Pick's formula. Definition (Notations). For any polygon P in a lattice, we note BP the number of points on its border and IP the number of points strictly inside the ...
Episode 01 – Finding Pick’s formula European section – Season 3
Episode 01 – Finding Pick’s formula
What is a lattice ? Definition (Lattice) A lattice is a square grid, a discrete surface with a regular gap between any two points. A simple example is the set of the points with whole coordinates on a coordinate graph. Example bc
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Episode 01 – Finding Pick’s formula
Lattices polygons
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Episode 01 – Finding Pick’s formula
Pick’s formula Definition (Notations) For any polygon P in a lattice, we note BP the number of points on its border and IP the number of points strictly inside the polygon. The area of the polygon will be noted AP .
Episode 01 – Finding Pick’s formula
Pick’s formula Definition (Notations) For any polygon P in a lattice, we note BP the number of points on its border and IP the number of points strictly inside the polygon. The area of the polygon will be noted AP . Example
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BP IP
Episode 01 – Finding Pick’s formula
= =
Pick’s formula Definition (Notations) For any polygon P in a lattice, we note BP the number of points on its border and IP the number of points strictly inside the polygon. The area of the polygon will be noted AP . Example
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BP IP
Episode 01 – Finding Pick’s formula
= =
5
Pick’s formula Definition (Notations) For any polygon P in a lattice, we note BP the number of points on its border and IP the number of points strictly inside the polygon. The area of the polygon will be noted AP . Example
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BP IP
Episode 01 – Finding Pick’s formula
= =
5 13
Find a formula involving the numbers BP and IP to compute the area of any lattice polygon.
Episode 01 – Finding Pick’s formula
Theorem For any convex polygon P in a square two-dimensional lattice, AP =
1 BP + IP − 1. 2
Episode 01 – Finding Pick’s formula
Theorem For any convex polygon P in a square two-dimensional lattice, AP =
1 BP + IP − 1. 2
Example
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BP
=
5
IP
=
13
AP
=
Episode 01 – Finding Pick’s formula
Theorem For any convex polygon P in a square two-dimensional lattice, AP =
1 BP + IP − 1. 2
Example
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BP
=
5
IP
=
AP
=
13 1 BP + IP − 1 2
Episode 01 – Finding Pick’s formula
Theorem For any convex polygon P in a square two-dimensional lattice, AP =
1 BP + IP − 1. 2
Example
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BP
=
5
IP
=
AP
=
AP
=
13 1 BP + IP − 1 2 5 + 13 − 1 2
Episode 01 – Finding Pick’s formula
Theorem For any convex polygon P in a square two-dimensional lattice, AP =
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Proof. First, it's obvious from the definitions that sPT = sP + sT . Then, suppose that Pick's formula holds for the two polygons, and we use the same notations as ...
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