Studys on the laws of Gauss projection for area distortion

In the regions which has high altitude and is far from central meridian, Gauss projection has bigger area distortion and becomes main factors which influences the accuracy of area measurement. Through researching in detail the area distortion in the three respects that from ground to reference ellipsoid surface from reference ellipsoid surface to Gauss plane and from Gauss plane to compensating level surface, this paper finds the laws of Gauss projection for area distortion, and draws one computation model on 2000 coordinates system. This has contributed to restrict the influence of Gauss projection and choose suitable central meridian and compensating level surface.

Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for -quasiconformal harmonic mappings of unit disk

UDC 517.51 We study the hyperbolically Lipschitz continuity, Euclidean and hyperbolic area distortion theorem, and coefficient estimate for the classes of -quasiconformal harmonic mappings from the unit disk onto itself.

Weighted area/angle distortion minimization for Mesh Parameterization

Purpose Mesh Parameterization is central to reverse engineering, tool path planning, etc. This work synthesizes parameterizations with un-constrained borders, overall minimum angle plus area distortion. This study aims to present an assessment of the sensitivity of the minimized distortion with respect to weighed area and angle distortions. Design/methodology/approach A Mesh Parameterization which does not constrain borders is implemented by performing: isometry maps for each triangle to the plane Z = 0; an affine transform within the plane Z = 0 to glue the triangles back together; and a Levenberg–Marquardt minimization algorithm of a nonlinear F penalty function that modifies the parameters of the first two transformations to discourage triangle flips, angle or area distortions. F is a convex weighed combination of area distortion (weight: α with 0 ≤ α ≤ 1) and angle distortion (weight: 1 − α). Findings The present study parameterization algorithm has linear complexity [𝒪(n), n = number of mesh vertices]. The sensitivity analysis permits a fine-tuning of the weight parameter which achieves overall bijective parameterizations in the studied cases. No theoretical guarantee is given in this manuscript for the bijectivity. This algorithm has equal or superior performance compared with the ABF, LSCM and ARAP algorithms for the Ball, Cow and Gargoyle data sets. Additional correct results of this algorithm alone are presented for the Foot, Fandisk and Sliced-Glove data sets. Originality/value The devised free boundary nonlinear Mesh Parameterization method does not require a valid initial parameterization and produces locally bijective parameterizations in all of our tests. A formal sensitivity analysis shows that the resulting parameterization is more stable, i.e. the UV mapping changes very little when the algorithm tries to preserve angles than when it tries to preserve areas. The algorithm presented in this study belongs to the class that parameterizes meshes with holes. This study presents the results of a complexity analysis comparing the present study algorithm with 12 competing ones.

Explicit Bounds and Sharp Results for the Composition Operators Preserving the Exponential Class

Letf:Ω⊂Rn→Rnbe aquasiconformal mappingwhose Jacobian is denoted byJfand letEXP(Ω)be the space of exponentially integrable functions onΩ. We give an explicit bound for the norm of the composition operatorTf:u∈EXP(Ω)↦u∘f-1∈EXP(f(Ω))and, as a related question, we study the behaviour of the norm oflog⁡Jfin the exponential class. TheA∞property ofJfis the counterpart in higher dimensions of the area distortion formula due to Astala in the plane and it is the key tool to prove the sharpness of our results.