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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));y=r(1−cos(t)). Note: There is a small error
towards the end of the video. The top of
the curve is at (πr,2r), not at (πr,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−asin(t));y(t)=r(1−acos(t)). |