A cycloid generated by a rolling circle
A
cycloid is the curve defined by the path of a
point on the edge of circular wheel as the wheel rolls along a
straight line.It is an example of a
roulette, a curve generated by a curve
rolling on another curve.
The cycloid is the solution to the
brachistochrone problem
(
i.e. it is the curve of fastest descent under gravity)
and the related
tautochrone
problem (
i.e. the period of a ball rolling back and
forth inside this curve does not depend on the ballâ€™s starting
position).
History
The cycloid was first studied by
Nicholas of Cusa and later by
Mersenne. It was named by
Galileo in 1599. In 1634
G.P. de Roberval showed that the area under
a cycloid is three times the area of its generating circle. In 1658
Christopher Wren showed that the
length of a cycloid is four times the diameter of its generating
circle. The cycloid has been called "The Helen of Geometers" as it
caused frequent quarrels among 17th century mathematicians.
Equations

A cycloid generated by a circle of
radius
r = 2
The cycloid through the origin, generated by a circle of radius
r, consists of the points (
x,
y),
with
 x = r(t  \sin t)\,
 y = r(1  \cos t)\,
where
t is a real
parameter,
corresponding to the angle through which the rolling circle has
rotated, measured in
radians. For given
t, the circle's centre lies at
x =
rt,
y =
r.
Solving for
t and replacing, the
Cartesian equation would
be
 x = r \cos^{1} \left(1\frac{y}{r}\right)\sqrt{y(2ry)}
The first arch of the cycloid consists of points such that
 0 \le t \le 2 \pi.\,
The cycloid is
differentiable
everywhere except at the
cusps
where it hits the
xaxis, with the derivative tending
toward \infty or \infty as one approaches a cusp. It satisfies the
differential
equation
 \left(\frac{dy}{dx}\right)^2 = \frac{2ry}{y}.
Area
One arch of a cycloid generated by a circle of radius
r
can be parametrized by
 x = r(t  \sin t),\,
 y = r(1  \cos t),\,
with
 0 \le t \le 2 \pi.\,
Since
 \frac{dx}{dt} = r(1 \cos t),
we find the area under the arch to be
 \begin{align}
A &= \int_{t=0}^{t=2 \pi} y \, dx = \int_{t=0}^{t=2 \pi}
r^2(1\cos t)^2 \, dt \\&= \left. r^2 \left( \frac{3}{2}t2\sin
t + \frac{1}{2} \cos t \sin t\right) \right_{t=0}^{t=2\pi}
\\&= 3 \pi r^2.\end{align}
Arc length
The arc length
S of one arch is given by
 \begin{align}
S &= \int_{t=0}^{t=2 \pi}
\left(\left(\frac{dy}{dt}\right)^2+\left(\frac{dx}{dt}\right)^2\right)^{1/2}
\, dt \\&= \int_{t=0}^{t=2 \pi} 2r \sin\left(\frac{t}{2}\right)
\, dt \\&= 8r.\end{align}
Cycloidal pendulum
If its length is equal to that of half the cycloid, the bob of a
pendulum suspended from the cusp of an
inverted cycloid, such that the "string" is constrained between the
adjacent arcs of the cycloid , also traces a cycloid path. Such a
cycloidal pendulum is
isochronous,
regardless of amplitude. This is because the path of the pendulum
bob traces out a cycloidal path (presuming the bob is suspended
from a supple rope or chain); a cycloid is its own
involute curve, and the cusp of an inverted cycloid
forces the pendulum bob to move in a cycloidal path.
The 17th Century Dutch mathematician
Christiaan Huygens discovered this
property of the cycloid and applied it to the design of more
accurate clocks for use in navigation.
Related curves
Several curves are related to the cycloid. When we relax the
requirement that the fixed point be on the edge of the circle, we
get the
curtate cycloid and the
prolate
cycloid. In the former case, the point tracing out the
curve is inside the circle, and, in the latter case, it is outside.
A
trochoid refers to any
of the cycloid, the curtate cycloid and the prolate cycloid. If we
further allow the line on which the circle rolls to be an arbitrary
circle then we get the
epicycloid (circle rolling on outside of
another circle, point on the rim of the rolling circle), the
hypocycloid (circle on
the inside, point on the rim), the
epitrochoid (circle on the outside,
point anywhere on circle), and the
hypotrochoid (circle on the inside,
point anywhere on circle).
All these curves are
roulettes with
a circle rolled along a uniform
curvature.
The cycloid, epicycloids, and hypocycloids have the property that
each is
similar to its
evolute. If
q is the
product of that curvature with the
circle's radius, signed positive for epi and negative for hypo,
then the curve:evolute
similitude ratio is
1 + 2
q.
The classic
Spirograph toy traces out
hypotrochoid and
epitrochoid
curves.
Use in architecture
Cycloidal arches at the Kimbell Art
Museum
The
cycloidal arch was used by architect Louis
Kahn in his design for the Kimbell Art Museum in Fort Worth, Texas.It was also used in the
design of the Hopkins Center in Hanover, New Hampshire.
See also
References
 An application from physics: Ghatak, A. &
Mahadevan, L. Crack street: the cycloidal wake of a cylinder tearing through a sheet.
Physical Review Letters, 91, (2003).
http://link.aps.org/abstract/PRL/v91/e215507
External links