Also in 3D, two reflections make a rotation?

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nomadreid
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In the plane, two reflections across non-parallel lines make a rotation around the point of intersection of those two lines. But in 3D, do two reflections across non-parallel planes make a rotation around the line of intersection of the two planes?
The easiest proof I know for the 2D statement in the summary does not carry over nicely to the 3D statement since rotations in 3D don't necessarily commute (the 2D proof uses this commuting among rotations in the plane around a common point). Before I then try to modify the proof so that it works, I would like to know whether the statement for 3D is even true. Thanks in advance.
 
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  • #2
I would refer you to Linear and Geometric Algebra by Alan Macdonald, problem 7.3.9:
7.3.9 (1 rotation ##\equiv ## 2 reflections) a. Show that the composition of two reflections in hyperplanes is a rotation, with the angle of rotation twice the angle between the hyperplanes.
b. Show that a rotation is the composition of two reflections in hyperplanes, with the angle between the hyperplanes half the angle of rotation.

I am not sure how well this matches your question, but I think it does.
 
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Super! It answers my question very well. Thank you, FactChecker!
 
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FAQ: Also in 3D, two reflections make a rotation?

What does it mean that two reflections make a rotation in 3D?

In 3D geometry, when you perform two reflections across two different planes, the result is equivalent to a rotation around an axis. The angle of this rotation is twice the angle between the two planes of reflection. This property is a consequence of the way reflections and rotations are related in Euclidean space.

How do you determine the axis of rotation when reflecting in two planes?

The axis of rotation can be found by identifying the line that is perpendicular to both planes of reflection. This line serves as the axis about which the rotation occurs. The angle of rotation is determined by the angle between the two planes, with the rotation being twice that angle.

Can you provide an example of two reflections resulting in a rotation?

Consider two planes in 3D space: Plane A and Plane B, which intersect at a line. If you reflect a point across Plane A and then across Plane B, the resulting point will be rotated around the line of intersection of the two planes. If the angle between Plane A and Plane B is θ, the resulting rotation will be by an angle of 2θ.

Is this property unique to 3D space?

This property is not unique to 3D space; it also holds in 2D geometry, where two reflections across intersecting lines result in a rotation around the point of intersection. However, in higher dimensions, the relationship becomes more complex, and the concept of reflections and rotations can involve more intricate geometric structures.

What are the practical applications of this concept?

This concept is widely used in computer graphics, robotics, and physics, particularly in understanding transformations and symmetries. For instance, in computer graphics, reflections and rotations are fundamental operations for modeling and rendering objects. In robotics, understanding these transformations is crucial for motion planning and control of robotic arms.

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