An interactive N-Dimensional constraint system

Journal ArticleIn 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

Graphs of polyhedra; polyhedra as graphs

AbstractRelations between graph theory and polyhedra are presented in two contexts. In the first, the symbiotic dependence between 3-connected planar graphs and convex polyhedra is described in detail. In the second, a theory of nonconvex polyhedra is based on a graph-theoretic foundation. This approach eliminates the vagueness and inconsistency that pervade much of the literature dealing with polyhedra more general than the convex ones

Moving into higher dimensions of geometric constraint solving

Journal ArticleIn this paper, we present an approach to geometric constraint solving, based on degree of freedom analysis. Any geometric primitive (point, line, circle, plane, etc.) possesses an intrinsic degree of freedom in its embedding space which is usually two or three dimensional. Constraints reduce the degrees of freedom of an object (or a set of objects). We use graph algorithms to determine upper and lower bounds for the degrees of freedom of a set of constrained objects, symbolically. This analysis is then used to establish dependency graphs and evaluation schemes for symbolic or numeric solutions to constraint problems. The approach has been generalized for 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. Also, higher than three dimensional solutions can be interpreted as approaches to over- and under- constrained problems. In this paper, we will present the theoretical background of the approach, and demonstrate how it can be applied within an interactive design environment