# 10.4: Parametric Equations

When we computed the derivative \(dy/dx\) using polar coordinates, we used the expressions \(x=f(\theta)\cos\theta\) and \(y=f(\theta)\sin\theta\). These two equations completely specify the curve, though the form \(r=f(\theta)\) is simpler. The expanded form has the virtue that it can easily be generalized to describe a wider range of curves than can be specified in rectangular or polar coordinates.

Suppose \(f(t)\) and \(g(t)\)are functions. Then the equations \(x=f(t)\) and \(y=g(t)\) describe a curve in the plane. In the case of the polar coordinates equations, the variable \(t\) is replaced by \(\theta\) which has a natural geometric interpretation. But \(t\)in general is simply an arbitrary variable, often called in this case a **parameter**, and this method of specifying a curve is known as **parametric equations**. One important interpretation of \(t\) is time. In this interpretation, the equations \(x=f(t)\) and \(y=g(t)\) give the position of an object at time \(t\).

Example 10.4.1 |
---|

Describe the path of an object that moves so that its position at time\(t\)is given by\(x=\cos t$\),\( y=\cos^2 t\).
We see immediately that \( y=x^2\), so the path lies on this parabola. The path is not the entire parabola, however, since \(x=\cos t\) is always between\(-1\)and\(1\). It is now easy to see that the object oscillates back and forth on the parabola between the endpoints \((1,1)\) and \((-1,1)\), and is at point \((1,1)\)at time \(t=0\). |

It is sometimes quite easy to describe a complicated path in parametric equations when rectangular and polar coordinate expressions are difficult or impossible to devise.

Example 10.4.2 |
---|

A wheel of radius 1 rolls along a straight line, say the \(x\)-axis. A point on the rim of the wheel will trace out a curve, called a cycloid. Assume the point starts at the origin; find parametric equations for the curve.
Figure |