What is an ellipse?

Geometry defines an ellipse as the path described by a moving point, when the sum of its distances from two fixed points remains constant. Each of these fixed points is called a focus (plural focii). The diagram below explains it a bit more simply. As the point P moves, it traces out the orange line. At all positions of P, the sum of the distances a and b will always be constant.

a + b = constant

The ellipse was first studied by a Greek mathematician named Menaechmus (around 380 BC) who was said to have been the tutor of Alexander the Great.

In 1605 the astronomer Johannes Kepler famously demonstrated that the planets in our solar system move in ellipses with the sun at one focus. In fact, it was Kepler who first coined the word ‘focus’ in this context.

How to draw an ellipse

Some people may remember, from their school days, drawing an ellipse with a pencil, two thumbtacks and a loop of string. Place two thumbtacks a short distance apart into a piece of stiff paper or cardboard. Loop the string around the thumbtacks and, with the pencil always held tightly outwards against the string, draw a closed curve. The curve you have drawn is an ellipse. Each thumbtack is one of the focii of the ellipse.

Reflective properties of an ellipse

Any ray passing through one focus of an ellipse will will reflect off of the ellipse and pass through the other focus.

This has practical applications in physics. There are many famous buildings with elliptically shaped ‘whisper galleries’ or ‘whisper chambers’. St. Paul's Cathedral, designed by architect and mathematician Sir Christopher Wren, contains one such whisper gallery. If a person stands in this room at one of the foci and utters even the slightest whisper, another person standing at the other focus can hear it.