- Do you like to travel?
- Surely!
- Have you ever traveled by train?
- Yes! Of course!
- And do you know, that while the train rushes ahead, there are points on its wheel which are moving in the opposite direction?
- How could it be possible?- Well, let's make an experiment. Just imagine, that you are inside a circle and this circle rolls along a straight line. If you had a brush in your hand, what line could it leave on a vertical wall?
- Such a curve as we are imagining was first named cycloid by Galileo in 1599. He studied its properties for 40(!) years.
Cycloid, though one might find it simple, has attracted the attention of many famous scientists. Among them there were Jacob and Johann Bernoulli, Wilhelm Gottfried Leibniz, Blaise Pascal, Rene Descartes, and Pierre Ferma.Cycloid is a curve with some very interesting properties.
For example, the cycloid has the property that a particle P sliding on a cycloid will exhibit simple harmonic motion and the period will be independent of the starting point. This is the tautochrone property and it was discovered by Christian Huygens in 1673. He constructed the first pendulum clock with a device to ensure that the pendulum was isochronous by forcing the pendulum to swing in a cycloid arc.
Also, the cycloid satisfies the brachistochrone property.
During many years mathematicians had been trying to solve the following problem: what form should have a smooth trough connecting two points (one point is higher, than the other), that the metal ball would slide on this trough from one point to the other under action of the weight for the shortest time. One might think, it should be a straight line...- May be, an arc of a circle?
- You know, Galileo suggested the same, but he was wrong. And only in 1696 Johann Bernoulli showed, the cycloid's segment is the "fastest" way for a particle P sliding on a curve connecting two points. These investigations marked the opening of a new field in mathematics, and drastically influenced its further development.
But let's return to the "train's paradox".
As you know, the train's wheels are not just disks, there is a special ledge on each of them. The role of this ledge is very important, it lowers below rail's level and prevents the train from leaving the track. It is the lowest part of the wheel that is moving in the direction opposite to the train's heading.
Here are the parametrical equations, describing these trajectories.
R is the radius of the circle, d is the distance from the center to the point which trajectory is under the question. 
If d < R it is a curtate cycloid while if d > R it is a prolate cycloid.If you try and make the experiments in Logo, you'll come to the drawings like this one.
- And now tell me please, have you ever plaid with a Spirograph?
- Oh yes!
Spirograph... a fascinating way to draw a million marvelous designs! These words are written on the box of my toy. Really, I can draw so many different drawings just with the help of few tools!- ... and the drawings which you are getting, are the cousins of our modest cycloid, for they are traced by a point P on a circle of radius r which rolls on a fixed circle of radius R.
Ms. Cycloid has also other relatives, please meet them.
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Ms.Cycloid relatives