! Maximize the area of a polygon having n points such
that the distance between any pair is <= 1.
Based on an example of D. Gay.
Notes:
1) This does not mean that the polygon will
fit inside a circle of diameter 1,
2) For n odd, there may be multiple local optima.
For;
! Keywords: Polygon;
SETS:
point: rho, theta;
ENDSETS DATA:
! Best known solutions:
n Area
3 0.4330127
4 0.5
5 0.657164
6 0.674981;
! Note, the area of a unit diameter
circle is 0.7853982;
point = 1..5;
ENDDATA
n = @SIZE(point);
! Partition the polygon into triangles, each with
an apex at point n and
maximize the sum of their areas;
MAX = .5*@SUM( point(i)  i #gt# 1:
rho(i)*rho(i1)*@SIN(theta(i)theta(i1));
);
@FOR( point(i):
rho(i) <= 1;
@BND( 0, theta(i), @PI());
);
! Points i and j can be at most 1 apart;
@FOR( point(i) i #lt# n:
@FOR( point(j) j #gt# i:
[Dist] rho(i)^2 + rho(j)^2
 2*rho(i)*rho(j)*@COS(theta(j)  theta(i)) <= 1
);
);
! They must be ordered;
@FOR( point(i)  i #gt# 1:
theta(i) >= theta(i1)
);
! Kill some symmetry;
theta(n) = @PI();
rho(n) = 0;
