Geometric Constraint Solving With Solution Selectors

Abstract

Current parametric CAD systems are based on solving equality types of constraints between geometric objects and parameters. This includes algebraic equations constraining the values of variables, and geometric constraints constraining the positions of geometric objects. However, to truly represent design intent, next-generation CAD systems must also allow users to input other types of constraints such as inequality constraints. Inequality constraints are expressed as inequality expressions on variables, or as geometric constraints that force geometric objects to be on specific sides or have specific orientations with respect to other objects. The research presented here investigates whether the frontier algorithm can be extended to solve geometry positioning problems involving systems of equality- and inequality-based declarations in which the inequality-based declarations are used as solution selectors to choose from multiple solutions inherently arising when solving these systems. It is found that these systems can be decomposed by the frontier algorithm in a manner similar to purely equality-based constraint systems, however they require tracking and iterating through multiple solutions and in many cases may require backtracking through the solution sequence. The computational complexity of the new algorithm is found to be the same as the frontier algorithm in the planning phase and linear in the execution phase with respect to the size of the problem but exponential with respect to the distance of solution selection steps from the satisfaction steps they control.