Ellipse

An ellipse usually looks like a squashed circle.

 

It is defined by two special points called foci.

"F" is a focus, "G" is a focus,
and together they are called foci.
(pronounced "fo-sigh")

 

 

ellipse

 

The distance from F to P to G is always the same value

In other words, when you go from point "F" to any point on the ellipse and then go on to point "G", you will always travel the same distance.

 

You Can Draw It Yourself

Put two pins in a board, put a loop of string around them, and insert a pencil into the loop. Keep the string stretched so it forms a triangle, and draw a curve ... you will draw an ellipse.

It works because the string naturally forces the same distance from pin-to-pencil-to-other-pin.

A Circle is an Ellipse

In fact a Circle is an Ellipse, where both foci are at the same point (the center).

In other words, a circle is a "special case" of an ellipse. Ellipses Rule!

Definition

An ellipse is the set of all points on a plane whose distance from two fixed points F and G add up to a constant.

Major and Minor Axes

The Major Axis is the longest diameter. It goes from one side of the ellipse, through the center, to the other side, at the widest part of the ellipse. And the Minor Axis is the shortest diameter (at the narrowest part of the ellipse).

The Semi-major Axis is half of the Major Axis, and the Semi-minor Axis is half of the Minor Axis.

Calculations

Area is easy, perimeter is not!

ellipse axes

Area

The area of an ellipse is:

π × a × b

where a is the length of the Semi-major Axis, and b is the length of the Semi-minor Axis. Be careful: a and b go from the edge to the center.

(Note: for a circle, a and b are equal to the radius, and you get π × r × r = πr2, which is right!

Perimeter Approximation

Rather strangely, the perimeter of an ellipse is very difficult to calculate, so I created a special page for the subject: read Perimeter of an Ellipse for more details.

But a simple approximation that is within about 5% of the true value (so long as a is not more than 3 times longer than b) is as follows:

perimeter formula

Remember, this is only a rough approximation!

Tangent

A tangent is a line that just touches a curve at one point, without cutting across it. Here is a tangent to an ellipse:

Here is a cool thing: the tangent line has equal angles with the two lines going to each focus! Try bringing the two focus points together (so the ellipse is a circle) ... what do you notice?


conic section parabola

Section of a Cone

You can also get an ellipse when you slice through a cone (but not too steep a slice, or you get a parabola or hyperbola).

In fact the ellipse is a conic section (a section of a cone) with an eccentricity between 0 and 1.

 

ellipse on xy graph

Equation

By placing an ellipse on an x-y graph (with its major axis on the x-axis and minor axis on the y-axis), the equation of the curve is:

x2/a2 + y2/b2 = 1

(similar to the equation of the hyperbola: x2/a2 − y2/b2 = 1, except for a "+" instead of a "−")