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Surface Representation

The rational Bezier surface (Fig. 1b) has the form

(1)


where is the positional vector of a point on the surface, are Bezier control points, are weights of Bezier control points, and stand for Bernstein polynomials, and are the independent curvilinear coordinates of the surface ranging from to , and denote the surface degrees (orders are greater by ) in and parametric directions, respectively, and stands for the rational Bernstein polynomial of the form

(2)


If the control points are arranged in a matrix of type then the corner entries define the corner vertices which are generally the only control points interpolated by the surface, the side entries (including the corresponding side corner entries) define the control polygon of the boundary curve of the surface, and the remaining points control the bow of the surface. The boundary curves are the rational Bezier curves (Fig. 1a) that can be written in a similar form

(3)


where refers to a point on the curve, are Bezier control points,  are weights of Bezier control points, denote Bernstein polynomial,  stands for a curve parameter varying in range from to , is the curve degree and represents the rational Bernstein polynomial expressed as

(4)


Points and correspond to curve end vertices. The first and last segments of the control polygon coincide with the curve tangent at the starting and ending vertices, respectively. A curve may be degenerated into a single point (a collapsed curve) if all its control points including the end vertices are merged at this point. This allows to model ``triangular'' surfaces or even more degenerated surfaces. To prevent the degeneration of a surface into a curve or point, no two adjacent curves are allowed to be degenerated into a single point and no two opposite or adjacent curves are allowed to be the same.

The Bernstein polynomials can be expressed as

(5)


or recursively as

(6)


where . Note that the ordinary Bezier entities can be derived from the rational Bezier entities when all weights are set to .

The rational Bezier curve is generally differentiable up to its degree. The first derivative of with respect to the parameter may be written as

(7)


where

(8)


Similarly, the rational Bezier surface is differentiable in each parametric direction up to the degree in that direction. The first derivatives of with respect to parameters and are given by

(9)


(10)


where

(11)
   

(12)
   

The derivatives of Bernstein polynomials in Eqs (8), (12), and (13) can be expressed recursively as

(13)


where

(14)




Next: Surface Metric Tensor Up: Indirect Triangulation of 3D Surfaces Previous: Indirect Triangulation of 3D Surfaces

Daniel Rypl
2005-12-07