The model is described by a boundary representation and consists of the following model entities:

- vertices,
- curves,
- surfaces,
- patches,
- shells, and
- regions.

Currently, rational Bezier entities are employed for free-form curves and surfaces representation. This allows to represent exactly conics and quadrics by entities starting with an order of three (quadratic curves and biquadratic surfaces).

The rational Bezier curve has the form

where is the point on the curve, are Bezier control points, are weights of Bezier control points, stand for Bernstein polynomials, denotes an independent variable varying in range from to , and is the curve degree. The curve order is equal to . and correspond to model vertices while the remaining points form the control polygon of the curve. They determine the bow of the curve and need not generally lie on the curve. The first and last segments of the control polygon coincide with the curve tangent in the starting and ending vertices respectively.

The rational Bezier surface can be written in a similar form

where is the point on the surface, are Bezier control points, are weights of Bezier control points, and stand for Bernstein polynomials, and denote independent parameters varying in range from to , and and are surface degrees (orders are equal to and ) in and parametric directions, respectively. If the control points are arranged in a matrix then the corner points correspond to model vertices, the side points correspond to control polygons of model curves bounding the surface, and the remaining points form the control polygon of the surface and need not generally lie on the surface.

Bernstein polynomial can be expressed as

(3) |

or recursively as

(4) |

where .

The ordinary Bezier entities can be derived from rational Bezier entities when all weights are set to 1.

Two types of model entities are distinguished. The physical ones which are designated for the actual discretization and the virtual ones which serve as auxiliary for geometry description or mesh size specification. Note that there are some restrictions on the fixation between virtual and physical entities.

*Daniel Rypl
2008-06-04*