Research On Comformal Map Projection Based On Riemann Surface

Comformal projection is an important class of projection,it is widely used in surveying and mapping,remote sensing and Geographic Information Science.However,the traditional model of conformal map projection tied to Euclidean geometry,the lack of uniformity in the theory and algorithms,has been difficult to meet the needs of modern geographic information science.Direct conversion of complex map projection,Gauss law section oblique axis projection,the polar regions nonsingular isometric and other issues,the more apparent deficiencies.Comformal projection model for the existing problems in this paper noted that conformal geometry and contact isometric natural on the introduction of conformal geometry isometric research,try to build a unified expression based on the theory of Riemann surfaces and isometric operation framework for isometric model provides a unified mathematical theory underlying support to overcome the existing zoning isometric,exotic drawbacks,simplifying meter projection equations and analysis;to achieve a direct conversion isometric model,find a new category isometric formula to solve new application requirements."Projection Analysis>--theoretical basis->Projection Model>Projection expression" to form the basic framework to the theory and application of direct conversion of rich map projection.The main contents and findings include:(1)A detailed discussion of the intrinsic relationship between Riemannian geometry,Riemann surface theory and map projections.Gives a measure of the deformation tensor and projection,affine connection with equidistant projection,the conformal structure of the underlying mathematical basis isometric projection.Finally,the research status of conformal geometry and calculate the Ricci flow,with the Riemann surface theory popularized the isometric projection.(2)Based on the map projection model Riemannian measure the deformation.Map projection with Riemannian deduced deformation tensor form,avoiding the drawbacks of traditional projector complicated formula.Secondly,from the general surface mapping study of map projection,the projection range of promotional applications.Followed by the introduction of a parallel movement isometric conformal structure and geodesic,simplification of the conformal and equidistant map projection models.With common latitude,positive Mercator projection and Gauss,for example,demonstrate the superiority of the new theory.These results provide a theoretical basis for the derivation of a new map projection equations.(3)From the same constitutive model based isometric Mobius transformation.To achieve isometric direct conversion introduces conformal automorphism transformation group,on this basis will be positive,horizontal,oblique axis azimuth isometric projection unified projective special unitary group,and promotion for the Riemann sphere conformal automorphism group.Then,according to the topological details the different genus Riemann surface and the lower boundary of Mobius group analytic model prove any isometric isomorphic simply connected Riemann surface,parse out the Mobius group after mapping providers uniquely determined.The study shows that the model based on the Riemann surface isometric simplifies the traditional isometric expression,get a new class of isometric projection.(4)Gauss projection model based on elliptic functions.Proved intrinsic relationship elliptic functions and isometric projection,with the elliptic function reinterpreted a large common isometric projection.This chapter will Jacobi elliptic functions as a projection of land charts Mercator direct conversion chart Gauss projection model,with periodic and singular points of elliptic functions Jocabi no singular regardless drawn with a single cycle of Gauss projection,ie no singular point Gauss pole area projection.