Herbert B. Voelcker

Active zones in CSG for accelerating boundary evaluation, redundancy elimination, interference detection, and shading algorithms

Solids defined by Boolean combinations of solid primitives may be represented in constructive solid geometry (CSG) as binary trees. Most CSG-based algorithms (e.g., for boundary evaluation, graphic shading, interference detection) do various forms of set-membership classification by traversing the tree associated with the solid. These algorithms usually generate intermediate results that do not contribute to the final result, and hence may be regarded as redundant and a source of inefficiency. To reduce such inefficiencies, we associate with each primitive A in a tree S an active zone Z that represents the region of space where changes to A affect the solid represented by S, and we use a representation of Z instead of S for set-membership classification. In the paper we develop a mathematical theory of active zones, prove that they correspond to the intersection of certain nodes of the original trees, and show how they lead to efficient new algorithms for boundary evaluation, for detecting and eliminating redundant nodes in CSG trees, for interference (null-set) detection, and for graphic shading.

A computing strategy for applications involving offsets, sweeps, and Minkowski operations

Abstract Offsets, sweeps, and Minkowski operations (M-ops) are easy to define in the existential (representation-free) mathematics of point sets, but computing ‘values’ for offset, swept, and M-summed entities is thought to be difficult by many. This article argues that such computations may be easy if (1) they are cast in specific application contexts, and (2) relevant mathematical definitions are discretized and implemented directly. The argument is based on 10 years of research on a range of motional, process-modeling, and visualization problems that involved offsetting, sweeping, and M-ops; the solution paradigm common to all was direct approximation of mathematical definitions, using ray representations and parallel computation as primary media. This article presents no new results; it merely summarizes a body of well documented research that illustrates an approach to problem solving, whose primary tenets are: compute only what you need to solve the problem at hand, and do that as directly as possible.

The current state of affairs in dimensional tolerancing: 1997

This paper summarizes the evolution of mechanical tolerancing practices, the general character of tolerances as they are currently understood, the current state of tolerancing technologies, and the recent surge of activity aimed at rationalizing and “mathematizing” form tolerancing. The paper concludes with two examples that illustrate current research frontiers.

The Ray casting engine and Ray representatives

Solid modeling is computationally intensive. Thus far its use in industry has been limited mainly to simple parts and simple applications, and this is not likely to change much until ‘massive’ computing power can be made available at an affordable cost. The RayCasting Engine described in this paper is one specialized source of ‘massive’ computing power for solid modeling, and it is but the simplest member of a potentially large family of ‘classification computers’ for solid modeling. The Ray Casting Engine (RCE) is a highly parallel, custom-VLSI computer that classifies grids of parallel lines against solids represented in CSG. The sets of parallel ‘in’ segments that the RCE produces are called ray representations (ray-reps); they can be thought of as sampled boundary representations. Ray-reps are obviously useful for graphics and mass-property calculation. Less obviously, they are surprisingly versatile if one exploits special properties -for example, boolean combination of solids by interval operations on ray-reps -and the fact that ray-reps are cheap to compute. Overall, the combination of a ‘new’ representation scheme (ray-reps) and a fast custom processor (the RCE) is changing our approach to solid modeling. We are now seeking ‘brute force’ solutions to problems, and are finding that some previously intractable problems -for example, those involving spatial sweeping and offsetting -are effectively computable and easy to program. This paper summarizes the genesis and principles of the RCE, some important properties of ray representations, and some exemplary applications of the (ray-rep, RCE) combination.

On the completeness and conversion of ray representations of arbitrary solids

A ray representation (ray-rep) of a solid is a set of line segments that lie inside the solid, obtained by classifying a grid of parallel lines with respect to the solid. Enhanced rayreps contain tags, which are symbolic data associated with each line segment. Tags may carry properties of the interior of a solid or may descrilx characteristics of a solid’s txrundary locaf to a segment’s endpoint. Ray-reps, enhanced or not, are proving to be powerful tools in attacking some difficult applications, such as those involving general sweeping and Minkowski operations. Ray-reps are popularly perceived as ‘approximate’ representations, incapable of representing a solid completely (unambiguously). This paper shows, in its first set of results, that ray-reps can be complete representations of solids when enhanced with h-fags, and when generated with suitable ray grids. h-tags designate halfspaces that model the local geometries of solids at ray-segment endpoints. Ray grid suitability is determined from mathematical conditions for disambiguating spatial decompositions induced from halfspaces identified by h-tags. Complete ray-reps can be viewed as a new representation scheme within a family of complete sampling represenfafions not covered by Requicha’s 1980 six-family taxonomy of complete representation schemes for solids. The paper’s second set of results covers representation conversion. Conditions and procedures are given for bilateral ray-rep–to/from-Brep and ray-rep-to/from-CSG conversion; these conversions enable ray-reps to be used as working representations in CAD/CAM systems having boundary or CSG primary representations. Methods for using ray-reps to accelerate conversions (e.g. Brep-to-CSG) are discussed.

Geometric Modelling Systems for mechanical design and manufacturing

Software systems for creating, editing, and maintaining geometric representations of solid objects are coming to play an important role in mechanical design and manufacturing. This paper discusses the generic characteristics that such Geometric Modelling Systems should exhibit, and proposes a high-level architecture for an industrially viable system that may be built by safely extending state-of-the-art geometric modelling technology.

Research In Statistical Tolerancing: Examples of Intrinsic Non-normalities, and their effects

This paper summarizes the results of exploratory studies of the statistics of actual values (minimal enclosing zones) of essentially two-dimensional geometric position, circularity. and runout tolerances. Data for the studies were collected via CMM measurements of IOO-sample lots of commercial washers and automotive valve seats. Phenomenological models were devised to predict the measured statistics, and agree reasonably well with the experimental data.

Remarks on the Essential Elements of Tolerancing Schemes

Tolerancing schemes for controlling the geometric variability of mechanical parts and assemblies are growing in number and variety. This Short Communication discusses the elements (logical components) that seem to be common to the major known tolerancing schemes, and proposes a classification for schemes currently in use, under discussion, or open for development.