Geometry for CAAD

Abstract

A geometric system for CAAD is described. The system handles geometric information and relations in the way that a relational database handles data; that is, new relations can be defined from old ones using the standard logical operations of the first-order predicate calculus. The advantage of this approach is that a large number of relations can be defined by the user from a few operations. Implementation of the logical operations is reduced to implementation of four processes: ALGEBRA, GEOMETRY, TRIANGULATION and SYNTHESIS. ALGEBRA takes a finite set of points in N-dimensional space, and constructs an algebraic description of its convex hull. GEOMETRY is the inverse of ALGEBRA. TRIANGULATION breaks a complex shape into convex parts, and SYNTHESIS recombines them. ALGEBRA and GEOMETRY are dual processes; therefore there are three algorithms only to be given. All these algorithms have been studied mathematically, and have also been used separately, on a small scale, in various CAAD systems; but there does not yet exist a large-scale implementation of any of them.