Combining constructive and equational geometric constraint-solving techniques

Abstract

In the past few years, there has been a strong trend towards developing parametric, computer aided design systems based on geometric constraint solving. An efective way to capture the design intent in these systems is to de ne relationships between geometric and technological variables. In general, geometric constraint solving including functional relationships requires a general approach and appropiate techniques to achieve the expected functional capabilities.This work reports on a hybrid method which combines two geometric constraint solving techniques: Constructive and equational. The hybrid solver has the capability of managing functional relationships between dimension variables and variables representing conditions external to the geometric problem. The hybrid solver is described as a rewriting system and is shown to be correct.

Revisiting variable radius circles in constructive geometric constraint solving

“…Joan-Arinyo and Soto described in [10] a hybrid technique that allows to solve constraint problems involving geometric elements with more than two degrees of freedom. In particular it is shown how the method solves variable radius circles attached to one geometric object which is determined up to position and orientation, from now on referred to as a cluster [3], through three constraints.…”

“…Of them, variable radius circles are common constructs in two dimensional constraint solving and are usually not handled fully by constructive solvers. Probably they are the most useful extension as they permit auxiliary construction in addition, as explained by Hoffmann and Vermeer, [9], Hoffmann and Joan-Arinyo, [8], and JoanArinyo and Soto, [10].…”

GA and CHC. Two Evolutionary Algorithms to Solve the Root Identification Problem in Geometric Constraint Solving

“…Figure 2 shows a construction plan generated by the ruler-and-compass geometric constraint solver reported in [14] for the problem depicted in 1. After assigning specific values to the parameters, the constructor interprets the construction plan and builds an object instance, provided that no numerical incompatibilities arise.…”

“…In general, a well constrained geometric constraint problem, [10,13,18], has an exponential number of solutions. For example, consider a geometric constraint problem that properly places n points with respect to each other.…”

“…In what follows, we assume that the set of geometric constraints coherently defines the object under design, that is, the object is generically well constrained and that a ruler-and-compass constructive geometric constraint solver like that reported in [14] is available. In this solver, intersection operations where circles are involved, rc and cc, may lead to up to two different intersection points, depending on whether the second degree equation to be solved has no solution, one or two different solutions in the real domain.…”