We have thus far encountered curves as the graphs of functions f (x). Such a graph consists of all points (x, f (x)) in the plane with x in the domain of f. In this chapter, we explore two further ways of describing curves that may be more convenient under certain circumstances.
A parametric curve is described by a pair of functions x(t) and y(t) and consists of the set of points (x(t), y(t)), where t varies over a specified domain. These functions describe how the x- and y-coordinates depend upon a third variable (t, often thought of as representing time), allowing somewhat more freedom than when y is forced to depend upon x directly. Parametric curves can be used, for example, to describe the motion of an object in a plane over a period of time. After a more thorough introduction, we will show how to calculate the velocity and acceleration of an object whose motion is given by parametric functions. We will then investigate how to find the length of a parametric curve.
A polar curve is described by a function r(θ) and consists of the points [r(θ), θ] at a counterclockwise angle of θ radians from the positive x- axis and a distance r from the origin, where θ varies over the domain of the function r(θ). A polar curve is therefore essentially the graph of a function, but in polar coordinates rather than the usual Cartesian coordinates. Polar curves can be used to give relatively simple expressions for intricate and interesting curves. Our focus with polar curves will be to find the area enclosed by a polar curve between two values of θ.