HIDE AND SEEK

There has been a robbery in your very precious house. You have to search the criminal. How are you going to do that using mathematics?
Prof. Bernard Silverman, a chief scientific advisor of London revealed a very stunning secret-the game of hide and seek is played at the basis of mathematics. Cool, isn’t it?

CONIC SECTION-It is a curve obtained as an intersection of a cone and a plane. These are the conic sections we usually study about- Circle, Ellipse, Parabola and a Hyperbola. This simple figure below shows how these conic sections are formed.

If the intersection of the plane is perpendicular to the axis of double cone, we obtain a circle. If it intersects at an angle less than the slope of the cone, we get an ellipse. If the plane is angled parallel to the side of the cone the intersection is a parabola and if it is angled so as to cut through both cones the intersection is a hyperbola. Now this is the geometrical meaning. If we see through coordinate geometry, we might define conic sections in the following manner. We would obtain a circle if the locus of points is at the same distance from the focus. In case of ellipse, the sum of the distances from the foci must be a constant (|x+y|=c). In hyperbola, the difference of distances from the foci must be a constant (|x-y|=c). As in the case of parabola, it is the locus of points which have the same distance from focus and a line called directrix.

Now, wondering why we’re briefing about conic sections? It is very simple. We use these awesome geometrical figures to trace out a person’s location. For this purpose, we can use circles, ellipses or hyperbolae.

If we are talking about the GPS, there are nine satellites overhead at any time which are constantly sending out signals giving us precise locations of the exact position of the object. Now, the GPS on phone receives the signal at some time ‘t’ and the distance ‘d’ travelled can be calculated by multiplying ‘t’ with the speed of light. This gives a circle with radius ‘d’. By using two more circles, we can find the exact location from the intersection of three circles. This method is called trilateration.

A multireceiver radar uses ellipses to find the location of the target. We send signal to the target, which replies. Then at another location, we have a receiver that picks the reply. Suppose the distance travelled by the signal from the transmitter to the target is ‘x’ and the distance travelled by it from the target to the receiver is ‘y’, we can find the sum of the two distances by multiplying the total time taken for the signal to be read and the speed of light. This sum of ‘x’ and ‘y’ is ‘d’ and the locus gives us an ellipse. So the target is somewhere on the ellipse with transmitter and the receiver as foci. Now, by using three different signals from three different locations, we get a single point of intersection of the three ellipses formed, giving us a precise location of the target. However, this method seems to have some loopholes. The target might not reply immediately. In that case, we get ‘x+y’ >’d’.

Hyperbolic trilateration is instead used in order to listen to the signal that replies silently. When a transmission is received in two different locations we know it has travelled an unknown distance, ‘x’, from the target to the first receiver and an unknown distance, ‘y’, from the target to the second receiver. Now by taking the difference between the time taken for the signal to reach both the receivers and multiplying it with the speed of light, we obtain the value of |x-y|. This is some constant value ‘d’ and we receive a point on a hyperbola. Therefore, the target is on a hyperbola with the two receivers on the focus. Now, if we bring two more hyperbolae and determine the point of intersection of the three, we get the exact point of location.

And that’s how the police catch criminals and all the objects can be traced easily. A very interesting invention, I must say.