Solving global geometric constraints on free-form curves

Abstract

Constraint-based geometric modeling is the standard modeling paradigm in current modern CAD systems. Generally, the user defines constraints on the geometric objects and a solver is applied to find a configuration of the geometry, which satisfies these constraints. Proper application of these constraints allows rapid modification of the geometry without loss of design intent. However, in current CAD systems, constraint solving for free-form geometric objects is generally limited. In particular, constraining global features such as limits on a curve's curvature values, are not supported. In this paper we present a general method, within the constraint-based framework, to construct global constraints on free-form curves. The method starts by defining sufficient conditions on the curves in terms of an inequality expression, unlike local constraints the global constraint expression will be defined for all the domain of the curves. We then transform the expression into a symbolic polynomial, whose coefficients are symbolic expressions of the original curves. In the final step, a set of inequality constraints is applied in terms of the symbolic coefficients. These inequality constraints enforce the positivity of the symbolic polynomial. The final inequality constraints are fed into the solver along with any other local constraints, which the user has provided on the curves. Therefore, the solution returned by the solver satisfies both the global constraints and any other local constraints the user supplies. We have implemented a prototype of our method using existing commercial constraint solvers. We present results on several problems, which are handled as global geometric constraints using our method.