Parametrized Plane Curves

Abstract

We start with some definitions. Definition 1.1. A (parametrized plane) curve is a continuous mapping m : I → ℝ², where I = [a, b] is an interval. The curve m is closed if m(a) = m(b). A curve m is a Jordan curve if it is closed and m has no self-intersection: m(x) = m(y) only for x = y or {x, y} = {a, b}. The curve is piecewise C ¹ if m has everywhere left and right derivatives, which coincide except at a finite number of points. The range of a curve m is the set m([a, b]). It will be denoted R _m. Notice that we have defined curves as functions over bounded intervals. Their range must therefore be a compact subset of ℝ² (this forbids, in particular, curves with unbounded branches).