Moving into higher dimensions of geometric constraint solving

In 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.