Concave Random Quadrilaterals from Four Points in a Disk

Four random points are chosen inside a circle. A quadrilateral is formed using the points as its vertices. Is concave or convex?

The probability that is concave is known to be . The plot shows the four points and the convex hull of the points; the convex hull is the smallest convex polygon that encloses all of the points. If has four vertices, is convex, but if has only three vertices, is concave and one of the fours points is inside .

If is concave, it is not unique: the quadrilateral can be formed in three ways; in the concave case, we show and not .

Snapshot 1: a convex quadrilateral: the convex hull has four vertices

Snapshot 2: a concave quadrilateral: the convex hull has only three vertices

This Demonstration illustrates the four-point problem of J. J. Sylvester (1814–1897). He presented the problem in the April 1864 issue of Educational Times; see [1], [2, pp. 15–21]. The problem was as follows: if we pick four points at random inside some given convex region, what is the probability that the four points are the vertices of a concave quadrilateral? Wilhelm Blaschke (1885–1962) showed that the probability is at least and at most , depending on the form of the convex region. W. S. B. Woolhouse (1809–1893) showed that the lower bound is achieved if the convex region is a circle; for a square, the probability is and for a regular hexagon, . The Demonstration is based on [2], where a method is presented to generate random points in a circle.

References

[1] R. E. Pfiefer, "The Historical Development of J. J. Sylvester's Four Point Problem," Mathematics Magazine, 62(5), 1989 pp. 309–317.

[2] P. J. Nahin, Digital Dice: Computational Solutions to Practical Probability Problems, Princeton, NJ: Princeton University Press, 2008.