Computing parameter ranges in constructive geometric constraint solving: Implementation and correctness proof

doi:10.1016/j.cad.2012.02.012

Computer-Aided Design

Volume 44, Issue 7, July 2012, Pages 709-720

https://doi.org/10.1016/j.cad.2012.02.012 Get rights and content

Abstract

In parametric design, changing values of parameters to get different solution instances to the problem at hand is a paramount operation. One of the main issues when generating the solution instance for the actual set of parameters is that the user does not know in general which is the set of parameter values for which the parametric solution is feasible. Similarly, in constraint-based dynamic geometry, knowing the set of critical points where construction feasibility changes would allow to avoid unexpected and unwanted behaviors.

We consider parametric models in the Euclidean space with one internal degree of freedom. In this scenario, in general, the set of values of the variant parameter for which the parametric model is realizable and defines a valid shape is a set of intervals on the real line.

In this work we report on our experiments implementing the van der Meiden Approach to compute the set of parameter values that bound intervals for which the parametric object is realizable. The implementation is developed on top of a constructive, ruler-and-compass geometric constraint solver. We formalize the underlying concepts and prove that our implementation is correct, that is, the approach exactly computes all the feasible interval bounds.

Highlights

► We describe an implementation of an algorithm to compute critical points in constructive geometric constraint solving. ► A fully developed case study illustrates how the algorithm works. ► We prove that the algorithm is correct.

Introduction

Many applications in computer-aided design, computer-aided manufacturing, kinematics, robotics or dynamic geometry are conveniently modeled by geometric problems defined by geometric constraints with parameters, some of them representing dimensions. These generic models allow the user to easily generate specific instances for various parameter and constraint values.

When parametric models are used in real applications, it is often found that instantiation may fail for some parameter values. Assuming that failures are not due to bugs in the system, they should be attributed to a more basic problem, that is, a certain combination of constraints in the model and values of parameters do not define a valid shape. We consider parametric models of geometric objects in the Euclidean plane with one degree of freedom corresponding to a variant parameter, after discounting rotations and translations of the entire object.

In general, the set of values of the variant parameter for which the parametric model is realizable and defines a valid shape is a set of intervals on the real line. The goal of this work is to implement the van der Meiden et al. approach [1], [2], to figure out the set of values of the variant parameter that bound these intervals. The approach is built on top of a constructive ruler-and-compass solver. We prove that the algorithm is correct in the sense that it yields all and only bounds of the interval feasible values.

Computing the set of parameter values for which a parametric object is realizable is a long standing problem. However, the literature published concerning this problem is scarce. Shapiro and Vossler, [3], and Raghothama and Shapiro, [4], [5], [6], developed a theory on validity of parametric family of solids by investigating the relationship between Brep and CSG schemas in systems with dual representations for solid modeling. The formulation is built on formalisms of algebraic topology. Unfortunately, it seems a rather difficult problem transforming these formalisms into effective algorithms.

Joan-Arinyo and Mata [7] reported on a method to compute feasible ranges for parameters in geometric constraint solving under the assumption that values assigned to parameters are non-trivial-width intervals. The method applies to complex systems of geometric constraints in both 2D and 3D and has been successfully applied in the dynamic geometry field, [8]. It is a general method, the main drawback, however, is that it is based on numerical sampling.

Hoffmann and Kim [9] developed a constructive approach to calculate parameter ranges for systems of geometric constraints that include sets of isothetic line segments and distance constraints between them. Model instantiation for distance parameters within the ranges output by the method preserve the topology of the set of isothetic lines.

In an illuminating work, van der Meiden [1], and van der Meiden and Bronsvoort, [2], reported on a constructive method to calculate parameter ranges for systems of geometric constraints. Constraint systems are restricted to systems of distance and angle constraints on points and straight lines in 2D or 3D spaces that are well constrained and decomposable respectively into triangular and tetrahedral subproblems. The method automatically determines the allowable range for a single parameter of the system, called variant parameter, that an actual solution exists for any value in the range. The method consists of two steps. First a set of values for the variant parameter, called critical points, [10], for which some well defined subproblem feasibility changes is computed. Once sorted, critical points define a sequence of intervals and their feasibility is established by checking feasibility at some point within each interval.

