DimBulb
Joined: 07 May 2009, 16:45 Posts: 62

Exercise [11.06]
"Find the geometrical nature of the transformation, in Euclidean 3space, which is the composition of the two reflections in planes that are not perpendicular."
This is supposed to be a sort of tough one, but it seems simple to me, which means I probably don't understand something. First of all, I assume the "two reflections in planes that are not perpendicular" means one reflection in each plane, not two reflections in each plane. Two reflections in a plane just gets you back where you started.
So a reflection in a plane, say like a mirror, produces a mirror image. A second reflection in a second plane produces a mirror image of a mirror image, which is a regular image, meaning righthandedness and lefthandedness are preserved and not reversed. The situation is like a periscope, which usually uses two offset but parallel mirrors, and produces a translation that allows you to look over fences or at an enemy's ship on the surface when you are in a submarine below the surface. The view through a periscope is not a mirror image.
Now, what happens when the mirrors are not parallel? The image is rotated. That is, you're in your submarine looking through your periscope, but the upper mirror is tilted slightly upward from parallel with the lower mirror, so you don't see the enemy ship, you see the a cloud in the sky. Still, the image is not a mirror image, just one rotated up. You are looking horizontally, but you are seeing upward.
So the geometrical nature of the transformation, in Euclidean 3space, which is the composition of two reflections in planes that are not perpendicular is a rotation.
More specifically the rotation is about the line where the planes intersect, and the rotation is twice the angle between the two planes. You might be able to see that thinking about the mirrors above, but what follows is another way to look at it.
Here is a drawing representing the two intersecting planes, drawn orthographically onend, so we just see two intersecting lines. I am reducing the three dimensional problem to two dimensions because obviously reflecting points in the two planes is not going to transform the position of the points in the direction perpendicular to the plane of the drawing (the direction of the line of planes' intersection). The planes and thus the lines in the drawing intersect at some angle .
Since the transformation is a rotation which preserves all angles, we need to show how one point is rotated and that will characterize the transformation. An easy point to pick is a point on the first plane, since reflection in the first plane of that point leaves the point where it is. Then reflection in second plane moves that point so it is rotated twice the angle between the two planes. See? I added another point not on the first plane to show how it is also rotated by the two reflections the same angle
