HW 11/28/2016    COSC 681   FALL 2016
Due 12/12/2016

1. Write a program to calculate pi using Monte Carlo technique.

Using the same algorithm (ratio of area of circle inscribed in square), obtain
an estimate of pi.  Plot pi vs number of iterations. 

We expect a rapid convergence to asymptotically approaching pi.

Repeat the program using two different random number generators.  One should
be a very poor random number generator. E.g. implement random0 from 
https://en.wikipedia.org/wiki/Linear_congruential_generator
You may wish to modify a, c, and m to get worse performance.

The other random number generator should be a much better one. You don't have
to implement this second random number generator, you can, for example, use
java.util.Random.

2. Using Monte Carlo technique, write a program to calculate the intersection 
area of the following on the x-y plane:

  - circle, radius 6, centered at (0, 0)
  - circle, radius 4, centered at (3, 3)
  - points less than the line y = -2 x + 7
  
Give a plot that shows the asymptotic approach to the solution vs iteration #.

3. Luke Skywalker needs to shoot a projectile down a ventilation shaft of the
DeathStar. The ventilation shaft is a cylinder with radius 0.5 and height 30 m. 
The target is at the bottom of the 30m shaft; it is a circle centered at 
(0, 0, -30) with radius = 0.1 meter.

Luke's aim is perfect -- he can hit the opening of the shaft with the perfect
lateral velocity to hit the target at the bottom of the shaft.
To simplify the problem, you can assume that Luke hits the center of the
top of the shaft with no lateral velocity: (x, y, z) = (0, 0, 0) and 
(vx, vy, vz) = (0, 0, -15), in meters/second. In the absence of other factors,
it will take 2 seconds for the projectile to go from the top of the shaft to
the target.

Unfortunately, there are random external gravitational fluctuations that 
perturb Luke's projectiles in their vx and vy as they move down the shaft.

Model the perturbation in vx by adding a factor with uniform distribution
over the range [-0.05, +0.05). Use a similar distribution for the perturbation in
vy.  Model the gravitational fluctuation by changing the perturbation amount
every k time units. k is uniformly distributed over the range [0, 2].  For a given
projectile, gravity can fluctuate multiple times before the projectile hits
the bottom of the shaft (or a side of the shaft).

As stated above, if there is no perturbation, a projectile will hit the target 
in 2 seconds ( -15 * 2 = -30). That means probability is pretty good that there will
be at least one change in velocity as the projectile moves down the shaft.
To further simplify the simulation, assume the time step is 0.1 second.

Run a Monte Carlo simulation of Luke's target shooting. Give plot(s) showing
the projectile impacts at the bottom of the shaft (if a projectile hits the
side of the shaft, indicate this with a mark on or outside the limit of the
target.

Answer this question:  How many shots will Luke have to take to have a 75% or
better chance of hitting the target?

Support your answer with the results of the simulation.