Dominique Michelucci

Solving geometric constraints by homotopy

Numerous methods have been proposed in order to solve geometric constraints, all of them having their own advantages and drawbacks. In this article, we propose an enhancement to the classical numerical methods, which, up to now, are the only ones that apply to the general case.

Bridging the gap between CSG and Brep via a triple ray representation

Computing intersections between algebraic surfaces is an essential issue for Brep-based modellers, and a very difficult one. The more often, existing methods are not reliable, and reliable ones are hairy. We think there is another and simple-minded way which avoids this problem without loss of practicalities. The key idea is computing a triple ray representation by zbuffer, raytracing or whatever, and then using the popular marchiny cubes algorithm with some local improvements. 1 The gap between CSG and Brep Breps [Hof89] describe solid objects by their boundary: surface patches, edges and vertices with their connectivity relations. They typically use free-form patches, carefully sewn together lo form the consistent, boundary of a solid which is then called a free-form (or sculptured) object. The high geometric coverage of free-form surfaces and their design flexibility are very appealing. In the other hand, Boolean operations on solid objects are an essential practicality for end users. Unfortunately, performing Boolean operations on Breps involve computing the intersection between algebraic surfaces, which is a very difficult task. Existing methods are often not reliable, and when they are, they are anyway exceedingly complicated: see [Pat93, KM96, HPY96]. The CSG model [Hof89] represent solid objects by a tree whose nodes carry Boolean operators and leaves carry algcbraic half-spaces (algebraic inequalities: f(~, y, Z) 5 0). In contrast to Breps, the CSG representation does not suffer from reliahilit,y problems, and the surface to surface intersection problem ‘is not a crucial issue. The raytracing method permits to visualize CSG objects and to convert them to ray representations (rayrep for short). The recursive space subdivision method permits to evaluate (ie to voxelize, or tessel&e) them as in the SVLIS modeller [Bow951 or in Taubin’s method [Tau93]. As long as a CSG modeller does not rely on tesselation, the latter can even be locally inconsistent without afffscting thr modellrr. Note the divide and conquelapproach basically relies on the possibility of quickly and simply classifying a point with respect to an algebraic half-space (by evaluating and testing the sign of the corresponding formula f(~, y, 2)). It is then possible to compute, by an interval arithmetic (or some variant), ranges of the function f for boxes (a box is a point whose coordinates are intervals): a box B is classified inside if f(B) < 0 and outside if f(B) > 0. Otherwise the box is subdivided (into 2 or 8 smaller ones, according to implementations). Such a classification test is not available for free-form objects. Unfortunately CSG does not support the full range of free-form objects. Several attempts have been made to combine appeals of CSG and Brep: l Using soft objects is mainly restricted to the Animation field for the moment [IS’95]. l In t,he CAD/CAM field, J. Menon & B. Guo [MG96] use a restricted set of free-form surfaces, with a low degree implicit form (2 or 3): each free-form patch is assigned a companion tetrahedron which contains the patch, and whose vertices are, in some way, its control points. These tetrahedra permit to edit patches in an intuitive and interactive way. This modeller allows a bilateral CSG/Brep conversion. l A. Pasko & V. Adzhiev & A. Sourin & V. Savchenko [PASS931 describe the interior of all geometric objects: algebraic half-spaces, Boolean operations, sweeps, some kind of deformations and blends, free-form volumes (sometimes called cuboids) [MPS96]. . . , by a semi-algebraic inequality f(x, Y> 2) I 0. None of the previous solutions fully integrate in the CSG model all the free-form objects used in Brep-based modellers. l A last approach combines CSG and Brep in that frerform primitives are accepted at leaves of the CSG tree. However, the simplicity of the pure-CSG scheme is lost: these modellers face the surface to surface intersection and the robustness problems. Recent works illustrating this tendency are due to S. Krishnan & D. Manocha [KM96], and to C.Y. Hu & N. Patrikalakis & X. Ye [HPY96]. No doubt for us that these modellers are masterpieces, tours de force of geometric computing. But they are too much complicated. Moreover, they do not cover all the possible cases: sweeps (occurring for instance in NC-milling), blends or Minkowski sums. .so/rdhfocfe/rf7~ ’ 97 Atlnlltn GA 1 ISA Copyright 1997 ACM 0-X9791-946-7107102 .,

Error-free boundary evaluation based on a lazy rational arithmetic: a detailed implementation

3.X)

Qualitative Study of Geometric Constraints

Abstract A new boundary-evaluation method is presented. It is based on error-free Boolean operations on polyhedral solids. An intersection algorithm that handles in a straightforward way all the possible geometric cases is described in detail. A general data structure is also described that allows the unified storage of solid boundaries. The intersection algorithm always runs to completion, producing consistent solids from consistent operands. Numerical errors are handled at an algorithm-independent level, which is an original exact arithmetic that performs only the necessary precise computations. Results from the implementation of the CSG solver are discussed.

