Rotoreflections in Conformal Geometric Algebra
Why think about rotoreflections?
Assuming we want to understand Conformal GA, we have to understand the transformations we can perform in it (conformal transformations!), which are its versors (compositions of 1-vectors/sphere reflections).
Rotoreflections (a rotation followed or preceded by a reflection) are a good pedagogic example of versors. This not because they are useful - they aren’t particularly. But they give you an idea of how some the weird transformations can work. If you want to work in many dimensions, most transformations are weird transformations!
The simplest transformations (“1-reflections”) are planar and sphere reflections (grade 1 only). In 3D the most complicated include roto-scale-flections (grade 1, 3, 5 mixture) and hyperbolic screws (grade 0, 2, 4 mixture).
The first example of a mixture you get introduced to is a quaternion/rotation, which is a mixture of grades 0 and 2. This throws you in at the deep end, in a sense. The grade 2 part of a quaternion is its axis, you visualize that as a line - fine. So how do you visualize the grade 0 part of a quaternion? Scalars are algebraically easy but geometrically a bit strange: the way to visualize a scalar is to visualize an infinite-volume block (because it is a multiple of the identity transformation/0-degree-rotation, “1”).
Rotoreflections: a mixed-grade object where both components have a conceptually-simple visualization with a clear relationship to the transform
In a rotoreflection, the grade 1 part a sphere/plane and the grade 3 part is a point pair/point. Alright, alright, the point pair may have a point at infinity - but that you can get used to!
Alright here we go, some rotoreflections. Each of these is a reflection in the sphere followed by a rotation around the blue circle:
Red point pair: trivector part. Note that it is preserved by both the rotation and the reflection
Black sphere: 1-vector part (ignore the dotted equator, it’s just there to make it look 3D)
Again the blue circle is what the rotation goes around. If you were to just do a rotation around the blue circle, you’d find that it would happen to preserve the sphere.
It’s not like an ordinary rotation axis (bivector) though, because it’s had a reflection performed on it by the sphere. If you wanted its representation it would be the trivector part divided by 1-vector part. So it’s in some sense “implicit”, both geometrically and algebraically.
Are all grade-1 + grade-3 mixes rotoreflections?
No!
First, the point pair must lie on the sphere, otherwise the sum is not a versor (you can’t decompose it into a composition of three 1-vectors). A more abstract way of saying this is that the wedge product of the grade 1 and grade 3 part has to be 0.
So, assuming we maintain that, we find there are grade-1 grade-3 mixes which are valid transformations but are not rotoreflections.
If you send both points in the pair infinitely far away, the rotoreflection becomes a euclidean transflection / glide reflection
If the point pair’s radius goes to 0 it becomes a parabolic transflection. Sorry no pictures of this right now but it looks cool!
If the sphere’s radius goes to 0 it becomes nonsense, because you can’t reflect in a 0 radius sphere
If radius of the point pair of sphere flip over to being negative, the transformation becomes a hyperbolic transflection or scaleflection. Here’s a hyperbolic transflection in 2D:
A hyperbolic transflection in 3D is harder.
Starting with the animation above, imagine it taking place in a plane hovering in front of you.
Take the arc with the points on it and construct the sphere containing it which is cut in half by the plane.
The circle that encloses the animation - that circle stays a circle, and is the trivector part of the hyperbolic transflection. Instead of having a blue implicit circle, we find we have a blue implicit point pair where this trivector circle meets the 1-vector sphere.