You are currently browsing the tag archive for the ‘radar’ tag.

Screen Shot 2022-11-05 at 3.42.23 PM

Finding planes with radar

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

Screen Shot 2022-11-05 at 2.55.49 PM

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

Screen Shot 2022-11-05 at 4.23.51 PM

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

Screen Shot 2022-11-05 at 4.23.57 PM

Rotating an ellipse

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

Screen Shot 2022-11-05 at 4.25.53 PM

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:

Screen Shot 2022-11-05 at 4.26.00 PM

This therefore gives:

Screen Shot 2022-11-05 at 4.26.03 PM

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

Screen Shot 2022-11-05 at 4.26.07 PM

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

Screen Shot 2022-11-05 at 3.07.11 PM

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

Screen Shot 2022-11-05 at 4.32.20 PM

Screen Shot 2022-11-05 at 4.32.23 PM

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

Screen Shot 2022-11-05 at 4.32.27 PM

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:

Screen Shot 2022-11-05 at 4.35.48 PM

So we can therefore translate our ellipse:

Screen Shot 2022-11-05 at 4.26.07 PM

Which becomes:

Screen Shot 2022-11-05 at 4.35.53 PM

When we plot this we get:

Screen Shot 2022-11-05 at 4.38.54 PM

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

Finding points of intersection

Screen Shot 2022-11-05 at 4.05.42 PM

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

Screen Shot 2022-11-05 at 4.42.06 PM

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:

Screen Shot 2022-11-05 at 4.42.27 PM

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

Screen Shot 2022-11-05 at 4.45.51 PM

So the coordinates of intersection are given by:

Screen Shot 2022-11-05 at 4.46.51 PM

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.

Website Stats

  • 9,171,843 views

About

All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner).

New website for International teachers

I’ve just launched a brand new maths site for international schools – over 2000 pdf pages of resources to support IB teachers.  If you are an IB teacher this could save you 200+ hours of preparation time.

Explore here!

Free HL Paper 3 Questions

P3 investigation questions and fully typed mark scheme.  Packs for both Applications students and Analysis students.

Available to download here

IB Maths Super Exploration Guide

A Super Exploration Guide with 168 pages of essential advice from a current IB examiner to ensure you get great marks on your coursework.

Available to download here.

Recent Posts

Follow IB Maths Resources from Intermathematics on WordPress.com