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
Is the sum of dot products of all interior angles towards a hollow space zero?
Author
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.
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.
Author
Then What about
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))
Sorry about my notations.
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
How about you explain what problem you are trying to solve, ideally with some example, and we'll go from there?
Are you thinking of Exterior Angles?
see Figure 2 of the following page:
https://www.cliffsnotes.com/study-guides/geometry/polygons/angle-sum-of-polygons
see Figure 2 of the following page:
https://www.cliffsnotes.com/study-guides/geometry/polygons/angle-sum-of-polygons
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