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There are several motions that create a cycloid. I have some examples here. Are there any others?

  1. Trace of a fixed point on a rolling circle
  2. Evolute of another cycloid (the locus of all its centers of curvature)
  3. Involute of another cycloid (trace of a pendulum constrained to another cycloid)
  4. Envelope of a family of lines with uniformly varying angle and intercept
Erfan Salavati
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  • Items 1,2 and 3 are known. The property 4) is not immediately recognizable (by me) without a sketch or better geometrical description. – Narasimham Aug 07 '15 at 13:55
  • I just conjectured item 4 and then verified it analytically. I don't know if it is new or not. – Erfan Salavati Aug 07 '15 at 14:01
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    (4) is an interesting observation. For specificity, the family of lines can be written as $$\frac{x}{t} + \frac{y}{t\tan t} = 1 \qquad\text{or}\qquad y = (t-x)\tan t$$ so that the $x$-intercept matches the measure of an angle made with the $x$-axis. Using the standard envelope-finding technique, one obtains $$x = \frac12(u+\sin u) \qquad\qquad y = -\frac12 (1-\cos u)$$ (where $u=2t$, but that doesn't matter). Nice! – Blue Aug 07 '15 at 19:07
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    @Blue, have a look at the cartoon in my answer. :) – J. M. ain't a mathematician Dec 11 '15 at 20:15

3 Answers3

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May I interest y'all in a short cartoon?

two cycloids

If you look carefully at the cartoon, you'll see two cycloids being generated by the same rolling circle. The first one is the usual case, where a point on the rolling circle's circumference traces out the cycloid.

The other, smaller cycloid is being generated by a related mechanism: it is the envelope of the diameter of the rolling circle!

Skipping the details, it can be shown that if the larger cycloid has the parametric equation $\left(t-\sin t\quad 1-\cos t\right)^\top$ the smaller cycloid has the corresponding equation $\left(\frac{2t-\sin 2t}{2}\quad\frac{1-\cos 2t}{2}\right)^\top$.

J. M. ain't a mathematician
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  • +1. Very nice. It's perhaps worth noting that the length of each smaller arc is $4$, while the length of the diameter is (of course) $2$. Therefore, the diameter "slips" as it brushes along the smaller cycloids. – Blue Dec 11 '15 at 22:58
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The Brachistochrone curve between two points at the same height is a cycloid.

Seven
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In dynamics, time taken for rolling oscillation of a small heavy marble irrespective of amplitude in such a shaped trough.. is constant $( = 2 \pi \sqrt {\frac{4 a}{g}}) $.. Tautochrone property.

EDIT1:

Distance of any cycloid point to x-axis ( on which the circle rolls) along its normal is half the radius of its curvature...one of its properties.

Narasimham
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