By Erik W. Grafarend, Rey-Jer You, Rainer Syffus (auth.)
In the context of Geographical info platforms (GIS) the publication deals a well timed evaluation of Map Projections. the 1st chapters are of foundational variety. We introduce the mapping from a left Riemann manifold to a correct one precise as conformal, equiaerial and equidistant, point of view and geodetic. specifically, the mapping from a Riemann manifold to a Euclidean manifold ("plane") and the layout of assorted coordinate structures are reviewed . A speciality is the remedy of surfaces of Gaussian curvature 0. the most important half is dedicated to the mapping the field and the ellipsoid-of-revolution to tangential airplane, cylinder and cone (pseudo-cone) utilizing the polar element, transverse in addition to indirect point. a number of Geodetic Mappings in addition to the Datum challenge are reviewed. within the first extension we introduce optimum map projections through variational calculus for the field, respectively the ellipsoid producing harmonic maps. the second one extension stories substitute maps for constructions , specifically torus (pneu), hyperboloid (cooling tower), paraboloid (parabolic mirror), onion form (church tower) in addition to clothoid (Hight velocity Railways) utilized in undertaking Surveying. 3rd, we current the Datum Transformation defined by means of the Conformal staff C10 (3) in a threedimensional Euclidean house , a 10 parameter conformal transformation. It leaves infinitesimal angles and distance ratios equivariant. Numerical examples from classical and new map projections in addition to twelve appendices record the fantastic global of Map Projections.
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Sample text
15 (Orthogonal projection S2R+ onto P2O , Cartesian coordinates, the second problem). 13. 100) (R − x )(R − y ) − x y R = 2 , 2 2 2 2 [R − (x + y )] R − (x2 + y 2) (x2 + y 2 )2 2 −1 ]) − 4 det[C G ] = , (tr[Cr G−1 r r r [R2 − (x2 + y 2 )]2 R2 R λ21 = λ2+ = 2 , = λ = + , λ 1 + R − (x2 + y 2) R2 − (x2 + y 2 ) det[Cr G−1 r ] = det[Cr ] = 2 2 2 2 2 2 2 λ22 = λ2− = 1, λ2 = λ− = +1. 101) g11 (C12 − λ22 g12 )2 − 2g12 (C11 − λ22 g11 )(C12 − λ22 g12 ) + g22 (C11 − λ22 g11 )2 = = x2 + y 2 , R2 − (x2 + y 2 ) 1 −y .
We pose four problems. (i) Prove that the images of meridians and parallel circles are straight lines. Prove the half length condition between the images of the central meridian and the pole, respectively, and the equator. (ii) Derive the left Cauchy–Green deformation tensor. (iii) Solve the left general eigenvalue-eigenvector problem. Prove the condition of an equiareal mapping Λ1 Λ2 = 1. (iv) Prove that at {Λ = 0, Φ = 0} the special pseudo-cylindrical projection is not an isometry. Fig. 9. Special pseudo-cylindrical projection of the sphere of type Eckert II (Eckert 1906), Tissot ellipses of distortion Solution (the first problem).
4 demonstrates regions which are not simply connected. Those regions are characterized by closed curves which can be laid around the inner holes and which cannot be contracted to a point within the region. The holes are against contraction. The 1 From Riemann Manifolds to Riemann Manifolds 5 Fig. 3. Simply connected regions mapping f : M2l → M2r is usually called deformation. In addition, the mappings f∗ (pullback) versus f ∗ (pushforward) of the left tangent space T M2l onto the right tangent space T M2r , also called pullback (right derivative map, Jacobi map Jr ), and of the right tangent space T M2r onto the left tangent map T M2l , also called pushforward (left derivative map, Jacobi map Jl ), are of focal interest for the following discussion.