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 epicycloid. In both cases the fixed circle is called the deferent, and the moving circle is the epicycle.
Some special names are given to the curves created in particular situations. For example, when the
two circles have the same radius, the epicycloid produced 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 APOLLONIUS OF
PERGA 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 confined 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. CAVALIERI’S PRINCIPLE then shows that the volume V of a
cylinder is given by V = Ah, where h is the height of the
cylinder and A is the area of its base.
ARCHIMEDES OF SYRACUSE (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 formula 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 P can
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 P onto the xyplane, and z is the height of P above the xy-plane. Cylindrical 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 θ varies between zero and 360°, and the height
z varies through all values, points on an infinitely long
cylinder are described.) The surface defined by the equation θ = c, for some constant c (allowing r and z to vary),
is a vertical HALF-PLANE with one side along the z-axis,
and the surface z = c is a horizontal plane.
A point P with cylindrical coordinates (r, θ, z) has
corresponding CARTESIAN COORDINATES (x, y, z) given by:
x = r cos θ
y = r sin θ
z = z
These formulae follow the standard conversion formulae for polar coordinates.
It is usual to present the angle θ in RADIAN MEASURE. In this case, a triple integral of the form
∫∫ v
∫ f(x,y,z)dx dy dz over a volume V described in Cartesian coordinates converts to the corresponding integral
∫∫ v ∫ f(r cosθ, r sinθ,z) r dr dθ dz in cylindrical coordinates.
The appearance of the term r in the integrand follows
for the same reason that r appears in the conversion of
a DOUBLE INTEGRAL from planar Cartesian coordinates
to polar coordinates.
See also ANGLE; SPHERICAL COORDINATES.
cylindrical coordinates 115
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 epicycloid. In both cases the fixed circle is called the deferent, and the moving circle is the epicycle.
Some special names are given to the curves created in particular situations. For example, when the
two circles have the same radius, the epicycloid produced 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 APOLLONIUS OF
PERGA 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 confined 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. CAVALIERI’S PRINCIPLE then shows that the volume V of a
cylinder is given by V = Ah, where h is the height of the
cylinder and A is the area of its base.
ARCHIMEDES OF SYRACUSE (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 formula 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 P can
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 P onto the xyplane, and z is the height of P above the xy-plane. Cylindrical 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 θ varies between zero and 360°, and the height
z varies through all values, points on an infinitely long
cylinder are described.) The surface defined by the equation θ = c, for some constant c (allowing r and z to vary),
is a vertical HALF-PLANE with one side along the z-axis,
and the surface z = c is a horizontal plane.
A point P with cylindrical coordinates (r, θ, z) has
corresponding CARTESIAN COORDINATES (x, y, z) given by:
x = r cos θ
y = r sin θ
z = z
These formulae follow the standard conversion formulae for polar coordinates.
It is usual to present the angle θ in RADIAN MEASURE. In this case, a triple integral of the form
∫∫ v
∫ f(x,y,z)dx dy dz over a volume V described in Cartesian coordinates converts to the corresponding integral
∫∫ v ∫ f(r cosθ, r sinθ,z) r dr dθ dz in cylindrical coordinates.
The appearance of the term r in the integrand follows
for the same reason that r appears in the conversion of
a DOUBLE INTEGRAL from planar Cartesian coordinates
to polar coordinates.
See also ANGLE; SPHERICAL COORDINATES.
cylindrical coordinates 115