PlusMaths recently did a nice post about the link between ellipses and radar (here), which inspired me to do my own mini investigation on this topic.  We will work in 2D (with planes on the ground) for ease of calculations!  A transmitter will send out signals – and if any of these hit an object (such as a plane) they will be reflected and received by a receiver.  This locates the object as somewhere on the ellipse formed with the receiver and transmitter as the 2 foci.  When we add a second receiver as shown above then if both receivers receive a signal, then we can narrow down the location of the object as the intersection of the 2 ellipses.

So, for this mini exploration I wanted to find the equations of 2 ellipses with a shared focus so that I could plot them on Desmos.  I then would be able to find the intersection of the ellipses in simple cases when both ellipses’ major axis lies on the x axis.

Defining ellipses

For an ellipse centred at the origin shown above, with foci at c and -c we have:

where c is linked to a and b by the equation:

Rotating an ellipse

Next we can imagine a new ellipse in a coordinate system (u,v)

This coordinate system is created by rotating the x and y axis by an angle of theta radians anticlockwise about the origin.  The following matrix transformation achieves this rotation:

This therefore gives:

and we can substitute this into our new coordinate system to give:

When we plot this we can therefore rotate our original ellipse by any given theta value:

We can use basic Pythagoras to see that the focus point c will become the point c1 shown above with coordinates:

By the same method we can see that the point c2 will have coordinates:

Transformation

Next we want to translate this new ellipse so that it shares a focus point with our original green ellipse.  To do this we need to translate the point c2 to the point c.  This is given by the translation:

So we can therefore translate our ellipse:

Which becomes:

When we plot this we get:

This then gives the 2nd ellipse in blue which does indeed share a focus point at c:

Finding points of intersection

The coordinates of when the 2 ellipses intersect is given by the solution to:

This looks a bit difficult!  So let’s solve an easier problem – the points of intersection when the theta value is 0 (i.e when the ellipses both lie on the x axis).  This simplifies things to give:

and we can find the y coordinates by substituting this into the original ellipse equation.

So the coordinates of intersection are given by:

So – in the above case we would be able to narrow down the location of the plane to 2 locations.  With a 3rd ellipse we could pinpoint the location exactly.