Robert M. Guzzo Math 32a 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 Let’s look at an example.

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

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. parameter parametric equationsInstead, 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 2r r O P C Q  T X We would like to find the lengths of OX and PX, since these are the horizontal and vertical distances of P from the origin. rr

r O P C Q  T X The Parametric Equations |OX| = |OT| - |XT| rr rr r sin  r cos  r |PX| = |CT| - |CQ| = |OT| - |PQ| x(  ) = r  - r sin  y(  ) = r - r cos 

Graph of the Function If the radius r=1, then the parametric equations become: x(  )=  -sin , y(  )=1-cos 

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 CycloidHistory of the CycloidHistory of the CycloidHistory of the Cycloid