z

–

ρ

r

–

ρ

spherical geometry 473

described by three coordinates ρ,θ, and ϕ, called the

spherical coordinates of P, where ρis the distance of P

from the origin Oof a C

ARTESIAN COORDINATE

system, θ

is the angle between the x-axis and the projection of the

line segment connecting Oto Ponto the xy-plane (mea-

sured in a counterclockwise sense from the x-axis), and ϕ

is the angle from the z-axis to the line segment connect-

ing Oto P. The angle θ, called the longitude, takes a

value between zero and 360°, and the angle ϕ, called the

colatitude, takes a value between zero and 180°. The

angles can also be presented in

RADIAN MEASURE

.

Spherical coordinates are useful for describing sur-

faces with spherical symmetry about the origin. For

example, the equation of a sphere of radius 5 is simply

ρ= 5. (As the angles θand ϕvary in their allowed

ranges, the points of a sphere are described.) The sur-

face defined by the equation θ= c, for some constant c

(allowing ρand ϕto vary), is a vertical

HALF

-

PLANE

with one side along the z-axis, and the surface ϕ= cis

one nappe of a right circular

CONE

.

A point Pwith spherical coordinates (ρ,θ,ϕ) has

corresponding Cartesian coordinates (x,y,z) given by:

x= ρsin ϕcos θ

y= ρsin ϕsin θ

z= ρcos ϕ

To see this, project the line segment of length ρconnect-

ing Oto Ponto the xy-plane. Call the length of this line

segment r. Then, according to the

POLAR COORDINATE

conversion formulae, we have x= rcosθand y= rsinθ.

(Examine the right triangle in the xy-plane with the seg-

ment of length ras hypotenuse.) According to the right

triangle in a vertical plane, with this line segment of

length ras base, the line segment connecting Oto Pon

length ρas hypotenuse, and a vertical leg of length z,

we have cosϕ= and cosϕ= . Thus z= ρcosϕand

r= ρsinϕ, and the conversion formulae now follow.

To convert from Cartesian coordinates to spherical

coordinates use:

(The first equation follows from the standard

DISTANCE

FORMULA

.)

A triple integral of the form ∫∫

V∫f(x,y,z)dx dy dz over

a volume Vdescribed in Cartesian coordinates converts

to the corresponding integral:

∫∫

V∫f(ρsinϕcosθ, ρsinϕsinθ, ρcosϕ) ρ2sinϕdρdθdϕ

in spherical coordinates. The appearance of the term ρ2

sinϕin the integrand follows for reasons analogous to

the appearance of rin the conversion of a

DOUBLE

INTEGRAL

from planar Cartesian coordinates to polar

coordinates.

As an example, using radian measure, the volume

Vof a sphere of radius Rcan be computed as:

See also

CYLINDRICAL COORDINATES

.

spherical geometry (elliptic geometry, Riemannian

geometry) Discovered in 1856 by German mathe-

matician G

EORG

F

RIEDRICH

B

ERNHARD

R

IEMANN

(1826–66), and later slightly modified by F

ELIX

K

LEIN

(1849–1925), spherical geometry is a

NON

-E

UCLIDEAN

GEOMETRY

in which the famous

PARALLEL POSTULATE

fails in the following manner:

Through a given point not on a given line,

there are no lines parallel to that given line.

Riemann used the surface of a

SPHERE

as a model of this

geometry by interpreting the word line to mean a great

circle on the sphere. Given that in a theory of geometry

two lines are meant to intersect at just one point (yet

any two great circles intersect at two

ANTIPODAL

POINTS

), it is appropriate then to interpret the word

point in spherical geometry as an antipodal pair of

points on the surface. In this setting, it is now also true

that any two distinct points determine a unique line.