Solid Representation Schemes Research Articles

Finite Algebras for Solid Modeling using Julia’s Sparse Arrays

An early research in solid modeling led by Herbert Voelcker at the University of Rochester and later at Cornell suggested that every solid representation scheme corresponds to an algebra, where the elements of the algebra are solid representations constructed and edited using operations in the algebra. For example, every CSG representation describes an element in a finite Boolean algebra of closed regular sets, whereas every boundary representation describes an element of a vector space of 2-chains in an algebraic topological chain complex. In this paper, we elucidate the precise relationships (functors) between all algebras used for CSG and boundary representations of solids. Based on these properties, we show that many solid modeling operations, including boundary evaluation, reduce to straightforward algebraic operations or application of identified functors that are efficiently implemented using point membership tests and sparse matrix operations. To fully exploit the efficacy of the new algebraic approach to solid modeling, all algorithms are fully implemented in Julia, the modern language of choice for numerical and scientific computing. • New research direction in solid modeling. • Algebraic relationships between CSG and boundary representations of PL solids. • Arrangements are isomorphic to the finite Boolean algebra of regularized sets. • This algebra is isomorphic to a linear vector space of 3-chains. • Operations on CSG and b-reps reduce to composition of algebraic operations. operations.

Implementation of a 5-Axis Milling Simulation System Using Triple Dexel Models

In the triple dexel model, object shape is represented by 3 groups of axis aligned segments named x-axis dexels, y-axis dexels, and z-axis dexels. Triple dexel model is known as compact and precise representation compared with other discrete solid representation schemes. Based on the triple dexel representation, a new 5-axis milling simulation system is developed. Differently from prior systems, this system uses GPU technology for accelerating 3 important processing steps in the milling simulation, which are (1) cutter swept volume generation, (2) subtraction of the swept volume from the workpiece solid model, and (3) update of the workpiece picture for achieving smooth animation of the milling process. An experimental system is implemented and some computational experiments are performed. Our system can realize a simulation of a complex 5-axis milling process in a few minutes. The animation speed is generally more than 100 frames per second.

Octree-to-BRep conversion for volumetric NC simulation

User-friendliness such as ease of creation and modification and system-friendliness such as space and time complexities, accuracy etc. are two important criteria for the selection of a solid representation scheme. No single solid representation scheme fully meets all the requirements of user-friendliness and system-friendliness. Hence, any practical CAD/CAM system maintains the geometry of the objects in more than one solid representation scheme simultaneously. Therefore, it is often required to convert one solid representation scheme to the other. The authors have developed a volumetric NC simulation system in which the instantaneous blank is represented as an octree as its space and time complexities are independent of the number of NC blocks. However, since octree does not lend itself readily for operations such as arbitrary transformations, rendered display etc., it is not directly usable for downstream applications such as animated display, verification and optimization. Boundary representation is suitable for these downstream applications. Therefore, it is required to convert octree of the instantaneous blank into BRep. An efficient algorithm for this conversion is presented in this paper. This algorithm essentially splits the octree into three quadtrees that store the geometry along the three principal directions. This algorithm is fast as it involves only tree-traversals.

Approximating CSG trees of moving objects

A discussion of the relationship between two solid representation schemes is presented: CSG trees and recursive spatial subdivision exemplified by the bintree, a generalization of the quadtree and octree. Detailed algorithms are developed and analyzed for evaluating CSG trees by bintree conversion. These techniques are shown to enable the addition of the time dimension and motion to the approximate analysis of CSG trees. This facilitates the solution of problems such as static and dynamic interference detection. A technique for projecting across any dimension is also shown. For “well-behaved” CSG trees the execution time of the conversion algorithm is directly related to the spatial complexity of the object represented by the CSG tree (i.e., as the resolution increases, it is asymptotically proportional to the number of bintree nodes and does not depend on the size or form of the CSG tree representation). The set of well-behaved CSG trees include all trees that define multidimensional polyhedra in a manner that does not give rise to tangential intersections at CSG tree nodes.

A solid modelling system for robot action planning

The characteristics of several solid representation schemes are discussed with respect to their possible use in robot action planning systems. The World Modeler (WM), a solid modeler developed by combining the generalized cylinder approach with constructive solid geometry, is presented. An efficient algorithm for computing collision among convex polyhedrons, that utilizes the internal geometric data structure of the World Modeler is explained.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Motion Planning Algorithms for Mechanical Assemblies

After having compared several planning algorithms proposed for robot motion, we will discuss how the choice of the solid representation scheme and of the primitives which support the geometric reasoning influence the design of an automatic programming system for mechanical assemblies. To realize the key modules of a suitable robot programming environment, we have defined an efficient algorithm for collision detection between convex polyhedra and we want to value its effectiveness for integrating local and global motion planning.

CSG-EESI: a new solid representation scheme and a conversion expert system

A representation scheme is proposed for describing 3-D mechanical parts and structural bodies which are formed from planes and quadratic faces. Since the method combines features of the constructive solid geometry (CSG) representation and an extension of the enhanced (EESI) spherical image representation (ESI), it is designated the CSG-EESI. In this scheme, the body model is roughly divided into two levels: the higher level corresponds to a restricted CSG tree that contains the structural information describing how the various subparts form the body; the lower level contains the geometric information for those simple subparts and represents them by an extension of enhanced spherical images. The scheme can be used both as the medium between pictorial models and relational models and as an internal model to facilitate the recognition of bodies. An expert system written in C-PROLOG on a VAX 11/750 is presented that converts BR-like models into the CSG-EESI representation is presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An approach to geometric modeling of solids bounded by sculptured surfaces

In this paper we present the design and implementation of a geometric modeling system for modeling solids bounded by sculptured surfaces. The three most important solid representation schemes—constructive solid geometry, boundary representation, and octree representation—are combined together in our system in such a way that the resulting scheme enjoys many of the advantages of its individual schemes. We also developed algorithms for conversion of objects from their boundary representation to octree representation, and for the boundary evaluation of octree encoded objects. The system was implemented on a DEC 1090 computer in PASCAL, and we have presented a simple illustrative example to show the use of the system to create solid objects bounded by sculptured surfaces.

Bintrees, CSG trees, and time

A discussion is presented of the relationship between two solid representation schemes: constructive solid geometry (CSG trees) and recursive spatial subdivision exemplified by the bintree, a generalization of the quadtree and octree. Detailed algorithms are developed and analyzed for evaluating CSG trees by bintree conversion, i.e., by determining explicitly which parts of space are solid and which empty. These techniques enable the addition of the time dimension and motion to the approximate analysis of CSG trees in a simple manner to solve problems such as dynamic interference detection. For "well-behaved" CSG trees, the execution time of the conversion algorithm is directly related to the spatial complexity of the object represented by the CSG tree (i.e., asymptotically it is proportional to the number of bintree nodes as the resolution increases). The set of well-behaved CSG trees includes all trees that define multidimensional polyhedra in a manner that does not give rise to tangential intersections at CSG tree nodes.