√b2– c2

√a2– b2

ellipsoid 161

Conversely, one can show that any equation of this

form, with a> b, does indeed yield an ellipse with foci

at positions (±, 0), and whose points Phave

distances from the foci a constant sum 2a. If, on the

other hand, b> a, the equation is again an ellipse, but

this time with foci along the y-axis at (0,±).

The common sum of distances is 2b. (If aequals b, the

curve is a

CIRCLE

.)

The equation shows that an ellipse crosses the x-

axis at x= ±aand the y-axis at y= ±b. The numbers a

and bare called the semimajor axis and the semiminor

axis, respectively.

Ellipses have the following reflection property: any

ray of light emanating from one focus is reflected off the

side of the ellipse directly toward the other focus. This

can be proved by solving an

OPTIMIZATION

problem.

Any room with walls curved in the shape of an ellipse

has the property that any whisper uttered at one focus

can be heard by anyone located at the second focus: not

only do sound waves bounce off the curved wall directed

from one focus to the other, but they also travel the same

distance and so arrive synchronized at the second focus.

The Mormon Tabernacle in Salt Lake City, Utah, and

the Whispering Gallery in the U.S. Capitol building in

Washington, D.C., are built to have this property.

Elliptical mirrors are used for the treatment of kid-

ney stones. By positioning a mirror so that the kidney

stone lies at one focus, medical practitioners can place

a high-intensity sound wave generator at the second

focus. Waves from the generator pass harmlessly

through the patient’s body to then concentrate at the

stone and destroy it.

An ellipse can be drawn using a pencil, a string and

two thumbtacks. Tacking each of the two ends of the

string at fixed locations (the foci), one pulls the string

taut with the tip of the pencil, and then slowly moves

the pencil around, all the while keeping the string taut.

The curve traced is an ellipse, with constant sum of the

distances from the foci being the length of the string.

In the process of deriving the equation of an

ellipse, we presented the equation

Set . This is called the

ECCENTRICITY

of the ellipse and has value between zero

and 1. The above equation can be rewritten:

The numerator of the quantity on the left side is the dis-

tance of a given point Pfrom a focus, and the denomina-

tor is the distance of the point Pfrom the vertical line x=

–a/e, called a directrix of the ellipse. This formulation

provides an alternative characterization of the ellipse:

An ellipse is the set of all points Psuch that the

ratio of its distance from a fixed point (the

focus) to its distance from a fixed line (the direc-

trix) equals a constant ewith value 0 < e< 1.

The

ECCENTRICITY

of a circle is defined to be e= 0.

If e= 1, this characterization gives a

PARABOLA

. For

e> 1, we have a

HYPERBOLA

.

See also A

POLLONIUS

’

S CIRCLE

;

PROJECTION

.

ellipsoid Any geometrical surface or solid sitting in

three-dimensional space possessing the property that

any plane that slices it produces a cross-section that is

either an

ELLIPSE

or a circle is called an ellipsoid. Such a

figure has three axes of symmetry.

An ellipsoid, centered about the origin (0,0,0) has

equation:

+ + = 1

The points (±a,0,0), (0,±b,0) and (0,0,+c) are the loca-

tions where the ellipsoid crosses the x-, y-, and z-axes,

respectively.

One can create an ellipsoid by rotating an ellipse

about one of its axes. This produces a figure with two of

the three quantities a, b, c equal in value. An ellipsoid

produced in this way is called a spheroid, but not every

ellipsoid is a spheroid. If all three quantities a, b, and c

have the same value r, the ellipsoid is a

SPHERE

of radius r.

Mathematicians have shown that the volume of an

ellipsoid is given by (4/3)πabc. (Compare this with the

equation for the volume of a sphere.) Since the time of

L

EONHARD

E

ULER

(1707–83), mathematicians have

attempted to find a simple formula for the surface area

z2

–

c2

y2

–

b2

x2

–

a2