#### ELLIPSES

Many
mathematical textbooks contain comprehensive accounts of what an ellipse is and
what its properties are. A recommended reference text is “Basic Calculus: from
Archimedes to Newton to its Role in Science” written by Alexander J Hahn, and
in particular, pages 54 – 55 and 90 – 94.

Here,
some of the properties have been selected and stated below for easy reference.

Figure
1

With
reference to Figure 1, the **standard equation of the ellipse** is

Eccentricity,
*e*, is defined as the ratio of the distance between the
centre of the ellipse and one of the foci to the semimajor axis, or

Then,

Since
the point *B *= (0,*b*)
is on the ellipse,
2*BF*_{1}
= *BF*_{1} + *BF*_{2} = 2*a*
and hence
*BF*_{1}
= *a.*
Similarly,
*BF*_{2}
= *a*.
By Pythagoras’ Theorem,

Figure
2

Referring
to Figure 2, **Cavalieri’s Principle** states that if
*d*_{x}
= *kc*_{x}
for all *x* and for a fixed
positive number *k*, then D =
*k*C.

Now, consider simultaneously the graph of the ellipse
and that of the circle
*x*^{2}
+ *y*^{2}
= *a*^{2}^{
}^{
} as shown in Figure 3.

Figure 3

Let *x* satisfy
-*a* ≤
*x* ≤ *a*
and, let (*x*, *y*) and
(*x*, *y*_{0})
be the indicated points on the ellipse and circle, respectively. Since (*x*,
*y*_{0})
satisfies *x*^{2}
+ *y*^{2}
= *a*^{2}
and *y*_{0}≥0
it follows that
.

Since (*x*,
*y*) is on the ellipse,

_{
}

The above relation is frequently used in later
calculations. In addition, if we suppose that the upper semicircles and the
upper part of the ellipse are separated as shown in Fig. 4(a), we would then
have demonstrated that
*d*_{x }= *kc*_{x} for all *x*.

Figure 4

Since
the area of a semicircle of radius *a *is
, it follows by Cavelieri’s principle that the area of the upper half of the
ellipse is equal to
.
Therefore, the full ellipse with semimajor axis *a* and semiminor axis *b
*has area
π*ab*. Note that Cavalieri’s principle also applies to Figure 4(b). In particular,
the area of the elliptical section has area
times that of the semicircular
section.