Cycloid
Download as PPT, PDF0 likes1,254 views
1) The document discusses parametric equations and uses the example of a wheel rolling across a surface to derive parametric equations. 2) As the wheel rolls, the location of a point on the wheel traces out a curve. To describe this curve, the document derives parametric equations that express the x and y coordinates of the point in terms of a third parameter, the angle of rotation. 3) The parametric equations derived are x(θ)=rθ - rsinθ and y(θ)=r - rcosθ, where r is the radius of the wheel and θ is the angle of rotation.
Cycloid
- 1. Robert M. Guzzo Math 32a Parametric Equations
- 2. 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 exampleLet’s look at an example.
- 3. 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.
- 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 parameterparameter to get a pair of parametric equationsparametric 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 r O P C Q θ TX We would like to find the lengths of OX and PX, since these are the horizontal and vertical distances of P from the origin.rθ
- 9. r O P C Q θ TX The Parametric Equations |OX| = |OT| - |XT| rθ rθ r sinθ r cosθ r |PX| = |CT| - |CQ| = |OT| - |PQ| x(θ) = rθ - r sinθ y(θ) = r - r cosθ
- 10. Graph of the Function If the radius r=1, then the parametric equations become: x(θ)=θ-sinθ, y(θ)=1-cosθ
- 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 CycloidHistory of the Cycloid