2020/04/10 by Anastasios Zachos, Zachos, Anastasios
Mathematics · #FOS: Mathematics #General Mathematics (math.GM) #math.GM
paper · pdf · doi:10.48550/arxiv.2004.05250
6 pages
arxiv created 2020/04/10 · arxiv updated 2020/04/14
We analyze Toponogov's sine theorem for an infinitesimal geodesic triangle ABC on a C2 regular surface M, which is given in his book [6, Problem 3.7.2] and we provide a generalization of the law of cosines for ABC on M. By replacing in the law of cosines B=\fracπ2 on M, we derive the generalized theorem of Pythagoras on a surface: AC2 = AB2 + BC2 + f(∠ A,\fracπ2,AB,BC)o(AC2) or AC2 = AB2 + BC2 + (∠ A + ∠ C-\fracπ2)2 where f(∠ A,∠ B,AB,BC) is a rational function w.r. to cosA; cosB, sinA, sinB, AB and BC.