MATH 131 -- Calculus for Physical and Life Sciences
The cycloids
September 17, 2007
The cycloids are the curves given by the parametric equations
If 
, the curve is an (ordinary) cycloid; if a > b, the curve
is called a curtate cycloid; if a < b, the curve is called a
prolate cycloid.
Here are the shapes. First a = b = 1
| > | ![plot([`+`(t, `-`(sin(t))), `+`(1, `-`(cos(t))), t = 0 .. `+`(`*`(4, `*`(Pi)))]); 1](images/cycloids_3.gif){kind=small} |
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Next, 
| > | ![plot([`+`(t, `-`(`*`(`/`(2, 3), `*`(sin(t))))), `+`(1, `-`(`*`(`/`(2, 3), `*`(cos(t))))), t = 0 .. `+`(`*`(4, `*`(Pi)))], x = 0 .. `+`(`*`(4, `*`(Pi))), y = 0 .. 2); 1](images/cycloids_6.gif){kind=small} |
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Finally, 
| > | ![plot([`+`(t, `-`(`*`(`/`(3, 2), `*`(sin(t))))), `+`(1, `-`(`*`(`/`(3, 2), `*`(cos(t))))), t = 0 .. `+`(`*`(4, `*`(Pi)))]); 1](images/cycloids_9.gif){kind=small} |
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| > |