A curtate cycloid, sometimes also called a contracted cycloid, is the path traced out by a fixed point at a radius , where is the radius of a rolling circle. Curtate cycloids are used by some violin makers for the back arches of some instruments, and they resemble those found in some of the great Cremonese instruments of the early 18th century, such as those by Stradivari (Playfair 1999).
A curtate cycloid has parametric equations
(1)
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(2)
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The arc length from is
(3)
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where is an incomplete elliptic integral of the second kind.