[enkimute]

Absolutely :

Algebraically we can always use the Hodge dual and its inverse to move to/from k-vectors from/to n-k vectors.

Geometrically we can always say two points define a line or two lines define a point.

Group Theory - here we can't swap. Two reflections make a rotation but two rotations do not make a reflection. So reflections are 'naturally' grade 1.

If you want everything to fit intuitively together, you want reflections to be grade 1, and by consequence hyperplanes (lines in 2D, planes in 3D, etc) to be grade 1 (i.e. vectors).

Doing it the other way around is possible but will be more verbose and less intuitive. (also with 'reflections' as natural grade 1 elements, the move from Euclidean to Conformal becomes trivial, simply reflect in (hyper)spheres instead, in this space it is also easier to see that the two approaches are not equivalent, it is natural to say that (in 2D) the 'meet' of two circles is a point pair, but not that the 'join' of two points is a point pair. (why would it not be a line just like in PGA?)).

Imho, one should learn to appreciate both halves of the picture, as for each specific problem one of them might be more natural. (for the popular Euclidean group and its associated geometry this view we're not used to may very well be more intuitive one, imho).

[cygx]

Isn't this just a consequence of phrasing things in terms of the wedge product instead of its 'pullback' via the Hodge? Because personally, I think the most natural representation of a reflection is the mirror plane, not its normal vector...

[enkimute]

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.

[cygx]

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-...

[enkimute]

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.