Concave Random Quadrilaterals from Four Points in a Disk

generate new points

Four random points are chosen inside a circle. A quadrilateral

Q

is formed using the points as its vertices. Is

Q

concave or convex?

The probability that

Q

is concave is known to be

35/(12

2

π

)=0.29552

. The plot shows the four points and the convex hull

C

of the points; the convex hull is the smallest convex polygon that encloses all of the points. If

C

has four vertices,

Q

is convex, but if

C

has only three vertices,

Q

is concave and one of the fours points is inside

C

.

If

Q

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

C

and not

Q

.