The Study Of Freeform Surface Optimization Parameterizations

Freeform curves and surfaces are always the focuses of Computer Aided Design(CAD),and play an increasingly significant role in computer animation and games,human-computer interaction and virtual reality.The result of surface parameterization greatly affects the outcome of surface texture mapping,visualization,surface discrete algorithm,and so on.Freeform surface has a rational polynomial mapping from three-dimensional surface to two-dimensional parameter domain.By the surface reparametrization,we can get infinitely various expressions of parameters.According to different reality applications,we are able to choose the most suitable parameterization to gain the best result[11-14].Because of the uniform mathematical model,Non-Uniform Rational B-Splines(NURBS)has become the most popular used surface presentation.However,the strict topological rule that the NURBS control points must lie in a rectangular grid always brings some problem.This means a lot of control points need adding to satisfy the rectangular topology without any other significance.Besides that,more control points could cause higher cost for the follow surface algorithm.In order to break the topological limits,Sederberg et al.proposed T-splines[48,49],which generalized from NURBS surfaces and added T-junctions.The T-junctions allow the surface control points not to be a rectangular grids,so that the T-splines are capable of reducing the number of redundant control points without changing the shape of surface.In this paper,several important parameterization characteristics of freeform surface are discussed.Also the mathematical definition of the orthogonality,the area-preserving property and the conformal property of freeform surface parameterization are given respectively by using the differential geometry of surface.Further more,the constraint conditions and optimization methods of specific parameterization are discussed for different surfaces.For the orthogonality of surface,Bezier surface is taken as an example.The rational bilinear surface with orthogonal parameterization is a rectangle.The explicit orthogonality condition of control points is given for Bezier surfaces of degree 2×2.And a scheme to construct surfaces with orthogonality is then presented.At last,some examples are given and the extension for Bezier surfaces of any degree is discussed.For the conformality of surfaces,an optimization algorithm are given to improve the conformal property of NURBS surface.Firstly,several common transformation in NURBS surface are introduced.Then the conformal energy based on the differential first fundamental form is derived in detail,and the numerical approximation of the result is computed by using the Simpson's rule.The general bilinear transformation and the optimizing method are discussed at last.For the area-preserving property,this paper firstly derives the Mobius transformations and the differential geometry of T-splines.Also a T-splines generating algorithm based on least square method is given.After that,the eqiuiareal energy and an optimizing method are discussed in detial.Some examples are given to show the results of the algorithm.