Vadim Shapiro

Separation for boundary to CSG conversion

important applications of b-rep CSG conversion arise in solid modeling, image processing, and c,lsewhere. In addition, the problem is of considerable theoretical interest. One of the most difficult steps in performing b-rep ~ CSG conversion for a curved solid object consists of determining o set of half-spaces that is sufficient for a CSG representation of the solid. This usually requires the construction of additional half-spaces whose boundaries do not contribute to the boundary of the solid. Such half-spaces are called separating half-spaces because their purpose is to separate certain subsets inside the solid from those outside of the solid, Construction of separating half-spaces is specific to a particular geometric domain, but several generic approaches are possible. We use the information present in the boundary of the solid being converted to study the constraints on the degree of separating half-spaces, and show that a suff]cicnt set of linear separating half-spaces exists for any solid whose boundary contains only planar edges. A compl[,te construction is given for solids whose faces lie in convex surfaces. Separation for more L,cneral solids, whose b-rep includes othm surfaces and nonplanar edges, is alsa discussed, but this general problem remains poorly understood. We apply the boundary-based separation to solids hounded by genera] quadric surfaces, Specifically, we prove that a sufficient set of linear separating half-spaces exists for any such solid and consider the required constructions in several common situations. The presented results allowed a successful implementation of an experimental b-rep + CSW conversion system that converts natural quadric b-reps in ]>:lras(J]ldTMt(} ~~clcnt C’SG representations in PADL-2

Implicit functions with guaranteed differential properties

Theory of R-functions [12] provides the methodology for constructing exact implicit functions for any semianalytic set. This paper systematically explores and compares the known constructions in terms of their differential properties and explains how such functions may be constructed automatically from CSG and boundary representations of solids. The constructed functions may be automatically differentiated and integrated and have many important applications in meshfree engineering analysis, motion planning, and scientific visualization.

Semi-analytic geometry with R-functions

V. L. Rvachev called R-functions ‘logically charged functions’ because they encode complete logical information within the standard setting of real analysis. He invented them in the 1960s as a means for unifying logic, geometry, and analysis within a common computational framework – in an effort to develop a new computationally effective language for modelling and solving boundary value problems. Over the last forty years, R-functions have been accepted as a valuable tool in computer graphics, geometric modelling, computational physics, and in many areas of engineering design, analysis, and optimization. Yet, many elements of the theory of R-functions continue to be rediscovered in different application areas and special situations. The purpose of this survey is to expose the key ideas and concepts behind the theory of R-functions, explain the utility of R-functions in a broad range of applications, and to discuss selected algorithmic issues arising in connection with their use.

Boundary representation deformation in parametric solid modeling

One of the major unsolved problems in parametric solid modeling is a robust update (regeneration) of the solids boundary representation, given a specified change in the solids parameter values. The fundamental difficulty lies in determining the mapping between boundary representations for solids in the same parametric family. Several heuristic approaches have been proposed for dealing with this problem, but the formal properties of such mappings are not well understood. We propose a formal definition for boundary representation. (BR-)deformation for solids in the same parametric family, based on the assumption of continuity: small changes in solid parameter values should result in small changes in the solids boundary reprentation, which may include local collapses of cells in the boundary representation. The necessary conditions that must be satisfied by any BR-deforming mappings between boundary representations are powerful enough to identify invalid updates in many (but not all) practical situations, and the algorithms to check them are simple. Our formulation provides a formal criterion for the recently proposed heuristic approaches to “persistent naming,” and explains the difficulties in devising sufficient tests for BR-deformation encountered in practice. Finally our methods are also applicable to more general cellular models of pointsets and should be useful in developing universal standards in parametric modeling.

What is a parametric family of solids

Classical solid modeling systems are being rapidly replacd by new parametric solid modeling systems, where solid models are defined and manipulated through high-level, pararneterized, and user-modifiable definitions. We show that such parametric solid modeling systems inherited the unsolved technical problems of earlier dual representation (boundary and CSG) systems which may limit their functionality and performance. In particular, the modern systems may not robustly support parametric and variational modeling, be cause the meaning of a “parametric family” is not always well-defined. We investigate existing practicea in parameb ric modeling, discuss corresponding mathematical models and limitations of parametric families, and explore technical difficulties that must be resolved to take full advantage of parametric solid modeling.

