Real Functions for Representation of Rigid Solids
A range of values of a real function f : Ed + Iw can be used to implicitly define a subset of Euclidean space Ed. Such “implicit functions” have many uses in geometric and solid modeling. This paper focuses on the properties and construction of real functions for the representation of rigid solids (compact, semianalytic, and regular subsets of Ed). We review some known facts about real functions defining compact semianalytic sets, and their applications. The theory of Rfunctions developed in (Rvachev, 1982) provides means for constructing real function representations of solids described by the standard (nonregularized) set operations. But solids are not closed under the standard set operations, and such real function representations are rarely available in modem solid modeling systems. More generally, assuring that a real function f represents a regular set may be difficult. Until now, the regularity has either been assumed, or treated in an ad hoc fashion. We show that topological and extremal properties of real functions can be used to test for regularity, and discuss procedures for constructing real functions with desired properties for arbitrary solids. 

Chain Models of Physical Behavior for Engineering Analysis and Design
The relationship between geometry(form) and physical behaviour(function) dominates many engineering activities. The lack of uniform and rigouros computational models for this relationship has resulted in a plethora of inconsistent(and thus usually incompatibe) computer aided design (CAD) tools and systems, causing unreasonable overhead in time, effort, and cost, and limiting to the extent to which CAD tools are used in practice. It seems clear that the formulization of the relationship between form and function is a prerequiste to taking full advantage of computers in automating design and analysis of engineering systems. We present a unified computational model of physical behaviour that explcitly links geometric and physical representations. The proposed approach characterizes physical systems in terms of their algebraictopological properties: cell complexes, chains and, operations on them. 

Separation for Boundary to CSG Conversion
Important applications of brep – CSG conversion arise in solid modeling, image processing, and elsewhere. In addition, the problem is of considerable theoretical interest. One of the most difficult steps in performing brep ~ CSG conversion for a curved solid object consists of determining o set of halfspaces that is sufficient for a CSG representation of the solid. This usually requires the construction of additional halfspaces whose boundaries do not contribute to the boundary of the solid. Such halfspaces are called separating halfspaces because their purpose is to separate certain subsets inside the solid from those outside of the solid, Construction of separating halfspaces 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 halfspaces, and show that a suff]cicnt set of linear separating halfspaces 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 brep includes othm surfaces and nonplanar edges, is alsa discussed, but this general problem remains poorly understood. We apply the boundarybased separation to solids hounded by genera] quadric surfaces, Specifically, we prove that a sufficient set of linear separating halfspaces exists for any such solid and consider the required constructions in several common situations. The presented results allowed a successful implementation of an experimental brep + CSW conversion system that converts natural quadric breps in Parasolid to efficient CSG representations in PADL2 

Efficient CSG Representations of 2Dimensional Solids
Good methods are known for converting a Constructive Salid–Geometry (CSG) representation of a solid into a boundary representation (brep) of the solid, but not for performing the inverse conversion, brep CSG, which is the subject of this paper. Important applications of brepCSG conversion arise in solid modeling, image processing, and elsewhere. The problem can be divided into two tasks: (1) finding a set afhal/spaces that is necessarya nd sufficient (but not unique) to represent a given solid, and (2) constructing an efficient CSG representation using those hal/spaces. This paper solves the problem for curved planar solids, i.e., rsets in £2, with or without holes, whose boundary is given by a collection of edges. The edges may be subsets of straight lines or convex curves .(i.e., curves which intersect any line in at most two points). We prove a number of results and describe algorithms that have been fully implemented for solids bounded by line segments and circular arcs. Empirical results show that the computed CSG representations are superior to those produced by earlier algorithms, and produce superior threedimensional CSG representations for mechanical parts defined by contour sweeping. A companion paper generalizes the results to higher dimensional solids. 

Construction and Optimization of CSG Representations
Boundary representations(Breps) and Constructive solid geometry(CSG) are widely used representation scheme for solids. While the problem of computing a Brep from a CSG representation is relatively well understood, the inverse problem of Brep to CSG conversion has not been addressed in general. The ability to perform Brep to CSG conversion has important implications for the architecture of solid modelling systems, and in addition, is of considerable theoretical interest. This paper presents a general approach to Brep to CSG conversion based on the partition of Euclidean space by surfaces induced from a Brep, and on the well known fact that clsoed regualr sets and regularised 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 Brep plus additional nonunique separating halfspaces. 

On the role of geometry in mechanical design
A complete design usually specifies a mechanical system in terms of component parts and assembly relationships. Each part has a fully defined nominal or ideal form and well defined material properties. Tolerances are used to permit variations in the form and properties of the components, and are used also to permit variations in the assembly relationships. Thus the geometry and material properties of the system and all of its pieces are fully defined (at least in principle). Henceforth we shall focus on geometry and, for reasons that will become evident, will not deal with materials despite their obvious importance 

Theory of Rfunctions and Applications: A Primer
A Rfunction is a real valued function characterized by some property that is completely determined by the corresponding property of its arguments,e.g., the sign of some real function is completely determined by the sign of their arguments. This primer summarizes some basic results from the theory of Rfunctions and describes some of the applications. 