Lab 12: Monte Carlo fun!

Monte-Carlo Integration extravaganza! In this problem you have to compute the area of complex surfaces.

• Consider a surface made as a sphere with a cylinder cut through it. The sphere is a unit sphere centered at the origin. The cylinder has radius 0.5 that is centered along the z-axis.

  Two views of a points randomly distributed on the surface of a sphere cut by a cylinder

  The shape of the "sphere - cylinder" has surface parts that are from the sphere and from the cylinder. You need to compute:
  • The volume of this shape. The correct answer is near 2.7.
  • The surface area of this shape (this might be best done by computing the surface area of the sphere that is not inside the cylinder, and the surface of the cylinder that is not outside the sphere). The correct answer is near 16.33.
  • When you get to the correct answer, Show the TA a plot of your points randomly distributed within the volume (for the first part), and on the surface (for the second part), and answers to the questions at the bottom of this page
• Consider a cylinder of radius 1 aligned along the z axis which goes from z= -2 to z = 2. Consider another cylinder of radius 1 aligned along the x-axis with x going from -2 to 2.

  A picture of the shape, and random points on the surface of the two cylinders making up the shape, color coded by which cylinder they come from (red or blue) if they are valid, and color coded green if they are *inside* the other cylinder (and thus not valid points on the surface).

  Estimate
  • The volume of this shape
  • The surface area of this shape, the answer should be near 34.2
  • When you get to the correct answer, Show the TA a plot of your points randomly distributed within the volume (for the first part), and on the surface (for the second part), and answers to the questions at the bottom of this page

  For each problem, you need to specify (these can be well written comments in your code)

  1. Over what volume/region/surface are you generating random numbers?
  2. What is the area of the region or surface?
  3. Justify why you are sampling points evenly over this surface.
  4. what is your criteria for accepting a sample?
  5. For your number of samples, what is the standard deviation of your estimate?
  Hints: To compute the surface area of part of a sphere, you first need to create random points on the sphere. In class we talked about an inefficient method to find points "near" the surface of the sphere (find points whose distance from the origin was "close enough" to 1. Another method is to randomly create a point (x,y,z) inside the sphere. Then think of that as a vector from the origin, and normalize that vector to have length 1. That gives you a random points on the sphere.

  Then, you should know or be able to look up the surface area of a complete sphere. Once you have random points on the sphere, and the ration of (points in the region you care about / all the points), then the monte-carlo estimate of the surface is the area of the complete sphere multiplied by the ratio.