Problem 16: Point Picking

Given a geometric figure like a circle, how would you randomly pick points in a uniform disrtibution on the circle? In the circle? How about on a sphere? These are the topics of this problem. Let's first consider producing a uniform probability distribution of points on a circle. One way to do this is by picking a random number in the interval and then plotting the point at that angle with respect to the center. Below are graphs that represent 100, 500, and 1000 points picked randomly with this method.

Another method for generating random points on a circle is accomplished by picking a random point from the interval (-1,1). Let (x₁, x₂) be the Cartesian coordinates of this point. We must discard all points that satisfy (why?). Otherwise, the Cartesian coordinates of the random point on the circle is given by


		(1)

Problem 16:

Show that Cartesian coordinates above represent points on the unit circle.

Solution

Below are three graphs that represent 100, 500, and 1000 points picked randomly using this method.

Now let's consider methods that would produce the same uniform probability distribution of points on and within a circle. Previously, the radius of every point was fixed at 1, whereas now the radius of a point from the origin is different from one point the next. One might first think to randomly pick an angle from as before and then randomly pick a radius from [0,1]. Thus, we have


		(2)

Consider the diagram below. Since the radii are randomly chosen from [0,1] under a uniform probability distribution, then half of all radii picked lie in [0,1/2] and the other half lie in [1/2,1]. Then, do half of the points chosen lie in the circle of radius 1/2 and the other half lie in the remaining region of the unit circle?

Observe what happens when Maple plots random points using equation (2).

Why is there a larger concentration of points near the origin as opposed to the region near the perimeter? Let A₁ denote the area of the circle of radius 1/2 and let A₂ denote the area of the ring having outer radius 1 and inner radius 1/2. Thus, we have


		(3)

Thus, under this method, we have half of all randomly chosen points being placed in a region three times smaller than the region into which the other half of points are placed. Hence, a large concentration of points near the origin. In order to correct this, consider the following transformation:


		(4)

Let A₁ and A₂ denote the same areas that they did previously. We see that


		(5)

Hence, A₁=A₂and thus when Maple makes a plot of random points in this fashion, we see that:

And so we have the uniform distribution we always wanted.

Let's now consider the unit sphere (you might want review this link on spherical coordinates). Observe a plot of random points where is uniformly distributed on , is uniformly distributed on , and we plot these points with the spherical coordinates equation .

The points appear to be clustered near the poles. Look what happens when we use r and as before and plot using


		(6)

where u,v are chosen uniformly from (0,1), and again using the in the spherical coordinates equation mentioned above.

References: Eric Weisstein's Mathworld