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Robert M. Guzzo Math 32a Parametric Equations

Robert M. Guzzo Math 32a Parametric Equations. 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 . only to be crushed by a rolling wheel.

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Robert M. Guzzo Math 32a Parametric Equations

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  1. Robert M. Guzzo Math 32a Parametric Equations

  2. 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.

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

  4. 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?

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

  6. Picture of the Problem

  7. 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! WARNING: Trigonometry ahead! WARNING: Trigonometry ahead! WARNING: Trigonometry ahead! WARNING: Trigonometry ahead!

  8. Diagram of the Problem 2r 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 C q P Q rq O X T

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

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

  11. Real-World Example: Gears

  12. 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

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