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I have a list of triangles of the mesh and point cloud. the question is :how to determine each point belongs to a triangle

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amina lk
amina lk 2016-2-26
评论： amina lk 2016-2-28
I have a list of triangles of the mesh and point cloud. the question is :how to determine each point belongs to a triangle
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Roger Stafford
Roger Stafford 2016-2-26
Let the points (x1,y1), (x2,y2), and (x3,y3) be the three vertices of a triangle, and let (x,y) be some arbitrary point. Define:
c1 = (x2-x1)*(y3-y1)-(x3-x1)*(y2-y1);
p1 = (x2-x) *(y3-y) -(x3-x) *(y2-y) ;
c2 = (x3-x2)*(y1-y2)-(x1-x2)*(y3-y2);
p2 = (x3-x) *(y1-y) -(x1-x) *(y3-y) ;
c3 = (x1-x3)*(y2-y3)-(x2-x3)*(y1-y3);
p3 = (x1-x) *(y2-y) -(x2-x) *(y1-y) ;
Then the point (x,y) lies inside or on the triangle if:
sign(c1)*sign(p1) >= 0 & sign(c2)*sign(p2) >= 0 & sign(c3)*sign(p3) >= 0
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Roger Stafford
Roger Stafford 2016-2-27
编辑：Roger Stafford 2016-2-27
Two points.
1) My apologies! I made the computations more complicated than they need be. The quantities c1, c2, and c3 are all identically equal, so you need only calculate one of them. They are each twice the signed area of the (straight-lined) triangle. with the sign determined by whether (x1,y1), (x2,y2), (x3,y3) are in counterclockwise or clockwise order around the triangle.
2) In answer to your question of triangles in three-dimensional space, that is a somewhat different situation. Your arbitrary point (x,y,z) has to be shown to lie in the same plane as that of the triangle, if that is what you mean by "belongs to", in addition to satisfying three inequalities. Do you want me to show how this is to be determined?

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