Displacement maps and procedural displacement shaders are a widely used approach of specifying geometric detail and increasing the visual complexity of a scene. While it is relatively straightforward to handle displacement shaders in pipeline based rendering systems such as the Reyes-architecture, it is much harder to efficiently integrate displacement-mapped surfaces in ray-tracers. Many commercial ray-tracers tessellate the surface into a multitude of small triangles. This introduces a series of problems such as excessive memory consumption and possibly undetected surface detail. In this paper we describe a novel way of ray-tracing procedural displacement shaders directly, that is, without introducing intermediate geometry. Affine arithmetic is used to compute bounding boxes for the shader over any range in the parameter domain. The method is comparable to the direct ray-tracing of B'ezier surfaces and implicit surfaces using B'ezier clipping and interval methods, respectively. Keyw...

This paper compares the performance and e#ciency of di#erent function range interval methods for plotting f(x, y) = 0 on a rectangular region based on a subdivision scheme, where f(x, y) is a polynomial. The solution of this problem has many applications in CAGD.

We describe a variant of a domain decomposition method proposed by Gleicher and Kass for intersecting and trimming parametric surfaces. Instead of using interval arithmetic to guide the decomposition, the variant described here uses affine arithmetic, a tool recently proposed for range analysis. Affine arithmetic is similar to standard interval arithmetic, but takes into account correlations between operands and sub-formulas, generally providing much tighter bounds for the computed quantities. As a consequence, the quadtree domain decompositions are much smaller and the intersection algorithm runs faster. keywords: surface intersection, trimming surfaces, range analysis, interval analysis, CAGD.

. After reviewing three classical sampling methods for implicit objects, we describe a new sampling method that is not based on scanning the ambient space. In this method, samples are "randomly" generated using physically-based particle systems. Introduction In computer graphics, an object is described either by a set of sample points or by an analytic scheme that uses mathematical equations to define its geometry and topology. Descriptions based on samples occur in areas such as medical images and terrain models. Analytical descriptions are usually found in applications of geometric modeling, such computeraided design and manufacture. When an object is described by samples, a reconstruction scheme is needed to recover its geometry and topology from the samples. This problem, called structuring, consists of providing a combinatorial structure to the samples in order to (ideally) recover the exact topology of the object and an approximation of its geometry. When the object is describe...

We show how to use affine arithmetic to represent a parametric curve with a strip tree. The required bounding rectangles for pieces of the curve are computed by exploiting the linear correlation information given by affine arithmetic. As an application, we show how to compute approximate distance fields for parametric curves.

In this paper we give mathematical proofs of two new results relevant to evaluating algebraic functions over a box-shaped region: (i) using interval arithmetic in centered form is always more accurate than standard a#ne arithmetic, and (ii) modified a#ne arithmetic is always more accurate than interval arithmetic in centered form. Test results show that modified a#ne arithmetic is not only more accurate but also much faster than standard a#ne arithmetic. We thus suggest that modified a#ne arithmetic is the method of choice for evaluating algebraic functions, such as implicit surfaces, over a box.

This review is a response to recent discussions on the reliable computing mailing list, and to continuing uncertainties about the properties and merits of Taylor forms, multivariate higher degree generalizations of centered forms. They were invented around 1980 by Lanford, documented in detail in 1984 by Eckmann, Koch and Wittwer, and independently studied and popularized since 1996 by Berz, Makino and Hoefkens. A highlight is their application to the verified integration of asteroid dynamics in the solar system in 2001, although the details given are not sufficient to check the validity of their claims.

Affine arithmetic is a model for self-validated numerical computation that affine arithmetic keeps track of first-order correlations between computed and input quantities. We explain the main concepts in affine arithmetic and it handles the dependency problem in standard interval arithmetic. We also describe some of its applications.

This paper first recalls by some examples the damages that the numerical inaccuracy of the floating-point arithmetic can cause during geometric computations, and it intends to explain why damages for geometric computations differ from those met in numerical computations. Then it surveys the various approaches proposed to overcome inaccuracy difficulties; conservative approaches use classical geometric methods but with `exotic' arithmetics instead of the standard floating-point one; radical ones go farther and reject classical techniques, considering them not robust enough against inaccuracy. 1 Introduction Geometric modellers provided by commercial CADCAM softwares, and methods from the more theoretical field of Computational Geometry all perform geometric computations: for instance triangulating or meshing geometric domains for finite elements simulation, or computing intersections between geometric objects. Inaccuracy is a crucial issue for geometric computations. Not only the numer...

Computing intersections between algebraic surfaces is an essential issue for Brep-based modellers, and a very difficult one. The more often, existing methods are not reliable, and reliable ones are hairy. We think there is another and simple-minded way which avoids this problem without loss of practicalities. The key idea is computing a triple ray representation by zbuffer, raytracing or whatever, and then using the popular marching cubes algorithm with some local improvements. 1 The gap between CSG and Brep Breps [Hof89] describe solid objects by their boundary: surface patches, edges and vertices with their connectivity relations. They typically use free-form patches, carefully sewn together to form the consistent boundary of a solid which is then called a free-form (or sculptured) object. The high geometric coverage of free-form surfaces and their design flexibility are very appealing. In the other hand, Boolean operations on solid objects are an essential practicality for end use...