Representation in CAD

GAIA Home Intersections in CAD Representation in CAD CAD boundary structures Algebraic curves and surfaces Parametric curves and surfaces Intersection challenges Self-intersection challenges Change of representation Intersection algorithms in GAIA Algebraic geometry for CAD Applications Software The GAIA project Computational Methods for Algebraic Spline Surfaces II	Representation in CAD 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 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. Share Published July 6, 2005

The CAD-boundary structure

Algebraic curves and surfaces

Parametric curves and surface