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, 2BF₁ = BF₁ + BF₂ = 2a and hence BF₁ = a. Similarly, BF₂ = 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 = kC.

Now, consider simultaneously the graph of the ellipse and that of the circle x² + y² = a² as shown in Figure 3.

Figure 3

Let x satisfy -a ≤ x ≤ a and, let (x, y) and (x, y₀) be the indicated points on the ellipse and circle, respectively. Since (x, y₀) satisfies x² + y² = a² and y₀≥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 πab. 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.