# 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 |
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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 |
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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 |

### Exercises 10.4

**Ex 10.4.1 **What curve is described by \( x=t^2\),\( y=t^4\)? If \(t\) is interpreted as time, describe how the object moves on the curve.

**Ex 10.4.2 **What curve is described by \(x=3\cos t\),\(y=3\sin t\)? If \(t\) is interpreted as time, describe how the object moves on the curve.

**Ex 10.4.3** What curve is described by \(x=3\cos t\),\(y=2\sin t\)? If \(t\) is interpreted as time, describe how the object moves on the curve.

**Ex 10.4.4** What curve is described by \(x=3\sin t\),\(y=3\cos t\)? If \(t\) is interpreted as time, describe how the object moves on the curve.

**Ex 10.4.5** Sketch the curve described by \( x=t^3-t\),\( y=t^2\). If \(t\) is interpreted as time, describe how the object moves on the curve.

**Ex 10.4.6** A wheel of radius 1 rolls along a straight line, say the\(x\)-axis. A point\(P\)is located halfway between the center of the wheel and the rim; assume\(P\)starts at the point \((0,1/2)\). As the wheel rolls, \(P\) traces a curve; find parametric equations for the curve.(answer)

**Ex 10.4.7**A wheel of radius 1 rolls around the outside of a circle of radius 3. A point\(P\)on the rim of the wheel traces out a curve called a **hypercycloid**, as indicated in figure __10.4.3__. Assuming\(P\)starts at the point \((3,0)\), find parametric equations for the curve. (answer)

**Ex 10.4.8**A wheel of radius 1 rolls around the inside of a circle of radius 3. A point\(P\)on the rim of the wheel traces out a curve called a **hypocycloid**, as indicated in figure __10.4.3__. Assuming\(P\)starts at the point\((3,0)\), find parametric equations for the curve. (answer)

**Ex 10.4.9**An **involute** of a circle is formed as follows: Imagine that a long (that is, infinite) string is wound tightly around a circle, and that you grasp the end of the string and begin to unwind it, keeping the string taut. The end of the string traces out the involute. Find parametric equations for this curve, using a circle of radius 1, and assuming that the string unwinds counter-clockwise and the end of the string is initially at\((1,0)\). Figure __10.4.4__ shows part of the curve; the dotted lines represent the string at a few different times. (answer)