Gao and Sitharam, in [11], [12], described a general result concerning the computation of critical points for 2D problems with one degree of freedom which include just points and distance constraints that can be abstracted as one degree of freedom Henneberg graphs. Here we consider problems including distance and angle constraints that can be abstracted as tree decomposable graphs, a superset of Henneberg graphs.

The van der Meiden et al. method is the subject of our study. In Section 2 we recall basic concepts on constructive geometric constraint solving. Section 3 formalizes the geometric constraint problem with one variant parameter. The van der Meiden method and our implementation are described in Sections 4 The van der Meiden method, 5 Our implementation respectively. Section 6 is devoted to prove that the method is correct. Section 7 describes a case study to illustrate how the approach works. Finally, in Section 8 we offer a short discussion.

Section snippets

Preliminaries

First we recall some basic concepts related to geometric constraint solving in general. Then we focus on the constructive technique. For an in depth discussion on this topic see, for example, [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23].

Problems with one variant parameter

In this section we present basic concepts concerning geometric constraint problems for which the value of a given constraint parameter is not fixed.

The van der Meiden method

The van der Meiden et al. method to compute the domain of the variant parameter has two steps, [1], [2]. In the first step the critical values for the variant parameter, that is, variant parameter values for which the construction plan feasibility might change, are computed. Then, in each interval defined by two subsequent critical values, pick a value for the variant parameter and determine whether the construction plan is feasible. Since intervals may be open or closed at each bound,

Our implementation

Algorithms have been implemented in Java(TM) SE 1.6.0_24, running on a personal computer featuring an Intel(R) Xeon(R) E5530 CPU running at 2.40 GHz and 3 GB RAM. For the sake of making it easier to read, we report them here in pseudo-code.

Let $T$ be the decomposition tree that solves the constraint problem at hand. In our implementation, each node in $T$ stores:

1.
$T . built$ : A Boolean flag that takes value true whenever the coordinates of the geometric object have been actually computed with respect

Algorithm correctness

In this section we show the correctness of our algorithm to compute the domain of the variant parameter of a geometric constraint solving problem with one variant parameter. A general result concerning the computation of critical points of 2D constraint systems including only points and point–point distance constraints with one degree of freedom that can be abstracted as one degree of freedom Henneberg graphs can be found in [11], [12]. Problems considered here include points, straight lines

Case study

To further illustrate how our algorithm works, we develop a case study, depicted in Fig. 15. From left to right, the figure shows a piston and connecting rod crankshaft, an abstraction of the piston represented as a geometric constraint problem, and an actual construction plan. The set of geometric elements includes four points ${p_{0}, p_{1}, p_{2}, p_{3}}$ and a straight line $l$ . The set of constraints includes four point–point distances, $d (p_{1}, p_{0}) = d_{0}, d (p_{1}, p_{2}) = d_{1}, d (p_{2}, p_{3}) = d_{2}$ , and $d (p_{0}, p_{3}) = d_{3}$ ; and three

Discussion

In parametric geometric modeling and its applications in fields like computer aided design and dynamic geometry, knowing the set of parameter values for which the model is realizable is paramount. This is a long standing open problem.

Van der Meiden in [2] and van der Meiden in [1] reported an approach to compute feasible parameter ranges for 2D and 3D geometric constraint-based, parametric problems. The problems considered in these works include points and straight lines as geometric elements

Acknowledgments

We thank the editor and four anonymous referees for their valuable comments and constructive suggestions which helped to improve the manuscript.

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View full text

Computing parameter ranges in constructive geometric constraint solving: Implementation and correctness proof

Abstract

Highlights

Introduction

Section snippets

Preliminaries

Problems with one variant parameter

The van der Meiden method

Our implementation

Algorithm correctness

Case study

Discussion

Acknowledgments

Computer-Aided Design

Nonlinear Analysis

Computer-Aided Design

Computer-Aided Design

Computer-Aided Design

Computers & Graphics

Computers & Graphics

Artificial Intelligence

What is a parametric family of solids?

Necessary conditions for boundary representations variance

Boundary representation deformation in parametric solid modeling

ACM Transactions on Graphics

Consistent updates in dual representation systems

Towards valid parametric CAD models

Computer-Aided Design