# Is the sum of dot products of all interior angles towards a hollow space zero?

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Imagine there is a hole and a number of edges are facing towards it,
When I take 2 consecutive edges and calculate the dot products of them,
I will get some value back, if I add all subsequent dot products until
the 1st edge is met again, will I get a sum of zero?
Thanks
Jack

Edited by lucky6969b

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I would say that most likely not.

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I am not sure I understand the question exactly, but for any interpretation of the question I can think of, the answer is "no". Just think of the case where your edges form a regular polygon that is not a square. The dot products will all be equal and not zero, therefore their sum won't be zero either.

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The edges will be concave and as such positive dot values returned. You will get some positive total. Only if the edges were perpendicular would you get 0.

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For every pair of edges,
since dot(edge_i, edge_i+1) = norm(i) * norm(i+1) * cos(angle)
then
angle_i = acos(dot(edge_i, edge_i+1) / norm(i) / norm(i+1))

As the sum of all interior angles of a polygon is 360 degrees,
If the sum is indeed 360 degrees, then it is hollow?
Thanks
Jack

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How about you explain what problem you are trying to solve, ideally with some example, and we'll go from there?

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