102 Lattice polygons
1.1 Pick’s Theorem: area of lattice polygon ....
A lattice point is a point with integer coordinates. A lattice polygon is
one whose vertices are lattice points (and whose sides are straight line
segments). For a lattice polygon, let
I=the number of interior points, and
B=the number of boundary points.
Theorem 1.1(Pick).The area of a lattice polygon isI+B 2 − 1.
If the polygon is a triangle, there is a simple formula to find its area
in terms of the coordinates of its vertices. If the vertices are at the points
(x 1 ,y 1 ),(x 2 ,y 2 ),(x 3 ,y 3 ), then the area is^1
1
2∣ ∣ ∣ ∣ ∣ ∣
x 1 y 1 1
x 2 y 2 1
x 3 y 3 1∣ ∣ ∣ ∣ ∣ ∣
.
In particular, if one of the vertices is at the origin(0,0), and the other
two have coordinates(x 1 ,y 1 ),(x 2 ,y 2 ), then the area is^12 |x 1 y 2 −x 2 y 1 |.
Given a lattice polygon, we can partition it intoprimitivelattice tri-
angles,i.e., each triangle contains no lattice point apart from its three
vertices. Two wonderful things happen that make it easy to find the area
of the polygon as given by Pick’s theorem.
(1) There are exactly 2 I+B− 2 such primitive lattice triangles no
matter how the lattice points are joined. This is an application of Euler’s
polyhedral formula.
(2) The area of a primitive lattice triangle is always^12. This follows
from a study of the Farey sequences.
(^1) This formula is valid for arbitrary vertices. It is positive if the vertices are traversed counterclockwise,
otherwise negative. If it is zero, then the points are collinear.