Angles and Deformations

Abstract

Any point in the plane can be expressed in polar coordinates, i.e., it can be written in the form (r cos(ϑ), r sin(ϑ)) for some r≥0 and some real number ϑ. The radius r is unique, being the distance from the origin, or the square root of the sums of the squares of the Cartesian coordinates. At the origin, r = 0, and ϑ can be any number. We often denote the origin simply by 0 instead of (0,0). Except for the origin, the angle ϑ is determined only up to adding integral multiples of 2π. We call any of these numbers an angle for the point.