Lattice Points. (1 Viewer)

[seanieg89]

On grid paper, a lattice point is a point which is at the intersection of two perpendicular lines.

Prove by induction or otherwise that a polygon drawn on grid paper with lattice point vertices has area given by:

A = i + b/2 -1

where i is the number of lattice points interior to the polygon and b is the number of lattice points on its boundary.

 

[Carrotsticks]

Will put up a solution for this later if nobody else does.

But for those who need a rough skeleton, first define two polygons A_1 and A_2 with areas obeying the formula in question. Show that A_1 + A_2 satisfies the formula too (easy) and then proceed to prove that any polygon can be decomposed into 'unit polygons' (ie: triangles with no interior lattice points) if correct diagonals are constructed, and use the previously proven fact that A_1 + A_2 satisfies the formula to complete the proof.

 

[Carrotsticks]

Whilst doing this problem, I found a couple of nice properties.

1. Find the minimum area of any polygon with lattice points.

2. When does the minimum area occur?