An interactive N-Dimensional constraint system

In this paper, we present a graph-based approach to geometric constraint solving. Geometric primitives (points, lines, circles, planes, etc.) possess intrinsic degrees of freedom in their embedding space. Constraints reduce the degrees of freedom of a set of objects. A constraint graph is created with objects as the nodes, and the constraints as the arcs. A graph algorithm transforms the undirected constraint graph into a directed acyclic dependency graph which can be directly used to derive a sequence of construction operations as a symbolic solution to the constraint problem. The approach has been generalized to an n-dimensional space, which, among other things, allows for a uniform handling of 2-D and 3-D constraint problems or algebraic constraints between scalar dimension. Solutions of arbitrary dimensions can be interpreted as approaches to over- and under- constrained problems. In this paper, we present the theoretical background of the approach, and report the results of it's application within an interactive modeling system.