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

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

[amina lk]

thank youvery much but,if there is a coordinates of triangle (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) how to determine the points of the triangle object belongs

[John D'Errico]

Stop asking the same question a few hours later. You got perfectly good answer. Asking the same question every few hours will not help.

[amina lk]

excuse me but i mean if i added the coordinate z

[amina lk]

thank you i resolt my prob of projection by this file joint

[amina lk]

thank you i resolt my prob of projection by this file joint

Answer 1

[Roger Stafford]

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

[amina lk]

thank you but if i added the coordinate z

[Roger Stafford]

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?