forced to follow a cycloid-shaped path always has the
same period irrespective of the length of the pendulum.
This is called the tautochrone property of the cycloid.
(It is also the case that if a ball starts at rest at any
point of an inverted cycloid and travels along the curve
under the force of gravity, then the time it takes to
reach the lowest point of the curve is independent of
the starting location of the ball.)
Related curves can be considered by following the
path traced by a point on the circumference of a circle
as that circle rolls along another circle. If the circle rolls
on the inside of a fixed circle, then the curve traced is
called a hypocycloid. If a circle rolls on the outside of a
fixed circle, then the curve traced is called an epicy-
cloid. In both cases the fixed circle is called the defer-
ent, and the moving circle is the epicycle.
Some special names are given to the curves cre-
ated in particular situations. For example, when the
two circles have the same radius, the epicycloid pro-
duced is heart-shaped and is called a
CARDIOID
. When
the rolling outer circle has diameter one-fourth that of
the fixed circle, the four-pointed curve produced is
called an astroid. An epicycloid with five cusps is
called a ranunculoid.
The epicycloid was known to A
POLLONIUS OF
P
ERGA
of the third century
B
.
C
.
E
., who used it in his
descriptions of planetary motion.
cylinder In three-dimensional space, a cylinder is the
surface formed by an infinite collection of parallel
straight lines, each passing through one point of a fixed
closed curve drawn in a plane. The closed curve is
called the directrix of the cylinder, and the lines drawn
are called the generators of the cylinder. Often the term
cylinder is used for the solid figure of finite volume con-
fined between two parallel planes. In this setting, the
cylinder has three faces: the two parallel planar regions,
each called a base of the figure, and the lateral surface
given by the straight lines that generate the cylinder. The
base of a cylinder need not be a circle. For example, a
CUBE
satisfies the definition of being a cylinder.
If the lateral surface is at right angles to the base,
then the cylinder is called a right cylinder. All other
cylinders are called oblique. The height of a cylinder is
the perpendicular distance between the two bases.
All horizontal cross-sections of a cylinder are the
same size and shape as the base of the cylinder. C
AVA
-
LIERI
’
S PRINCIPLE
then shows that the volume Vof a
cylinder is given by V= Ah, where his the height of the
cylinder and Ais the area of its base.
A
RCHIMEDES OF
S
YRACUSE
(ca. 287–212
B
.
C
.
E
.)
showed that the volume of a
SPHERE
is two-thirds that
of the volume of the cylinder that contains it. The for-
mula for the volume of a sphere readily follows.
See also
CONE
.
cylindrical coordinates (cylindrical polar coordinates)
In three-dimensional space, the location of a point Pcan
be described by three coordinates—r, θ, and z—called
the cylindrical coordinates of P, where (r, θ) are the
POLAR COORDINATES
of the projection of Ponto the xy-
plane, and zis the height of Pabove the xy-plane. Cylin-
drical coordinates are useful for describing surfaces with
circular symmetry about the z-axis. For example, the
equation of a cylinder of radius 5 with a central axis, the
z-axis can be described by the simple equation r= 5. (As
the angle θvaries between zero and 360°, and the height
zvaries through all values, points on an infinitely long
cylinder are described.) The surface defined by the equa-
tion θ= c, for some constant c(allowing rand zto vary),
is a vertical
HALF
-
PLANE
with one side along the z-axis,
and the surface z= cis a horizontal plane.
A point Pwith cylindrical coordinates (r, θ, z) has
corresponding C
ARTESIAN COORDINATES
(x, y, z) given by:
x= rcos θ
y= rsinθ
z= z
These formulae follow the standard conversion formu-
lae for polar coordinates.
It is usual to present the angle θin
RADIAN MEA
-
SURE
. In this case, a triple integral of the form
∫∫
v∫f(x,y,z)dx dy dz over a volume Vdescribed in Carte-
sian coordinates converts to the corresponding integral
∫∫
v∫f(rcosθ,rsinθ,z) r dr dθdz in cylindrical coordinates.
The appearance of the term rin the integrand follows
for the same reason that rappears in the conversion of
a
DOUBLE INTEGRAL
from planar Cartesian coordinates
to polar coordinates.
See also
ANGLE
;
SPHERICAL COORDINATES
.
cylindrical coordinates 115