MathDB

MTRPia in

2044

is highly advanced and a lot of the work is done by disc-shaped robots, each of radius

1

unit. In order to not collide with each other, there robots have a smaller

360

-degree camera mounted on top, as shown in the figure (robot

r_1

'sees' robot

r_2

). Each of there cameras themselves are smaller discs of radius

c

. Suppose there are three robots

r_1, r_2, r_3

placed 'consecutively' such that

r_2

is roughly in the middle. The angle between the lines joining the centres of

r_1, r_2

and

r_2, r_3

is given to be

\theta

. The distance between the centres of

r_1,r_2 =

distance between centres of

r_2,r_3 = d

. Show (with the aid of clear diagrams) that

r_1

and

r_3

can see each other iff

\sin{\theta} > \frac{1-c}{d}

. As a bonus, try to show that in a longer 'chain' of such robots (same

d

\theta

), if

\sin{\theta} > \frac{1-c}{d}

then all robots can see each other.

4