Equations for an Ellipse

We can use the pin and string construction to get a mathematical description for an ellipse. Think about the way that construction works:

  • We have two pins with a loop of string draped around them.
  • Using the tip of a pencil we pull the loop tight.
  • Keeping the string tight, we drag a pencil around the two pins and in so doing we trace out an ellipse.

Notice that at all times, the loop of string is pulled into the shape of a triangle, with the two pins and the pen at the three corners. Suppose that we label the lengths of the three sides A, B and D so that the letters stand for:

Drawing of Ellipse A=distance from pen to first pin
B=distance of pen to second pin
D=distance between pins

We conclude that A+B+D must always be equal to the total length of our string. If we denote this string length by S, then we can write A+B+D=S, or:


Of course only A and B change as we trace out the ellipse -- S and D remain fixed. Our equation thus shows that for all points on the ellipse, the sum of the distances to two fixed points (i.e. the pins) is a constant. In fact this is one way that an ellipse is defined:

An ellipse is the figure consisting of all those points for which the sum of their distances to two fixed points (called the foci) is a constant.

Different looking ellipses correspond to different choices for:

  • the distance D between the two fixed points
  • the (constant) sum of the distances to the fixed points

This description of an ellipse is not the most useful if we want to get detailed information about the ellipse such as the total length around its perimeter or the total area it encloses. For this level of detail we need to introduce the idea of coordinates to describe all points in a flat 2-dimensional plane. The cordinates we need are called Cartesian coordinates, in honour of their inventor, Rene Descartes.

Imagine that our ellipse lies in a flat plane. On this plane we can draw two straight lines which intersect at right angles. Call one of these lines the x-axis, and call the other one the y-axis (it's not too important which is which). Call the point where the intersect the origin, and label it with the pair (0,0). For any other point in the plane, measure its perpendicular distance to the two axes, and call these x and y, as in the diagram. Label the point by the pair (x,y). These are called the Cartesian coordinates of the point. You can think of them as giving the address of the point ("to get to point (x,y), start at the origin, go a distance x along the x-axis, then go a distance y parallel to the y-axis. You can't miss it.")

Our goal is to use Cartesian coordinates to get a convenient description of all the points lying on an ellipse. Given an ellipse, we can locate the coordinate axes anywhere we find convenient relative to the ellipse. As you'll see, it makes sense to draw the x-axis through the two foci of the ellipse, and to put the y-axis exactly midway between them, as in the diagram. If the distance between the foci is D, then their cartesian coordinates will be (-D/2,0) for the one and (+D/2, 0) for the other. Now consider any point on the ellipse, and denote its coordinates by (x,y). By the definition of an ellipse, we know that:

(distance from (x,y) to (-D/2,0) ) + (distance from (x,y) to (+D/2,0) )= a constant (denoted by L)

Before we can go any further, we need to understand how to compute the distance between any two points. Suppose the points have coordinates (x,y) and (a,b). Then:

distance from (x,y) to (a,b) = ((x-a)2+(y-b)2)1/2
(Remember that z1/2 means the square-root of z.)

If we apply this to our point on the ellipse, then the defining condition tells us that:

((x-D/2)2+y2)1/2 + ((x+D/2)2+y2)1/2 = L

We can rearange this to read:

((x-D/2)2+y2)1/2 = L - ((x+D/2)2+y2)1/2

Squaring both sides of this equation then leads to the formula:

2 L ((x+D/2)2+y2)1/2 = L2 + 2DX

If we again square both sides of the equation and rearrange things a bit, we get:

(4/L2) x2 + 4/(L2-d2) y2 = 1

This looks a little neater if we use a to denote the quantity L/2, and use b to denote the quantity (L2-d2)1/2/2. The equation then becomes:

x2/a2 + y2/b2 = 1

and we arrive at the following definition of an ellipse:

An ellipse is the figure consisting of all points in the plane whose cartesian coordinates satisfy the equation
x2/a2 + y2/b2 = 1


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