CYCLOID : Exploration

We have seen how the cycloid is generated by a point on a circle of radius a rolling along the x-axis. The cycloid illustrated above has a = 1. (How can we tell this?) This generation of the cycloid tells us that the distance between two adjacent cusps (where the curve meets the x-axis) is 2 a.

The cycloid has parametric equations

x = a (t – sin t), y = a(1 – cos t), 0 t < 2.

Let us prove this in the case a = 1. Since the circle has radius 1, and is rolling along the horizontal axis, OQ equals the length of arc PQ , and this is equal to the angle PCQ. Let us call this measurement t (for the angle, measured in radians). We now calculate the coordinates of the point P. Working from the point Q, and using the illustrated position of P, we obtain

          x = t – sin ( – t) = t – sin t, y = 1 + cos ( – t) = 1 – cos t, as required.

You can check that these parametric formulae continue to hold for other positions of P.

It can also be shown that the area of the region bounded by a single arch of the curve and the x-axis is 3 a2.

The cycloid occurs as the solution of a couple of interesting physical problems.



The tautochrone property

A tautochrone or isochrone curve is the curve for which the time taken by an object sliding without friction to its lowest point is independent of its starting point. It is unclear how this problem actually arose. Even the existence of such a curve is not obvious.

The solution of the tautochrone problem, that is, the attempt to identify this curve, was solved by Christiaan Huygens in 1659. He proved geometrically in his Horologium oscillatorium (‘The Pendulum Clock’, 1673) that the cycloid has this property. We might note that around this period, there was great interest in applying mathematics to all sorts of physical and mechanical problems.

You can find a mathematical solution to the problem in this Wikipedia reference:

http://en.wikipedia.org/wiki/Tautochrone_curve ,

but it is quite difficult. The above animated gif comes from this Wikipedia page.

We can also illustrate the property with an applet.

Click the diagram at right to activate the applet. Sliding the point ‘k’ along the top segment positions the point K on the inverted cycloid. Now clicking the ‘Animate’ button or manually dragging the point Q along the horizontal axis causes the points to slide. What do you notice?

We find that for all points K, the points P and K reach the lowest point of the cycloid at precisely the same time.

[Note on the applet: This applet illustrates the result because it has been constructed to do so! In fact, the velocities of the points P and K are not accurately portrayed here.]


The brachistochrone property

A property of the cycloid related to the tautochrone property is the brachistochrone problem:

Find the shape of the curve down which an object sliding from rest and accelerated by gravity will slip (without friction) from one point to another in the least time.

The term derives from the Greek brachistos (the shortest) and chronos (time). Isaac Newton was challenged to solve the problem in 1696, and did so the very next day! The solution is known to be an arc of the cycloid.

A proof of this result can be found in Wolfram MathWorld:

http://mathworld.wolfram.com/BrachistochroneProblem.html

The mathematics is quite difficult.


BIBLIOGRAPHY

A Book of Curves, Lockwood, E. H. C.U.P. 1967),

Wikipedia : http://en.wikipedia.org/wiki/Cycloid

Wolfram MathWorld : http://en.wikipedia.org/wiki/Cycloid