Theorem 1 Given a simple closed polygon whose vertices have integer coordinates, then
# interior lattice points# boundary lattice points
Clearly we can reduce Pick's Theorem to the case of triangles because
where area, # interior lattice points, and # boundary points. These all imply
We may further assume that there are no other lattice points other than its vertices. Now the idea is to embed the special triangles into rectangles.
For a triangle of case 1
By thinking of the triangle as a rectangle,
# of interior points
It's crucial that there are no lattice points in the hypotenuse. QED.
Going back to the case of an arbitrary polygon, there is an alogrithm that can be used to count the number of lattice points inside of it:
Definition 2 Let be a triangle whose vertices are integral. We say is primitive if it has no other lattice opints in its boundary or interior.
Lemma 3 Every lattice triangle can be divided into primitive triangles.
Lemma 4 A primitive triangle has area .
Lemma 5 Euler's formula: In a triangulation of a polygon,
where # vertices in primitive triangulation , # edge segments in prmitive triangulation , and # primitive triangles .
Corollary 6 The number of primitive triangles from the first lemma is
where # interior points and # boundary points .
The proof of the third lemma can be done using the corollary and the first lemma:
The proof of the corollary can be done using the second lemma.
The proof of the second lemma goes as follows:
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