Spline Functions On Spheres And Tori

Because all of the surfaces in the real world can be regarded as Riemannian Surfaces, constructing splines whose parametric domain is arbitrary Riemannian Surfaces and effectively computing such splines in real-world applications are of fundamental importance in solid and shape modeling, geometric design,graphics,etc. In paper[3],Gu proved that the existence of splines on Riemannian Surfaces is equivalent to the existence of an affine atlas and presented a theoretical framework. They also pointed that the existence of an affine atlas is soley determined by the topology of the Riemann Surfaces. In article [3], they triangulate the Riemann Surface, and then find the approximate affine atlas by variational method and conformal invariants.In this paper, the author has studied how to construct spline functions (Triangular NURBS) on the sphere with the north pole removed S~2 \ {(0,0,1)} and the standard torus T~2 in detail. Because the sphere with the north pole removed S~2 \ {(0,0,1)} and the standard torus T~2 admit standard complex analytic structures, we can obtain their holomorphic 1-forms without triangulating them. In this way, We also obtain the affine atlas and construct splines on them.This article contains three works as follows:1. This article has proved that the bi-holomorphic maps between S2 \ {(0,0,1)} and C are unique up to an affine transformation. Because the splines are invariant under the affine transformation, this means that there is the unique affine atlas on S2 \ {(0,0,1)} essentially. For the torus, this article also has proved that there is the unique affine atlas essentially.2. This article has described the sphere with the north pole removed S2\{ (0,0,1)}(§4.3) and the standard torus T2(§4.4) in detail and constructed splinefuntions on them.3. This article has characterized the approximation properties and the stability of the basis fuctions on compact support of the Riemann Surfaces. It follows that the spline functions defined on Riemann Surfaces has better approximation properties and the stability on compact support. Further more, this article concluded that the splines defined on torus are stable.