Theory of Map Projection: From Riemann Manifolds to Riemann Manifolds

The Theory of Map Projections is based here on the transformation of Riemann manifolds to Riemann manifolds. Section 2 offers some orientation based on simultaneous diagonalization of two symmetric matrices. We separate simply connected regions. In detail, we review the pullback versus pushforward operations.Section 3 introduces the first multiplicative measure of deformation: the Cauchy–Green deformation tensor, its polar decomposition, as well as singular value decomposition. An example is the Hammer retroazimuthal projection.A second multiplicative measure of deformation is presented in Sect. 4: stretch, or length distortion, and Tissot portrait as well. The Euler–Lagrange deformation tensor presented in Sect. 5 is the first additive measure of deformation based on the difference of the metrics { ds 2 , dS 2 } $$\{\mathrm{ds}^{2},\mathrm{dS}^{2}\}$$ . Section 6 introduces a review of 25 different measures of deformation. First, angular shear is the second additive measure, also called angular distorsion, left and right.Section 8 introduces a third multiplicative measure of deformation, called relative angular shear. In contrast, the equivalence theorem of conformal mapping in Sect. 9 is based on Korn–Lichtenstein equations. Areal distortion in Sect. 10 offers a popular alternative based on the fourth multiplicative and additive measure of deformation, namely, dual deformation called areomorphism. Section 11 offers an equivalence theorem of equiareal mapping. The highlight is our review of canonical criteria in Sect. 12: (i) isometry; (ii) equidistant mapping of submanifolds; (iii) in particular, canonical conformism, areomorphism, isometry, and equidistance; and finally, (iv) optimal map projections. Please study Sect. 13, the exercise: the Armadillo double projection.