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Order of a 3D Triangle

Date: 08/29/2001 at 07:40:38
From: vincent
Subject: Winding order af a 3d triangle


Do you know how to find the winding order of a triangle given three 
points in space (x,y,z)? I want to know if when I visit the vertex 
from a to b, then from b to c, and returning to a, I am going 
clockwise or anticlockwise.

Date: 08/29/2001 at 11:58:44
From: Doctor Rob
Subject: Re: Winding order af a 3d triangle

Thanks for writing to Ask Dr. Math, Vincent.

This depends on from which side of the plane containing a, b, and c 
you view this operation. Opposite sides of the plane give opposite 
answers. (Think about it!)

Suppose the three coordinate axes are arranged so that if the x-axis 
is pointing along your right thumb, and the y-axis is pointing along 
your right forefinger, then the z-axis is pointing along your right 
middle finger. If you have three points A(x1,y1,z1), B(x2,y2,z2), and 
C(x3,y3,z3), and you view the operation from a fourth point 
P(x4,y4,z4), then the sign of the following determinant will give you 
the information you want:

       |x1 y1 z1 1|
       |x2 y2 z2 1|
   d = |x3 y3 z3 1|.
       |x4 y4 z4 1|

If d > 0, then A to B to C to A will appear from P to be clockwise.  
If d < 0, then it will appear to be anticlockwise.
If d = 0, then P is in the same plane as ABC, so one cannot observe 
either clockwise or anticlockwise angular motion.

If your axes are arranged using the left hand instead of the right, 
then the two cases are reversed: d > 0 means anticlockwise and d < 0 
means clockwise.

- Doctor Rob, The Math Forum   
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Triangles and Other Polygons

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