Dynamic approximation of complex graphical constraints by linear constraints

Abstract

Current constraint solving techniques for interactive graphical applications cannot satisfactorily handle constraints such as non-overlap, or containment within non-convex shapes or shapes with smooth edges. We present a generic new technique for efficiently handling such kinds of constraints based on trust regions and linear arithmetic constraint solving. Our approach is to model these more complex constraints by a dynamically changing conjunction of linear constraints. At each stage, these give a local approximation to the complex constraints. During direct manipulation, linear constraints in the current local approximation can become active indicating that the current solution is on the boundary of the trust region for the approximation. The associated complex constraint is notified and it may choose to modify the current linear approximation. Empirical evaluation demonstrates that it is possible to (re-)solve systems of linear constraints that are dynamically approximating complex constraints such as non-overlap sufficiently quickly to support direct manipulation in interactive graphical applications.