Detailed analysis of many mathematical properties of sculptured models has been hindered by the fact that the properties do not have the same representation as the surface. For example, unit tangents, surface normals, and principal curvatures are typically computed at predefined discrete sets of points on the surface. As such, aliasing can occur and features between samples can be missed. Synthesizing information about the shape of an object and operating on the model, whether by physical machining tools, graphics display programs, or mathematical analysis, has been treated as either a discrete or local problem in general. The researchbeing reported on here has focused on another approach, that of creating algorithms that construct the mathematical properties in closed form, or construct approximations to those mathematical properties through symbolic computation. Global analysis can then be applied while an accurate error bound is obtained.

This paper discusses a new, symbolic approach to geometric modeling called generative modeling. The approach allows specification, rendering, and analysis of a wide variety of shapes including 3D curves, surfaces, and solids, as well as higher-dimensional shapes such as surfaces deforming in time, and volumes with a spatially varying mass density. The system also supports powerful operations on shapes such as "reparameterize this curve by arclength", "compute the volume, center of mass, and moments of inertia of the solid bounded by these surfaces", or "solve this constraint or ODE system". The system has been used for a wide variety of applications, including creating surfaces for computer graphics animations, modeling the fur and body shape of a teddy bear, constructing 3D solid models of elastic bodies, and extracting surfaces from magnetic resonance (MR) data. Shapes in the system are specified using a language which builds multidimensional parametric functions. The language is bas...

This paper presents a model fabrication scheme that automatically approximates a model whose boundary consists of several freeform surfaces by developable surfaces and then unroll these developable surfaces onto a plane. The model can then be fabricated by assembling the sets of developable surfaces which have been cut from planar sheets and rolled back to their proper Euclidean locations. Both the approximation and the rolling methods can be made arbitrarily precise. 1 Introduction It is common to find freeform surfaces manually approximated and assembled as sets of piecewise developable surfaces [9]. In general, freeform surfaces are not developable and cannot be exactly represented as piecewise developable surfaces. Yet, "developable surfaces are of considerable importance to sheet-metal- or plate-metal-based industries This work was supported in part by DARPA (N00014-91-J-4123). All opinions, findings, conclusions or recommendations expressed in this document are those of the au...

Cut Cell fs 0 1 fs 0 0 fs 5 2 fs 5 1 fs 5 0 T 1 T 2 T 0 T 3 tp 0 tp 1 tp 2 tp 3 fp 0 1 fp 3 0 fp 5 1 F [0-5] fp [face#] [poly#] fs [face#][seg#] T [0-n] tp[0-n] fp 0 0 x x fp 5 0 1 0 x 2 Figure 3-5: Anatomy of an abstract cut-cell.

This work documents a new method for rapid and robust Cartesian mesh generation for componentbased geometry. The new algorithm adopts a novel strategy which first intersects the components to extract the wetted surface before proceeding with volume mesh generation in a second phase. The intersection scheme is based on a robust geometry engine that uses adaptive precision arithmetic and which automatically and consistently handles geometric degeneracies with an algorithmic tie-breaking routine. The intersection procedure has worse case computational complexity of O(N logN) and is demonstrated on test cases with up to 121 overlapping and intersecting components including a variety of geometric degeneracies.

We present a system for fast and accurate display of CSG (constructive solid geometry) models. Such models have as primitives, polyhedra and solids whose boundaries can be represented using rational spline surfaces. As a part of pre-processing, we compute the Brep (boundary representation) from the CSG tree and represent the resulting solid using trimmed spline surfaces. No assumptions are made on the number of primitives or the degree of the primitives in the CSG tree. Given a trimmed spline model, we tessellate it into polygons as a function of the viewing parameters and render it using visibility culling and coherence between successive frames. The choice of a analytic representation of the model and the trimming curves is fundamental to the fast performance of the system. The system has been used to convert parts of a submarine storage and handling system model represented as more than 2; 000 CSG trees. The B-rep consists of more than 30; 000 trimmed spline surfaces and is displaye...