Pick's and Euler's TheoremsDate: 05/06/99 at 06:44:12 From: Charlie Subject: Pick's theorem What is Pick's theorem and how can it be linked with Euler's theorem? Thank you. Date: 05/06/99 at 10:50:06 From: Doctor Anthony Subject: Re: Pick's theorem Hi Charlie, Pick's Theorem says that the area of a simple lattice polygon P is given by I + B/2 - 1, where I is the number of lattice points in the Interior of P and B is the number of lattice points on the Boundary of P. Pick's Theorem is equivalent to Euler's formula and closely connected to the Farey series; it may be generalized to non-simple polygons and to higher dimensions. To help remember the correct formula, you can check it on easy cases (unit square, small rectangles, etc) or, better, you can view how it arises from additivity of area. One can view Pick's formula as weighting each interior point by 1, and each boundary point by 1/2, except that two boundary points are omitted. Now suppose we are adjoining two polygons along an edge as in the diagram below. Let's check that Pick's formula gives the same result for the union as it does for the sum of the parts (and thus it gives an additive formula for area, as required). 1/2 1/2 1/2 1/2 ... - @ @ - ... ... - @ @ - ... / \ / \ / \ / \ 0 @ @ 0 @ 0 | | . 1/2 @ @ 1/2 . . @ 1 . . | + | . => . . . 1/2 @ @ 1/2 . . @ 1 . | | 0 @ @ 0 @ 0 \ / \ / \ / \ / ... - @ @ - ... ... - @ @ - ... 1/2 1/2 1/2 1/2 The edge endpoints we choose as the two omitted boundary points. The inside points on the edge were each weighted 1/2 + 1/2 on the left, but are weighted 1 on the right since they become interior. All other points stay interior or stay boundary points, so their weight remains the same on both sides. So Pick's formula is additive. - Doctor Anthony, The Math Forum http://mathforum.org/dr.math/ |
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