CycloidsDate: 06/16/98 at 16:07:53 From: David Marek Subject: Cycloids What are cycloids, and how do they differ from sine curves? Do they have a general equation? Date: 06/17/98 at 12:53:11 From: Doctor Peterson Subject: Re: Cycloids Hi, David. Your question is phrased in an interesting way, because the difference between these two curves is worth thinking about. Both can be thought of in terms of a point on the circumference of a rolling circle. To make a cycloid, you just follow the path of the point itself: | 2r| *** *** +++ | * * * * * + |* * * * * + + + * * * + + 0*-----------------------*-----------------------*-----+++----- 0 2 pi r 4 pi r The equation is best expressed in parametric form, with both x and y depending on a "time" variable t: x = r(t - sin t) y = r(1 - cos t) To make a sine curve, you ignore the left-right motion of the point around the circle, and just plot its height, with the x-coordinate being the location of the center of the circle; you can think of it as projecting the point onto the vertical diameter of the circle and drawing a point there: | 2r| *** *** +++ | * * * * +--* + | * * * * + + + | * * * * + + 0**---------------------***---------------------***----+++----- 0 2 pi r 4 pi r The parametric equations here would be x = rt y = r(1 - cos t) Yes, this is really y = r(1 - cos x/r) - not the sine, but it's the same shape. Showing it this way makes the relation especially clear. For more about cycloids, see these sites on the Web: Cycloid - MacTutor Math History Archives, St. Andrews http://www-history.mcs.st-and.ac.uk/history/Curves/Cycloid.html (For Xah Lee's Special Plane Curves: Cycloid, don't miss the link at the bottom of this page under Other Web sites to California, USA.) A Java applet from International Education Software (IES) http://www.ies.co.jp/math/java/cycloid/cycloid.html Mechanics - School of Physics, Univ. of Melbourne: http://www.ph.unimelb.edu.au/lecdem/mg3.htm -Doctors Peterson and Sarah, The Math Forum http://mathforum.org/dr.math/ |
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