Constrained Biquadratic Polynomial Approximation Of Quadratic Surfaces Using M(o|¨)bius Transformation

Computer Aided Geometric Design(CAGD)mainly deals with free curves and surfaces.Rational curves and surfaces with Bernstein basis functions have good geometric properties,and they have been widely used in practical application.However,due to the complexity of the differential and integral calculation of the rational curves and surface,and the need of the data exchange and transfer in product design,rational curves and surfaces are needed to be approximated by polynomial curves and surfaces.Therefore,polynomial approximation of rational B(?)zier curves and surfaces is still a worthy research topic.Most of the previous methods for polynomial approximation of rational B(?)zier curves and surfaces are discussed in the L2 norm.In this paper,a new distance function based on M(o|¨)bius parameter transformation is proposed to approximate the Hausdorff distance,and we discuss the biquadratic polynomial approximation problem under this new distance for the quadratic surface(ie bi-quadratic rationa B(?)zier patches).The control points of the approximating surface are divided into two parts:boundary control points and interior control points.The four boundary curves and internally the quadratic surfaces with polynomial approximations are respectively performed.The control points of approximating surfaces can be expressed by the two parameters of the M(o|¨)bius transformation.Then,by solving the minimum value of the new approximation distance function,we obtain the optimal parameter values in the M(o|¨)bius parameter transformation,and also the control points of the biquadratic polynomials approximation surface.Finally,some numerical examples are given to demonstrate the effectiveness and correctness of the proposed method.