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Advanced - Geometry · 4:27
Polygon Area, Pick's Theorem & Lattice Points
The shoelace formula computes twice a polygon's area as a sum of cross products of consecutive vertices (signed trapezoids against the x-axis), exactly in integers, with the sign giving the orientation. For a lattice polygon the boundary count is one gcd per edge, and Pick's theorem S = I + B/2 - 1 then gives the interior count, a result with no 3D analogue. Lattice points under a line with fractional slope are counted by the floor sum in O(log) steps with a Euclid-like reduction.
You will learn
- Compute twice the area of a simple polygon with the shoelace formula in integers and read its orientation from the sign
- Count boundary lattice points with one gcd per edge and interior points with Pick's theorem
- Explain the proof ladder of Pick's theorem and why it fails in 3D (the Reeve tetrahedron)
- Evaluate the floor sum of floor((a·x + b)/m) in O(log m) with the Euclid-like reduction