VQ - Parametrically Defined Curves Lesson

Series and Convergence

Parameter

Curves in the plane can be thought of more generally than the graph of a function. If a curve is thought of as the path of a moving object whose position changes over time, the x- and y-coordinates of the particle's position are a function of a third variable t. In this situation t is the parameter, an arbitrary constant whose value affects the specific nature but not the formal properties of a mathematical expression. Although t represents time in many applications of parametrically defined functions, the parameter does not necessarily have to be t, nor does the parameter have to represent time. The parameter  is also used to represent an angle when parametric equations involve trigonometric functions. One such real-world example is the trajectory (path) that a satellite follows through space as a function of time.

Parametric Equations and Their Graphs

A set of equations expressing a number of quantities as explicit functions of the same set of independent variables (parameters) and equivalent to some direct functional relationship between these quantities is collectively referred to as parametric equations. For a set of parametric equations x = f(t) and y = g(t), each value of t determines a point (x, y). As t varies, the point (x, y) also varies and traces out a curve C. The parametric equations x = f(t) and y = g(t), where f and g are continuous functions of t, taken together with the graph of the parametric equations define a plane curve. One of the advantages of representing curves parametrically is the additional information provided such as the direction (path) of the curve as the parameter varies. In addition, knowing exactly when two different objects pass a given point is vital for airline routing and satellite orbits. The presentation below provides an introduction to parametric equations and their graphs.

The next presentation provides motivation for why parametric equations are helpful in representing real-world situations.

Graphing parametric equations using the parametric mode of a graphing device makes it possible to draw graphs of relationships that are not functions. View the presentation illustrating how to use a graphing calculator to graph a parametrically defined curve.

One of the most famous and interesting curves, traced by a point P on the circumference of a circle rolling along a straight line in a plane, is the cycloid. CLICK HERE to see the resulting path traced by P. Links to an external site.

Image Caption (to the right): Galileo (1564-1642) first called attention to the cycloid and named it in 1599. Many mathematicians studied the cycloid and investigated its properties. Roberval (1602-1675) in 1628 determined its area using the new method of indivisibles that had been developed by Cavalieri (1598-1647). Fermat (1601-1665) and Descartes (1596-1650), as well as Roberval, each came up with a different method for drawing the tangent lines to this curve. Torricelli (1608-1647), a pupil of Galileo, in 1644 published his own discoveries on areas and tangents of the cycloid. Many priority disputes arose in connection with these discoveries, and so the cycloid became known as the "Helen of Geometers".

The Cartesian equation of a cycloid is so complicated that it is rarely used. In contrast simple parametric equations are easily found. The parametric equations defining a cycloid are derived below.

Eliminating the Parameter

Finding a rectangular equation in xy-form that represents the graph of a set of parametric equations is described as eliminating the parameter. This process often makes curve recognition more obvious, although both the domain and range implied by the original set of parametric equations may be changed. The presentation below illustrates how to convert parametric equations to rectangular form by eliminating the parameter.

The next two videos provide multiple examples of eliminating the parameter and graphing a variety of parametrically defined curves.

It should be noted that it is not always possible to eliminate the parameter and produce a rectangular equation. 

Parametrically Defined Curves Practice

Match the parametric equations x = f(t) and y = g(t) with its graph, its equivalent Cartesian equation, or its motion description.