To Explain the World: The Discovery of Modern Science (48 page)

Note that where the ellipse intersects the horizontal axis we have
y
= 0, so
x
²
=
a
²
, and therefore
x
= ±
a
; thus Eq. (1) describes an ellipse whose long diameter runs from –
a
to +
a
along the horizontal direction. Also, where the ellipse intersects the vertical axis we have
x
= 0, so
y
²
=
b
²
, and therefore
y
= ±
b
, and Eq. (1) therefore describes an ellipse whose short diameter runs along the vertical direction, from –
b
to +
b.
(See Figure 12.) The parameter
a
is called the “semimajor axis” of the ellipse. It is conventional to define the eccentricity of an ellipse as

The eccentricity is in general between 0 and 1. An ellipse with
e
= 0 is a circle, with radius
a
=
b.
An ellipse with
e
= 1 is so flattened that it just consists of a segment of the horizontal axis, with
y
= 0.

Second Definition

Another classic definition of an ellipse is that it is the set of points in a plane for which the sum of the distances to two fixed points (the foci of the ellipse) is a constant. For the ellipse defined by
Eq. (1), these two points are at
x
= ±
ea
,
y
= 0, where
e
is the eccentricity as defined in Eq. (3). The distances from these two points to a point on the ellipse, with
x
and
y
satisfying Eq. (1), are

so their sum is indeed constant:

This can be regarded as a generalization of the classic definition of a circle, as the set of points that are all the same distance from a single point.

Since there is complete symmetry between the two foci of the ellipse, the average distances
and
of points on the ellipse (with every line segment of a given length on the ellipse given equal weight in the average) from the two foci must be equal:
=
, and therefore Eq. (5) gives

This is also the average of the greatest and least distances of points on the ellipse from either focus:

Third Definition

The original definition of an ellipse by Apollonius of Perga is that it is a conic section, the intersection of a cone with a plane at a tilt to the axis of the cone. In modern terms, a cone with its axis in the vertical direction is the set of points in three dimensions
satisfying the condition that the radii of the circular cross sections of the cone are proportional to distance in the vertical direction:

where
u
and
y
measure distance along the two perpendicular horizontal directions,
z
measures distance in the vertical direction, and
α
(alpha) is a positive number that determines the shape of the cone. (Our reason for using
u
instead of
x
for one of the horizontal coordinates will become clear soon.) The apex of this cone, where
u
=
y
= 0, is at
z
= 0. A plane that cuts the cone at an oblique angle can be defined as the set of points satisfying the condition that