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Ellipses (Mathematics Glossary)

What Is an Ellipse?

An ellipse is a shape containing a set of points whose distance from a two fixed points, the foci, add up to a constant.

An ellipse looks like a flattened circle.

Parts of an Ellipse

  • An ellipse has two points inside it, each called a focus. Together they are called foci.



    For every point on the ellipse, the sum of the distances to each focus is the same.

  • The major axis is the longest line segment through the ellipse.



    The major axis passes through both foci and the center of the ellipse.

  • The minor axis is the line segment perpendicular to the major axis, through the center of the ellipse.


  • The semi-major axis is half of the major axis and the semi-minor axis is half of the minor axis.

Properties of Ellipses

The Sum of the Distances from the Foci to Any Point on the Ellipse is Constant

Consider lines drawn from a point on the ellipse P to the two foci X and Y, with length x and y respectively.

The lengths x and y can be added. Let the sum be c:



It doesn't matter which point on the ellipse is chosen, the lengths of the lines drawn from the foci to the point always adds up to the same length:

Eccentricity

The eccentricity e of an ellipse gives a measure of how much the ellipse has been flattened from a circle.

0 < e < 1

The simplest definition of eccentricity is the distance between the two foci divided by the length of the semi-major axis:



Area of an Ellipse

The area of an ellipse is found using the formula:



where a is the semi-major axis and b is the semi-minor axis.

Read more about how to find the area of an ellipse

Equation of an Ellipse

The equation of an ellipse centred on the origin is:



where a is the semi-major axis and b is the semi-minor axis of the ellipse.



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Note

ELLIPTICAL

Anything shaped as an ellipse can be described as elliptical.

A CIRCLE IS A KIND OF ELLIPSE

A circle can be considered a type of ellipse.

The semi-major and semi-minor axes are equal and are called the radius.

(Exercises: Put a = b = r into:

  • the formula for the area of an ellipse

  • the equation of an ellipse
to see if you can get the equation of a circle and the equation of a circle).

The eccentricity of a circle is 0.

DRAWING AN ELLIPSE

The ellipse has the special property that when the lengths of the lines from the foci to a point on the ellipse are added, that length stays the same.

This gives a simple way to draw an ellipse.

Push two pins into a sheet of paper and place a loop of string around them. With a pencil, pull the string taut. Keeping the string taut, draw around.



Because the string length stays the same, the total distance from the pencil to the two foci remains constant. You will draw an ellipse.

This will not be a proper ellipse, as the string will stretch a little. But it will be good enough for some purposes.

If a gardener wishes to make an elliptical flower bed, they will mark out an ellipse using two stakes in the ground and rope. This is called a Gardeners Ellipse.

ELLIPSES AS CONIC SECTIONS

An ellipse is found by slicing through a cone at an angle to the base:



Other shapes, such as circles, parabolas and hyperbolas can also be found by slicing through a cone.

THE WHISPERING GALLERY AT ST. PAUL'S

There is an elliptical room under the dome of St. Paul's Cathedral in London, known as the Whispering Gallery.



If you stand in the focus of this ellipse and whisper, the sound gets reflected off the walls to the other focus of the ellipse.

Also, if you whisper near the wall, the sound will travel around the walls and can be heard at any other point on the wall. The sound has become known as whispering-gallery waves as the effect was first discovered at the St. Paul's Whispering Gallery.