hyperboloid 255

Taking the

DERIVATIVE

we see:

and

In analogy to the ordinary trigonometric functions, math-

ematicians define four additional hyperbolic functions:

The hyperbolic cosine curve arises in nature as the

shape of the curve formed by a chain hanging freely

between two points. This curve is called a

CATENARY

.

See also O

SBORNE

’

S RULE

.

hyperbolic geometry (Lobachevskian geometry) Inde-

pendently discovered in 1823 by Hungarian mathemati-

cian J

ÁNOS

B

OLYAI

(1802–60) and in 1829 by Russian

mathematician N

IKOLAI

I

VANOVICH

L

OBACHEVSKY

(1792–1856), hyperbolic 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 is more than one line parallel to that

given line.

French mathematician J

ULES

H

ENRI

P

OINCARÉ

(1854–1912) later provided a simple model for this

geometry and the means to easily visualize geometric

results in this theory. The “Poincaré disk” consists of

all the points in the interior of the

UNIT CIRCLE

. A

“point” in geometry is any point inside this circle, and

a “line” is to be interpreted as a circular arc within

the circle with endpoints perpendicular to the bound-

ary of the circle. Any diameter of the boundary circle

is also considered a line. Distances are not measured

with a traditional ruler: points on the boundary circle

are considered to be infinitely far from the center of

the circle.

Bolyai and Lobachevsky showed that all but the

fifth of E

UCLID

’

S POSTULATES

hold in the hyperbolic

geometry and, moreover, that this model of geometry is

consistent (that is, free of

CONTRADICTION

s). This

establishes that the parallel postulate cannot be logi-

cally deduced as a consequence of the remaining

axioms proposed by Euclid.

In hyperbolic geometry, all angles in triangles sum

to less than 180°, and the ratio of the circumference of

any circle to its diameter is less than π. (Moreover, the

value of this ratio is not the same for all circles.) Also,

it is possible for two perpendicular lines to be parallel

to the same line.

Physicists, following the work of A

LBERT

E

INSTEIN

,

suggest that the geometry of our universe is hyperbolic:

that it appears to us as Euclidean is a result of the fact

that we occupy such a small portion of it. (This is anal-

ogous to the fact that it is difficult to recognize the

Earth as round when living on it.)

See also E

UCLIDEAN GEOMETRY

; P

LAYFAIR

’

S AXIOM

;

SPHERICAL GEOMETRY

.

hyperboloid The

SOLID OF REVOLUTION

obtained by

rotating a

HYPERBOLA

about one of its axes is called a

hyperboloid. If the rotation is performed about the axis

that lies between the two branches of the curve, then

the resulting surface is called a hyperboloid of one

sheet. It resembles a cylinder “pinched” at the center so

as to curve inward. Points on this surface satisfy an

equation of the form + = + 1 for some

constants a, b, and c. If, instead, the rotation is per-

formed about the axis that connects the two foci of the

hyperbola, then the resulting surface has two distinct

parts, each resembling a bowl. This surface is called a

hyperboloid of two sheets. Points on this surface satisfy

the equation + = – 1.

One can construct a model of a hyperboloid of

one sheet by holding two metal rings, one directly

above the other, and tying vertical strings from points

on the lower ring to their corresponding points on the

top ring. If one then rotates the top ring so that the

vertical strings begin to tilt, the model begins to “con-

strict.” The resulting surface is a hyperboloid of one

sheet. One can also see this surface by tying a string

around a handful of uncooked spaghetti strands.

z2

–

c2

y2

–

b2

x2

–

a2

z2

–

c2

y2

–

b2

x2

–

a2