|
M408M Learning Module Pages
Main page Chapter 10: Parametric Equations and Polar CoordinatesLearning module LM 10.1: Parametrized Curves:3 kinds of functions, 3 kinds of curvesThe cycloid Visualizing parametrized curves Tracing curves and ellipses Lissajous figures Learning module LM 10.2: Calculus with Parametrized Curves:Learning module LM 10.3: Polar Coordinates:Learning module LM 10.4: Areas and Lengths of Polar Curves:Learning module LM 10.5: Conic Sections:Learning module LM 10.6: Conic Sections in Polar Coordinates:Chapter 12: Vectors and the Geometry of SpaceChapter 13: Vector FunctionsChapter 14: Partial DerivativesChapter 15: Multiple Integrals |
The cycloidOne of the most important examples of a parametrized curve is a cycloid. This is the path followed by a point on the rim of a rolling ball. If you've ever seen a reflector on the wheel of a bicycle at night, you've probably seen a cycloid. It is very difficult to describe a cycloid using graphs or level sets, but as a parametrized curve it's fairly simple. The following video derives the formula for a cycloid:$$x=r(t-\sin(t));\qquad y=r(1-\cos(t)).$$Please watch carefully, since this example will show up repeatedly in later learning modules. Note: There is a small error towards the end of the video. The top of the curve is at $(\pi r, 2r)$, not at $(\pi r, r)$. Also, you can't really mount a light or reflector on a bike tire completely on the rim. More realistically, if the reflector is mounted a fraction $a$ of the way to the rim, then the trajectory of the reflector is actually $$x(t) = r(t-a \sin(t));\qquad y(t)=r(1-a\cos(t)).$$ This gives a curve similar to a cycloid, but without the sharp cusps at the bottom. |