Planar Polygon

Flat polygons embedded in three-space can be transformed into a congruent planar polygon as follows. First, translate the starting vertex to (0, 0, 0) by subtracting it from each vertex of the polygon. Then find the normal to the polygon by taking the cross product of the first and last vertices. Now, let be the rotation matrix for Euler angles , , and , and solve

(1)

for and (after first expressing sines in terms of cosines using . The result is

			(2)
			(3)

The signs are chosen as follows:

			(4)
			(5)

Plugging these back in and applying to the original polygon then gives a polygon whose vertices all have one component zero. This component can then be dropped. The only special cases which need to be taken into account are , in which case the polygon is parallel to the -plane and the third components can be immediately dropped. The second occurs when , in which case there is no component of the normal vector along the x-axis, so the Euler rotation will not work. However, simply picking a different starting vertex from which to calculate the normal resolves this degenerate case.