# Ellipse

An ellipse usually looks like a **squashed circle**:

"F" is a **focus**, "G" is a **focus**,

and together they are called **foci**.

(pronounced "fo-sigh")

The total distance from **F to P to G** stays the same

In other words, we always travel **the same distance** when going from:

- point "F" to
- to
**any point on the ellipse** - and then on to point "G"

## You Can Draw It Yourself

Put two pins in a board, and then ...

put a loop of string around them,

insert a pencil into the loop,

stretch the string so it forms a triangle,

and draw a curve.

It is 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). So to draw a circle we only need one pin!

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.

### Major Axis Equals f+g

Remember from the top how the distance "f+g" stays the same for an ellipse?

Well f+g is equal to the **length of the major axis**.

Can you think why? (Try moving the point P at the top.)

## Calculations

Area is easy, perimeter is not!

## 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** are **from the center** outwards (not all the way across).

(Note: for a circle, **a** and **b** are equal to the radius, and you get π** × r × r = π r^{2}**, 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:

Remember this is only an approximation! (That is why the "equals sign" is squiggly.)

## Tangent

A tangent line 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?

## Reflection

Light or sound starting at one focus point reflects to the other focus point (because angle in matches angle out):

Have a play with a simple computer model of reflection inside an ellipse.

## Eccentricity

The eccentricity is a measure of how "un-round" the ellipse is.

The formula (using semi-major and semi-minor axis) is:

\frac{√(a^{2}−b^{2})}{a}

## Section of a Cone

We also get an ellipse when we **slice through a cone** (but not too steep a slice, or we 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.

## 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:

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

(similar to the equation of the hyperbola: **x ^{2}/a^{2} − y^{2}/b^{2} = 1**, except for a "+" instead of a "−")

**Or** we can use parametric equations, where we have another variable "t" and we calculate x and y from it, like this:

- x = a cos(t)
- y = b sin(t)

(Just imagine "t" going from 0° to 360°, what x and y values would we get?)