Return to my Mathematics pages
Go to my home page
The Cycloid
© Copyright 2002, Jim Loy
A cycloid (top left) is the path (locus) of a point on a rolling
circle. The other two pictures are the curtate cycloid (the path of a point
inside a rolling circle) and a prolate cycloid (the path of a point outside a
rolling circle). When an automobile moves, no matter what its speed, the
portion of the tire that is in contact with the road is, of course, motionless
for an instant (for zero sec.). The wheels of a train (locomotive) extend below
the top of the tracks. And we see from the third diagram that a point on the
outside edge of such a wheel goes backward for a brief moment before moving
forward again.
Galileo studied and named the cycloid. Various mathematicians found its area
(between the cycloid and the horizontal axis). Others found the arc length.
Gear teeth are often made of cycloids to reduce friction.
In 1696, Johann Bernoulli challenged other mathematicians to solve the
Brachistochrone Problem. He knew that the answer was portion of a cycloid
(inverted from the above diagram). This problem is to find the shape down which
an object can slide accelerated by gravity (ignoring friction) from one point
to another in the shortest time. It was solved by Leibniz, L'Hospital, Newton,
and Jakob Bernoulli. If a fighter pilot wants to reach a point (the safety of a
cloud perhaps) in the shortest time, he should start going straight down and
follow a cycloid which intersects with his intended target.
The cycloid (again inverted from the above diagram) also solves the
Tautochrone Problem. Find the curve down which an object can slide from any
point to the bottom (accelerated by gravity and ignoring friction), always in
the same length of time. Huygens solved this in 1673.
An epicycloid is the path of a circle rolling on the outside of a circle. A
hypocycloid is the path of a circle rolling on the inside of a circle (see
The Astroid). The above diagrams were
drawn using Geometer's Sketchpad.
Return to my Mathematics pages
Go to my home page