From: Dave Rusin
Date: Mon, 13 Jul 1998 17:28:33 -0500 (CDT)
To: atescomp@interpath.com
Subject: Re: Area of 3D Polygon
I forget exactly what your problem was, but the area of a polygon can be
computed by subdividing the region into triangles. The area of a triangle
can be computed in several ways, depending on what's easily known:
1. Heron's formula: A^2 = s(s-a)(s-b)(s-c) where A is the area,
a,b,c are the side lengths, s=(a+b+c)/2. In R^3 you can compute
the lengths of the sides with the distance formula, of course.
2. Let M be the matrix whose columns contain the vector displacements
of two vertices from the third; then (M transpose)*M is a 2x2 matrix
N and the area is (1/2) sqrt(det(N))
3. Given any planar polygon (not just a triangle, but must be planar)
you can move it to R^2 without changing area, then measure area
in the plane using formulas as I have suggested. You accomplish the
motion by (a) subtracting one point from the others to have point !
at the origin, (b) Use the Gram Schmidt process to form an
orthonormal basis in the plane of the next two points, (c) compute
coordinates with respect to this basis for the other points. Now
it's all in the plane.
For (2) and (3) consult your nearest linear algebra/matrix theory text.
dave