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Point in polygon

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take the point in question, and project a ray from it (any direction will do)

check to see if that ray crosses each of the lines in the poly. if you get an even number of crossings (0,2,etc), its not in the poly.

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You can get the equation of the 4 lines bounding the poly by solving the standard y=mx+b equation of a line. Now plug in the value of your point. If you get a value that either below or above an opposite pair of lines, then the point is out side the poly. If it is between the lines (above one, below the other), then check the other pair. A point is in the poly if and only if it is between both pairs of oppsite sides.

See crude Diagram:






y1,y2 are lines on opposite side of poly.
I, O1, and O2 are points.

O1 is outside the poly because y1(O1.x) and y2 (O1.x) are both positive
O2 is outside the poly because y1(O2.x) and y2 (O2.x) are both negative

I is between the lines because y1(I.x) is negative, but y2(I.x) is positive

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Just some clarification on tans'' method. There is at least one special case that I can see that must be taken into account. Assuming your polygon is limited to 4 verts than this works great however, there is a case where extending a line in one direction might run through a single vertex giving you the result of being inside the poly when in fact its not. Also dont forget the case where your point is ON a poly''s vertex

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