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WOLFRAM|DEMONSTRATIONS PROJECT

Concave Random Quadrilaterals from Four Points in a Disk

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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
.
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