solid of revolution 469

1

–

2

1

–

2

()

the angles αand βsatisfy the fundamental relationship

In general we have:

If the speed of travel through one medium is s

units per second, and through a second

medium is runits per second, then the optimal

route of travel across a boundary separating

the two media occurs when the angle of inci-

dence αand the angle of refraction βsatisfy

the relationship:

This principle is today called Snell’s law of refraction.

Snell observed that a beam of light moving from one

medium to another, say, from air to water, obeys this

law. Light of one color, that is, of one particular

wavelength, will travel through the same medium at a

speed different from that of a beam of a different

color, and so will have its own appropriate angle of

refraction. For this reason, white light—the composi-

tion of all colors—separates into a rainbow of colors

at the interchange of a new medium. Water droplets

suspended in the air after a rainstorm separate the col-

ors of sunlight in this way to produce a large rainbow

visible from the ground.

soap bubbles The film of a soap solution is elastic

and has the property that it always “pulls in on itself”

to create a surface of least possible surface area. Noting

that a soap bubble, containing a fixed volume of air,

always adopts the shape of a

SPHERE

provides a physi-

cal demonstration of the following three-dimensional

variation of the

ISOPERIMETRIC PROBLEM

:

Of all figures with a fixed volume, the sphere

has the least surface area.

Polish mathematician Hermann Schwarz (1843–1921)

provided a mathematical proof of this observation

in 1882.

Many basic questions about arrangements of bub-

bles separating more than one chamber of air are unre-

solved. In 2000 mathematicians Frank Morgan, Michael

Hutchings, Manuel Ritoré, and Antonio Ros managed

to prove that the familiar double bubble seen in nature

(two portions of a sphere attached to a curved boundary

separating the two volumes) is indeed the shape of mini-

mal total surface area for a configuration enclosing two

chambers of air. The corresponding triple-bubble prob-

lem remains open.

solid angle Just as an

ANGLE

in two-dimensional

space is the measure of the amount of length of a

UNIT

CIRCLE

“cut off” by two rays describing the angle, a

solid angle in three-dimensional space is the amount of

surface area of a unit

SPHERE

intercepted by a bundle of

rays emanating from the center of the sphere. A solid

angle need not have any particular shape; it is defined

by the nappe of a

CONE

, which can be of any shape.

The unit of measure of a solid angle is called a stera-

dian. Because the surface area of a sphere is 4π(1)2, there

are a total of 4πsteradians about a point in space. The

measure of a hemisphere is · 4π= 2πsteradians.

Solid angles are used in physics to measure the

fraction of the total emission of radiation from a

source in space. For example, the Earth, of radius

approximately 6370 km, exposes a total surface area of

· 4π(6370)2km2toward the Sun at any time. As the

distance of the Earth from the Sun is approximately

150 million km, we have that the Earth occupies the

fraction of the surface

area of sphere about the Sun of radius 150 million km.

The solid angle of the Earth with respect to the Sun is

this fraction of the full 4πsteradians of space about the

Sun. Notice that this fraction is exceedingly small. Life

on Earth is sustained by less than 1 billionth of the

total energy emitted by the Sun.

The polyhedral angle of a

POLYHEDRON

is the solid

angle of the region formed by projecting one face of the

polyhedron onto a unit sphere.

solid of revolution If the region under a curve y= f(x)

between x= aand x= bis rotated through one revolu-

tion about the x-axis, it generates a three-dimensional

figure called a solid of revolution. More generally, any

figure in the plane rotated about a line in the plane that

does not cut the figure produces a solid of revolution.