A Reparametrisation of the Sphere through a Conformal Mapping between the Sphere and a Riemann Surface

Conformal transformations are of much interest to modelers of physical phenomena as they provide many attractive mathematical properties such as locally preserving the isotropy of scales, invariance of the structure of the operators such as Laplacian under the transformation. It is known to atmosphere and ocean modelers as to generate coordinate transformations on the sphere using the analytic functions belonging to the class of Mobius transformations which are linear and one-to-one in the complex plane. This work describe the method to use the analytic function that belongs to the class other than the Mobius transformations. Especially the complex power function is used to generate a reparametrisation of the sphere so as to provide variable resolution geomtry on the sphere. It is shown how the High resolution Tropical Belt Transformation is generated from this analytic function. While it is not possible to generate coordinate transformations on the sphere with this class of functions, it is indeed possible to achieve reparametrisation of the sphere. Construction of the Riemann surface is used to achieve this reparametrisation.