemes. Since the curves more closely mimic the polygon than their counterparts employing the Bezier or B-spline schemes, the method is potentially useful for creating and manipulating geometric models. In connection with the logical extension of the approach from curves to surfaces, a discussion of the relationship between continuity and redundant data storage is included.

Henry G Timmer

Papers

Computation of global geometric properties of solid objects

Alternative representation for parametric cubic curves and surfaces