A Geometric relaxation solver for parametric constraint-based models

In this paper, a new relaxation algorithm for solving geometric constraint-based models is proposed. The algorithm starts from a constructive symbolic representation of objects (Constructive Parametric Solid Model, CPSM) and proceeds by iterative relaxation of the geometric constraints. Models that can be reduced to distance and angle constraints can be handled. A new algorithm based on an iterative global deformation of the system is presented and discussed, and its convergence is proved. The performance of hybrid algorithms involving global deformation and individual constraint relaxation is discussed on several practical cases.Postprint (published version

Combining constructive and equational geometric constraint solving techniques

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 define relationships between geometric and technological variables. In general, geometric constraint solving including functional relationships requires a general approach and appropiate techniques toachieve 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.Postprint (published version

A simple construction method for sequentially tidying up 2D online freehand sketches

This paper presents a novel constructive approach to sequentially tidying up 2D online freehand sketches for further 3D interpretation in a conceptual design system. Upon receiving a sketch stroke, the system first identifies it as a 2D primitive and then automatically infers its 2D geometric constraints related to previous 2D geometry (if any). Based on recognized 2D constraints, the identified geometry will be modified accordingly to meet its constraints. The modification is realized in one or two sequent geometric constructions in consistence with its degrees of freedom. This method can produce 2D configurations without iterative procedures to solve constraint equations. It is simple and easy to use for a real-time application. Several examples are tested and discussed

On tree decomposability of Henneberg graphs

In this work we describe an algorithm that generates well constrained geometric constraint graphs which are solvable by the tree-decomposition constructive technique. The algorithm is based on Henneberg constructions and would be of help in transforming underconstrained problems into well constrained problems as well as in exploring alternative constructions over a given set of geometric elements.Postprint (published version

Revisiting variable radius circles in constructive geometric constraint solving

Variable-radius circles are common constructs in planar constraint solving and are usually not handled fully by algebraic constraint solvers. We give a complete treatment of variable-radius circles when such a circle must be determined simultaneously with placing two groups of geometric entities. The problem arises for instance in solvers using triangle decomposition to reduce the complexity of the constraint problem.Postprint (published version

Searching the solution space in constructive geometric constraint solving with genetic algorithms

Geometric problems defined by constraints have an exponential number of solution instances in the number of geometric elements involved. Generally, the user is only interested in one instance such that besides fulfilling the geometric constraints, exhibits some additional properties. Selecting a solution instance amounts to selecting a given root every time the geometric constraint solver needs to compute the zeros of a multi valuated function. The problem of selecting a given root is known as the Root Identification Problem. In this paper we present a new technique to solve the root identification problem. The technique is based on an automatic search in the space of solutions performed by a genetic algorithm. The user specifies the solution of interest by defining a set of additional constraints on the geometric elements which drive the search of the genetic algorithm. The method is extended with a sequential niche technique to compute multiple solutions. A number of case studies illustrate the performance of the method.Postprint (published version