The representation of volume objects in Computer Aided Design (CAD) is a combination of data structures and mathematically described points, curves and surfaces. The data structures describe the topology of the CAD-model, how volumes are bounded by faces, faces bounded by edges and edges are bounded by vertices. The geometric description of a face is a mathematically described surface, the geometric description of an edge is a mathematically described curve, and the description of a vertex is a point described by coordinates.

The CAD boundary structure is described in the STEP standard (ISO 10303), and is a basis for the CAD-systems and the transfer of data models between different CAD-systems.

### Algebraic curves and surfaces

In CAD systems the curves and surfaces with an algebraic (implicit) representation are with minor exceptions curves and surfaces of total degree 1 or 2:

• Total degree 1: straight line and plane
• Total degree 2: circle, ellipse, hyperbola, parabola, sphere, ellipsoid,...
• Total degree 4: torus

### Parametric curves and surface

The parametric curves and surfaces in CAD are piecewise polynomial and rational curves of degree >0. The representation chosen in NURBS - NonUniform Rational B-splines, a representation format that is numerically stable (minimized rounding errors) and that has very stable numerical algorithms for evaluation, differentiation and more complex operations such as subdivision.

Published June 7, 2005