Curves

Abstract

A curve in either 2-space or 3-space may be given parametrically by specifying coordinate functions. For example, a circle x in the plane is defined by x₁(t) = cos(t) and x₂(t) = sin(t) for −π < t ≤ π; the argument t is called the parameter of the curve mapping x. The graph of x is thus (x₁(t),x₂(t)) | x₁(t) = cos(t), x₂(t) = sin(t), −π < t ≤ π. In general, a plane curve is a mapping from some interval [a,b] , R into R². A space curve is a mapping from some interval [a,b],R into R³. In either case, the domain of the curve mapping [a, b] may be open, half-open, or closed, and may be the entire real line or may be bounded above and/or below. We thus follow modern tradition that a curve is a mapping, and not a point set; however, we often use language that confuses the graph of a curve with the curve mapping itself. The parametric representation of (the graph of) a space curve is not unique. The circle above can also be represented by x₁(h) = (1 −h ²)/(l + h²) and x₂(h) = 2h/(l + h²) for −∞ < h ≤ ∞; this follows by introducing tan(t/2) for h.