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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
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.
Projectile Motion Investigation II
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Another example for investigating projectile motion has been provided by fellow IB teacher Ferenc Beleznay. Here we fix the velocity and then vary the angle, then to plot the maximum points of the parabolas. He has created a Geogebra app to show this (shown above). The locus of these maximum points then form an ellipse.
We can see that the maximum points of the projectiles all go through the dotted elliptical line. So let’s see if we can derive this equation.
Let’s start with the equations for projectile motion, usually given in parametric form:
Here v is the initial velocity which we will keep constant, theta is the angle of launch which we will vary, and g is the gravitational constant which we will take as 9.81.
We can plot these curves parametrically, and for each given value of theta (the angle of launch) we will create a projectile motion graph. If we plot lots of these graphs for different thetas together we get something like this:
We now want to see if the maximum points are in any sort of pattern. In order to find the maximum point we want to find when the gradient of dy/dx is 0. It’s going to be easier to keep things in parametric form, and use partial differentiation. We have:
Therefore we find the partial differentiation of both x and y with respect to t. (This simply means we can pretend theta is a constant).
We can then say that:
We then find when this has a gradient of 0:
We can then substitute this value of t back into the original parametric equations for x:
and also for y:
We now have the parametric equations in terms of theta for the locus of points of the maximum points. For example, with g= 9.81 and v =1 we have the following parametric equations:
This generates an ellipse (dotted line), which shows the maximum points generated by the parametric equations below (as we vary theta):
And here is the graph:
We can vary the velocity to create a new ellipse. For example the ellipse generated when v =4 creates the following graph:
So, there we go, we have shown that different ellipses will be created by different velocities. If you feel like a challenge, see if you can algebraically manipulate the parametric equations for the ellipse into the Cartesian form!
Essential Resources for IB Teachers
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Essential Resources for both IB teachers and IB students
1) Exploration Guides and Paper 3 Resources
I’ve put together a 168 page Super Exploration Guide to talk students and teachers through all aspects of producing an excellent coursework submission. Students always make the same mistakes when doing their coursework – get the inside track from an IB moderator! I have also made Paper 3 packs for HL Analysis and also Applications students to help prepare for their Paper 3 exams. The Exploration Guides can be downloaded here and the Paper 3 Questions can be downloaded here.
IB Maths Resources from Intermathematics 