### The geometry of Great Britain

A complicated coastline such as that of Great Britain provides an
interesting example for testing algorithms in computational geometry.
In the following examples I use the Mercator projection.

#### centroid

Homework exercise: show using Green's theorem that the centroid of
a uniform planar lamina is given by boundary integrals:
*(∫x*^{2}dy,∫y^{2}dx)/(2m),
where *m=∫∫dxdy* is the mass.

I thus computed the centroid of the mercator projection of England to be at
approximately longitude -1.46943, latitude 52.6016. In the national grid
coordinate system this is (435930.05,300595.2), or SK359006. This is near
Sibson, Leicestershire.

#### enclosing ellipse of minimum area

The computation of the enclosing ellipse of minimum area is an application of
semidefinite
programming. The figures show such ellipses for all of GB, including
islands, and for England alone.