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by enkimute 2028 days ago
The grade 1 element is not the normal vector of the plane, it is the plane itself (as a homogeneous linear form, it includes the distance from the origin as an extra coefficient, which the normal vector does not).

Linear equations form a linear space, you can add them and multiply them with scalars. (creating things called line-pencils, plane-bundles etc .. classic projective geometry). Hence you call the grade-1 elements of the graded version of such a linear space 'vectors'. (just like you can make vector spaces with functions or all other sorts of objects).

So the reflection formula from GA in general, which is to reflect an arbitrary element X w.r.t. a grade-1 element a :

-aXa

is simply to be read as reflecting the object 'X' in the plane (3D) or line (2D) or sphere (3D CGA) or circle (2D CGA) 'a'.

Such a reflection should modify all other reflections, except for itself (where it should only flip orientation) :

-aaa = -a

It is easy to verify that this holds for general Euclidean planes written as homogeneous linear equations in their grade-1 element form. And from that everything else follows. (composition of reflections gives you rotations/translations (leaving points invariant in 2D), etc).

Planes being grade 1 elements in no way implies characterizing them by a normal vector.
1 comments
cygx 2028 days ago
Another poster has already linked [1] and [2]. That's what I was getting at as well: A bias towards the wedge/'join' and against the 'anti-wedge'/'meet' crops in via the geometric product, when the situation should be symmetric due to Hodge duality. This leads to constructions I find rather unnatural.

[1] http://terathon.com/blog/projective-geometric-algebra-done-r...

[2] http://terathon.com/blog/symmetries-in-projective-geometric-...

link
enkimute 2028 days ago
A bias towards 'vectors must be points' crops in as an unwritten legacy axiom, leading to a broken correspondence with Group theory (which again cannot be turned around, discrete symmetries compose into continuous ones, but not the other way around). A further restriction to the unfortunately very symmetric 3D Euclidean space hides many of the problems with that approach. (for example only in this space the even subalgebra is shared, and bivectors are self-dual).

Consider for example R_(2,0,1) in both scenarios.

Using Geometric Product as Group Composition

  e1  = reflection w.r.t. x=0 axis
  e2  = reflection w.r.t. y=0 axis
  e12 = reflection w.r.t. y=0 axis followed by reflection w.r.t. x=0 axis
      = 180 rotation around origin.
Using Anti-Geometric Product as Group Composition

  e1  = not a group element, because anti-norm is zero.
  e2  = not a group element, because anti-norm is zero.
  e12 = not a group element, because anti-norm is zero.
So to get the same behavior you need the following elements when using the Anti-Geometric product as composition

  e02 = reflection w.r.t. x=0 axis
  e01 = reflection w.r.t. y=0 axis
  e0  = reflection w.r.t. y=0 axis followed by reflection w.r.t. x=0 axis
      = 180 rotation around origin.
So choosing the geometric anti-product as group composition operator is possible, but imho breaks the readability of your transformations completely.

All I can say is that I hope that those linked articles do not give people an excuse to not consider the model in which vectors, grade-1 elements, are reflections. This is the true half of the picture that is new and being missed.

link