Robust regularized set operations on polyhedra

Journal ArticleThis paper describes a provably correct and robust implementation of regularized set operations on polyhedral objects. Although the algorithm described here does not assume manifold polyhedra and handles all possible degenerate cases, it turns out to be quite simple. The geometric operations and relations are computed with floating point arithmetic which is efficient but not necessarily precise. To ensure that the results are still consistent we implemented a test that detects when dependent decisions contradict each other. The consistency test is relatively simple and can be carried out locally without having to reason about the logical dependencies of the geometric relations. The logical structure and the efficiency of the algorithm are barely influenced by the consistency test which makes this approach well suited for interactive modeling systems on modern workstations

An axiomatic approach for solving geometric problems symbolically

technical reportThis paper describes a new approach for solving geometric constraint problems and problems in geometry theorem proving. We developed a rewrite-rule mechanism operating on geometric predicates. Termination and completeness of the problem solving algorithm can be obtained through well foundedness and confluence of the set of rewrite rules. To guarantee these properties we adapted the Knuth-Bendix completion algorithm to the specific requirements of the geometric problem. A symbolic, geometric solution has the advantage over the usual algebraic approach that it speaks the language of geometry. Therefore, it has the potential to be used in many practical applications in interactive Computer Aided Design

Detecting ambiguities: an optimistic approach to robustness problems in computational geometry

Journal ArticleComputational geometry algorithms deal with geometric objects, usually represented by coordinates in an n-dimensional Euclidean space. Most efficient algorithms implement geometric operations as floating point arithmetic operations on the coordinates. Since floating point numbers can only approximate the "real" world, these operations often lead to topologically inconsistent results, especially when degenerate cases are handled. Recently, a variety of methods have been developed to cope with this, so called, robustness problem. This paper describes a new approach based on the optimistic assumption that in the majority of cases the decisions can be made consistently, even with imprecise data. Degenerate cases are decided with some tolerance. A test is applied for detecting when decisions made by the algorithm that logically depend on each other are inconsistent due to ambiguities arising from the approximation. In case of ambiguities, the inconsistencies can be resolved by increasing the tolerance. The proposed ambiguity test can be carried out in constant time whenever a decision is made during computation. Therefore, this method does not change the asymptotic complexity of the underlying algorithm in most practical cases, which is a clear advantage over previous approaches

GDI reference manual

Journal ArticleGDI is a dialog interface tool library for C + + applications. It facilitates the design and implementation of graphical, object-oriented user interfaces for workstations equipped with a graphical display, a mouse and a keyboard. GDI's design allows for its portability onto a multitude of platforms. This is achieved by separating the user interface from the application program, as well as the orderly detachment of system independent user- interface tools from the system dependent, low level, window operations

Robustness in geometric modeling - an intuitionistic and tolerance-based approach

Journal ArticleAn intuitionistic geometry approach is taken to develop two tolerance-based methods for robust geometric computation. The so called analytic model method and the approximated model method are developed independently of a specific application or a geometric algorithm. Geometric robustness is formally defined. Geometric relations are computed based on tolerances defined for geometric objects. Dynamic tolerance updating rules are given to preserve properties of the geometric relations. The two methods differ in the definition of robustness and they use different tolerance updating rules, and hence, they preseve different properties and are suitable for different kinds of applications. To handle the possibly occuring ambiguities dynamic ambiguity handling methods are described as well

Robustness in solid modeling - a tolerance based, intuitionistic approach

Journal ArticleThis paper presents a new robustness method for geometric modeling operations. It computes geometric relations from the tolerances defined for geometric objects and dynamically updates the tolerances to preserve the properties of the relations, using an intuitionistic self-validation approach. Geometric algorithms using this approach are proved to be robust. A robust Boolean set operation algorithm using this robustness approach has been implemented and test examples are described in this paper as well

Interaction with constraints in 3D modeling

Journal ArticleInteractive geometric modeling is an important part of the industrial product design process. This paper describes how constraints can be used to facilitate the interactive definition of geometric objects and assemblies. We have implemented a geometric modeling system that combines the definition of objects by interactive construction operations and specification of geometric constraints. The modeling operations automatically generate constraints to maintain the properties intended by their invocation, and constraints, in turn, determine the degrees of freedom for further interactive mouse-controlled modeling operations. A symbolic geometry constraint solver is employed for solving systems of constraints

Sketching as a solid modeling tool

Journal ArticleThis paper describes 'Quick-sketch', a 2-d and 3d modeling tool for pen based computers. Users of this system define a model by simple pen strokes drawn directly on the screen of a pen-based PC. Lines, circles, arcs, or B-spline curves are automatically distinguished and interpreted from these strokes. The system also automatically determines relations, such as right angles, tangencies, symmetry, and parallelism, from the sketch input. These relationships are then used to clean up the drawing by making the approximate relationships exact. Constraints are established to maintain the relationships during further editing. A constraint maintenance system, which is based on gestural manipulation and soft constraints, is employed in this system. Several techniques for sketch based definitions of 3d objects are provided as well, including extrusion, surface of revolution, ruled surfaces and sweep. Features can be sketched on the surface of a 3d object, using the same 2d and 3d techniques. This way, objects of medium complexity can be sketched in seconds. The system can be viewed as a front-end to more sophisticated modeling, rendering or animation environments, serving as a hand sketching tool in the preliminary design phase

Robust solid modeling by avoiding redundancy for manifold objects in boundary representation

Journal ArticleThis paper describes a new approach to the robustness problem in solid modeling. We identify as t h e main cause of t h e lack of robustness that interdependent topological relations are derived from approximate data. Disregarding the interdependencies very likely violates basic properties, such as reflexivity, and transitivity, resulting in invalid data representations, such as dangling edges, missing faces, etc. We show that the boundary of manifold objects can be represented without redundant relations which avoids inconsistencies. An algorithm for regularized set operations for manifold solids which is based on the principle of avoiding and eliminating redundancy is described. This algorithm has been implemented for objects bounded by planar and natural quadric surfaces; it handles coincidence and incidence cases between surfaces and curves robustly

Robust boolean set operations for manifold solids bounded by planar and natural quadric surfaces

Journal ArticleThis paper describes our latest effort in robust solid modeling. An algorithm for set operations on solids bounded by planar and natural quadric surfaces, that handles all geometrically degenerate cases robustly, is described. We identify as the main reason for the lack of robustness in geometric modeling, that dependent relations are handled inconsistently by disregarding the dependencies. Instead of using explicit reasoning to make dependent decisions consistent, we show that redundant computation can be avoided by correctly ordering the operations, and redundant data can be eliminated in the set operation algorithm, so that the result is guaranteed to be a valid two-manifold solid