EggMath: The Shape of an Egg
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:
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:
A+B=SD
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 2dimensional 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 xaxis, and call the
other one the yaxis (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 xaxis, then
go a distance y parallel to the
yaxis. 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 xaxis through the two
foci of the ellipse, and to put the yaxis
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) =
((xa)^{2}+(yb)^{2})^{1/2}
(Remember that z^{1/2} means the
squareroot of z.)
If we apply this to our point on the ellipse,
then the defining condition tells us that:
((xD/2)^{2}+y^{2})^{1/2} + ((x+D/2)^{2}+y^{2})^{1/2} = L
We can rearange this to read:
((xD/2)^{2}+y^{2})^{1/2} = L  ((x+D/2)^{2}+y^{2})^{1/2}
Squaring both sides of this equation then leads
to the formula:
2 L
((x+D/2)^{2}+y^{2})^{1/2}
= L^{2} + 2DX
If we again square both sides of the equation
and rearrange things a bit, we get:
(4/L^{2}) x^{2} +
4/(L^{2}d^{2}) y^{2} =
1
This looks a little neater if we use a
to denote the quantity L/2, and use
b to denote the quantity
(L^{2}d^{2})^{1/2}/2.
The equation then becomes:
x^{2}/a^{2} +
y^{2}/b^{2} = 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
x^{2}/a^{2} +
y^{2}/b^{2} = 1

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