Robert M Guzzo Math 32 a Parametric Equations

Robert M. Guzzo Math 32 a Parametric Equations

We’re used to expressing curves in terms of functions of the form, f(x)=y. What happens if the curve is too complicated to do this? Let’s look at an example

An ant is walking along. . . only to be crushed by a rolling wheel. Question: What is the path traced out by its bloody splat? Why would we ask such a question? Mathematicians are sick bastards!!!

Problem Posed Again (in a less gruesome manner) A wheel with a radius of r feet is marked at its base with a piece of tape. Then we allow the wheel to roll across a flat surface. a) What is the path traced out by the tape as the wheel rolls? b) Can the location of the tape be determined at any particular time?

Questions: • What is your prediction for the shape of the curve? • Is the curve bounded? • Does the curve repeat a pattern?

Picture of the Problem

Finding an Equation • f(x) = y may not be good enough to express the curve. • Instead, try to express the location of a point, (x, y), in terms of a third parameter to get a pair of parametric equations • Use the properties of the wheel to our advantage. The wheel is a circle, and points on a circle can be measured using angles. WARNING: Trigonometry ahead!

Diagram of the Problem 2 r r C q P Q rq O X T We would like to find the lengths of OX and PX, since these are the horizontal and vertical distances of P from the origin.

The Parametric Equations r C r q P r cosq |OX| = |OT| - |XT| = |OT| - |PQ| x(q) = rq - r sinq Q r sinq |PX| = |CT| - |CQ| y(q) = r - r cosq rq O T X rq

Graph of the Function If the radius r=1, then the parametric equations become: x(q)=q-sinq, y(q)=1 -cosq

Real-World Example: Gears

For Further Study • Calculus, J. Stuart, Chapter 9, ex. 5, p. 592: The basic problem. Stuart also looks at more interesting examples: • What happens if we move the point, P, inside the wheel? • What happens if we move P some distance outside the wheel? • What if we let the wheel roll around the edge of another circle? • History of the Cycloid