Solid modelers based on constructive solid geometry (CSG) typically generate shaded displays directly from CSG by using ray-casting techniques, which do not require informatin on the faces, edges, and vertices that bound a solid. This article describes an alternative-a simple new algorithm based on a depth-buffering or z-buffering approach. The z-buffer display algorithm operates directly on CSG, does not require explicit boundary data, and is easier to implement than ray casting. Ray-casting and z-beffering algorithms have comparable performances, but z-buffering is often faster for objects with complex surfaces, because it avoids expensive curve/surface intersection calculations. Because of their simplicity, depth-buffering algorithms for CSG are well-suited to hardware implementations, and may lead to machines simpler than those now being built for ray casting.

/pdf/depth-buffering-display-techniques-for-constructive-solid-1uzcr113td.pdf

Depth-Buffering Display Techniques for Constructive Solid Geometry

In adaptively weighted numerical integration, for a given set of quadrature nodes, order and domain of integration, the quadrature weights are obtained by solving a system of suitable moment fitting equations in least square sense. The moments in the moment equations are approximated over a simplified domain that is homeomorphic to the original domain, and then are corrected for the deviation from the original domain using shape sensitivity analysis. Using divergence theorem, the moments reduce to integrals over the boundary of the simplified domain. The proposed approach supports accurate and efficient computation of quadrature weights for integration of a priori unknown functions over arbitrary 2D and 3D solid domains. Experimental results (2D) indicate that adaptively weighted integration compares favorably with more traditional approaches. Because the adaptively weighted integration avoids excessive domain subdivision, it is useful in many applications and meshfree analysis in particular.

https://www.inf.usi.ch/hormann/nsfworkshop/posters/Thiagarajan.pdf

Adaptively weighted numerical integration over arbitrary domains

The surface area of a geometric model, like its volume, is an important integral property that needs to be evaluated frequently and accurately in practice. In this paper, we present a new quasi-Monte Carlo method using low-discrepancy sequences for computing the surface area of a 3D object. We show that the new method is more efficient than a Monte Carlo method using pseudo-random numbers. This method is based on the Cauchy ‐ Crofton formula from integral geometry, and it computes the surface area of a 3D body B by counting the number of intersection points between the boundary surface of B and a set of straight lines in E 3 . Low discrepancy sequences are used to generate the set of lines in E 3 to reduce the estimation errors that would be caused by using statistically uniformly distributed lines. We study and compare two different methods for generating 3D random lines, and demonstrate their validity theoretically and experimentally. Experiments suggest that the new quasi-Monte Carlo method is also more efficient than the conventional approach based on surface tessellation. q 2002 Elsevier Science Ltd. All rights reserved.

/pdf/using-low-discrepancy-sequences-and-the-crofton-formula-to-27svdd1gdo.pdf

Using low-discrepancy sequences and the Crofton formula to compute surface areas of geometric models

Shape sensitivity of constructively represented geometric models

The GMSolid system for modeling solids is being enhanced by the addition of free-form surfaces represented as B-splines. The major reason for this enhancement is that a wide variety of automotive parts cannot be modeled satisfactorily using existing GMSolid methods based on quadric surfaces. Free-form surfaces, which can be conformed to twisting, nonquadric surface regions, are required for efficient and accurate modeling. Representing free-form surfaces as B-splines provides a convenient mathematical foundation for many of the computational geometry algorithms needed in a solid modeler.

Free-Form Surfaces in GMSolid: Goals and Issues

When Applied to Combustion Chamber Design Analysis at General Motors, This Treebased Calculation Method Has Saved Time over the Boundary Representation Approach.

Sarraga

Papers

Computation of Surface Areas In GMSolid