Most geometries on the plane R 2 are non-Euclidean. Let s denote arc length. Then Euclidean geometry arises from the formula for curve length ds 2 = dx 2 + dy 2 , · · ds Ñ Ñ Ñ Ñ Ñ Ñ Ñ dx · dy which itself arises from applying the Pythagorean theorem to each segment in a piecewise-linear approximation to a curve. Given a definition of ds, one then measures… (More)