Construction and optimization of CSG representations

Boundary representations (B-reps) and constructive solid geometry (CSG) are widely used representation schemes for solids. While the problem of computing a B-rep from a CSG representation is relatively well understood, the inverse problem of B-rep to CSG conversion has not been addressed in general. The ability to perform B-rep to CSG conversion has important implications for the architecture of solid modelling systems and, in addition, is of considerable theoretical interest. The paper presents a general approach to B-rep to CSG conversion based on a partition of Euclidean space by surfaces induced from a B-rep, and on the well known fact that closed regular sets and regularized set operations form a Boolean algebra. It is shown that the conversion problem is well defined, and that the solution results in a CSG representation that is unique for a fixed set of halfspaces that serve as a ‘basis’ for the representation. The ‘basis’ set contains halfspaces induced from a B-rep plus additional non-unique separating halfspaces. An important characteristic of B-rep to CSG conversion is the size of a resulting CSG representation. We consider minimization of CSG representations in some detail and suggest new minimization techniques. While many important geometric and combinatorial issues remain open, a companion paper shows that the proposed approach to B-rep to CSG conversion and minimization is effective in E2, In E3, an experimental system currently converts natural-quadric B-reps in PARASOLID to efficient CSG representations in PADL-2.

Chain Models of Physical Behavior for Engineering Analysis and Design

The relationship between geometry (form) and physical behavior (function) dominates many engineering activities. The lack of uniform and rigorous computational models for this relationship has resulted in a plethora of inconsistent (and thus usually incompatible) computer-aided design (CAD) tools and systems, causing unreasonable overhead in time, effort, and cost, and limiting the extent to which CAD tools are used in practice. It seems clear that formalization of the relationship between form and function is a prerequisite to taking full advantage of computers in automating design and analysis of engineering systems.We present a unified computational model of physical behavior that explicitly links geometric and physical representations. The proposed approach characterizes physical systems in terms of their algebraic-topological properties:cell complexes, chains, and operations on them.

Errata: Maintenance of Geometric Representations Through Space Decompositions

The ability to transform between distinct geometric representations is the key to success of multiple-representation modeling systems. But the existing theory of geometric modeling does not directly address or support construction, conversion, and comparison of geometric representations. A study of classical problems of CSG ↔ b-rep conversions, CSG optimization, and other representation conversions suggests a natural relationship between a representation scheme and an appropriate decomposition of space. We show that a hierarchy of space decompositions corresponding to different representation schemes can be used to enhance the theory and to develop a systematic approach to maintenance of geometric representations.

Approximate distance fields with non-vanishing gradients

For a given set of points S, a Euclidean distance field is defined by associating with every point p of Euclidean space Ed a value that is equal to the Euclidean distance from p to S. Such distance fields have numerous computational applications, but are expensive to compute and may not be sufficiently smooth for some applications. Instead, popular implicit modeling techniques rely on various approximate fields constructed in a piecewise manner. All such constructions lead to sacrifices in distance properties that have not been properly studied or characterized. We show that the quality of an approximate distance field may be characterized locally near the boundary by its order of normalization and can be studied in terms of the field derivatives. The approach allows systematic quantitative assessment and comparison of various construction methods. In particular, we provide detailed analysis of several popular field construction methods that rely on set decompositions and R-functions, as well as identify the key factors affecting the quality of the constructed fields.

The Architecture of SAGE – A Meshfree System Based on RFM

Abstract. In a meshfree system, a geometric model of a domain neither conforms to, nor is restricted by a spatial discretization. Such systems for engineering analysis offer numerous advantages over the systems that are based on traditional mesh-based methods, but they also require radical approaches to enforcing boundary conditions and novel computational tools for differentiation, integration, and visualization of fields and solutions. We show that all of these challenges can be overcome, and describe SAGE (Semi-Analytic Geometry Engine) – a successful system specifically intended for meshfree engineering analysis. Our approach and individual modules are based on Rvachev’s Function Method (RFM) but the described techniques, algorithms, and software are applicable to all mesh-based and meshfree methods and have broad use beyond solutions of boundary value problems.