Estimate the distribution of the number of points in the convex hull of five points uniformly randomly picked in a triangle, parallelogram or ellipse; see http://math.stackexchange.com/questions/282147.

Question282147.java

public class Question282147 {
    final static int TRIANGLE = 0;
    final static int PARALLELOGRAM = 1;
    final static int ELLIPSE = 2;

    final static int shape = ELLIPSE;

    final static int ntrials = 1000000;

    public static void main (String [] args) {
        double [] [] x = new double [5] [2];

        int [] counts = new int [3];

        for (int n = 0;n < ntrials;n++) {
            for (int i = 0;i < 5;i++)
                for (;;) {
                    for (int j = 0;j < 2;j++)
                        x [i] [j] = Math.random ();
                    if (shape == PARALLELOGRAM)
                        break;
                    else if (shape == TRIANGLE) {
                        if (x [i] [0] > x [i] [1])
                            break;
                    }
                    else if (shape == ELLIPSE) {
                        double r = 0;
                        for (int j = 0;j < 2;j++) {
                            double d = x [i] [j] - 0.5;
                            r += d * d;
                        }
                        if (r < 0.25)
                            break;
                    }
                    else
                        throw new Error ();
                }

            int count = 0;
            for (int k = 1;k < 5;k++)
                outer:
                for (int l = 0;l < k;l++) {
                    boolean first = true;
                    boolean positive = true;
                    
                    for (int m = 0;m < 5;m++)
                        if (m != k && m != l) {
                            boolean orientation = (x [m] [0] - x [k] [0]) * (x [l] [1] - x [k] [1]) - (x [m] [1] - x [k] [1]) * (x [l] [0] - x [k] [0]) > 0;
                            if (first) {
                                positive = orientation;
                                first = false;
                            }
                            else if (positive != orientation)
                                continue outer;
                        }
                    
                    count++;
                }
            counts [count - 3]++;
        }

        for (int i = 0;i < 3;i++)
            System.out.println ((i + 3) + " : " + counts [i] / (double) ntrials);
    }
}