Error-free boundary evaluation using lazy rational arithmetic: a detailed implementation

Modeling by geometric constraints is a promising method in the field of CAD/CAM. However, this process, closely related to computer programming, is also error prone. A geometric constraints based modeler should help the end user to find his mistakes, or, better, not to commit ones by watching the building process. The well known main cause of errors with these methods is the specification of redundant constraints, and sometimes conflicting constraints. It’s also important to detect under-constrained parts of the system involving indecisiveness. This chapter alludes to some numerical and probabilistic tools that could be used for this goal. There’s also a decomposition method of constraints systems using tools of the graph theory. Since these two approaches have their own advantages it’s worth combining them. All these methods will be studied first for systems of equations and then extended to the more specific case, but also more interesting for modeling, of systems of geometric constraints.

Geometric constraint solving: The witness configuration method

A new boundary evaluation method is presented. It is based on error-free Boolean operations on polyhedral solids. W e describe, in detail, an intersection algorithm that handles, in a straightforward way, all the possible geometric cases. We also describe a general data structure that allows an unified storage of solid boundaries. The intersection algorithm always runs to completion, producing consistent solids from consistent operands. Numerical errors are handled at an algorithm independent level: an original exact arithmetic that performs only the necessary precise computations. Results from our implementation of this CSG solver are discussed.

Numerical decomposition of geometric constraints

Geometric constraint solving is a key issue in CAD, CAM and PLM. The systems of geometric constraints are today studied and decomposed with graph-based methods, before their numerical resolution. However, graph-based methods can detect only the simplest (called structural) dependences between constraints; they cannot detect subtle dependences due to theorems. To overcome these limitations, this paper proposes a new method: the system is studied (with linear algebra tools) at a witness configuration, which is intuitively similar to the unknown one, and easy to compute.

Nonlinear systems solver in floating-point arithmetic using LP reduction

Geometric constraint solving is a key issue in CAD/CAM. Since Owens seminal paper, solvers typically use graph based decomposition methods. However, these methods become difficult to implement in 3D and are misled by geometric theorems. We extend the Numerical Probabilistic Method (NPM), well known in rigidity theory, to more general kinds of constraints and show that NPM can also decompose a system into rigid subsystems. Classical NPM studies the structure of the Jacobian at a random (or generic) configuration. The variant we are proposing does not consider a random configuration, but a configuration similar to the unknown one. Similar means the configuration fulfills the same set of incidence constraints, such as collinearities and coplanarities. Jurzaks prover is used to find a similar configuration.

Geometric constraints solving: some tracks

This paper presents a new solver for systems of nonlinear equations. Such systems occur in Geometric Constraint Solving, e.g., when dimensioning parts in CAD-CAM, or when computing the topology of sets defined by nonlinear inequalities. The paper does not consider the problem of decomposing the system and assembling solutions of subsystems. It focuses on the numerical resolution of well-constrained systems. Instead of computing an exponential number of coefficients in the tensorial Bernstein basis, we resort to linear programming for computing range bounds of system equations or domain reductions of system variables. Linear programming is performed on a so called Bernstein polytope: though, it has an exponential number of vertices (each vertex corresponds to a Bernstein polynomial in the tensorial Bernstein basis), its number of hyperplanes is polynomial: O(n2) for a system in n unknowns and equations, and total degree at most two. An advantage of our solver is that it can be extended to non-algebraic equations. In this paper, we present the Bernstein and LP polytope construction, and how to cope with floating point inaccuracy so that a standard LP code can be used. The solver has been implemented with a primal-dual simplex LP code, and some implementation variants have been analyzed. Furthermore, we show geometric-constraint-solving applications, as well as numerical intersection and distance computation examples.

The ellipsoidal skeleton in medical applications

This paper presents some important issues and potential research tracks for Geometric Constraint Solving: the use of the simplicial Bernstein base to reduce the wrapping effect in interval methods, the computation of the dimension of the solution set with methods used to measure the dimension of fractals, the pitfalls of graph based decomposition methods, the alternative provided by linear algebra, the witness configuration method, the use of randomized provers to detect dependences between constraints, the study of incidence constraints, the search for intrinsic (coordinate-free) formulations and the need for formal specifications.