[enkimute]

> But to really make it popular and understandable, a "simple" version of GA that handles translation, rotation & non-uniform scaling would really help, without the group theory concepts, even better, make it in the context of a scene graph hiearchy, with a unified operator like "multiply".

The rich structure of GA (that ultimately follows from just one axiom extra) unifies a wide range of concepts and theories. Considering just one application, or one link, puts one at risk of arriving at a model that breaks these connections to other parts of mathematics. It seems unfair to expect to understand the why without considering the connections to Lie Groups, their associated geometries, differential forms, etc.

That said, I have some unpublished examples displaying and processing bvh (mocap) files that I'll try to cleanup and put online.

> Also is it possible to collapse a series of such transforms in a single versor like you can do with matrices without going into dual quaternion stuff? In generic game developemnt, translation, rotation, and non-uniform scaling are all extremely basic things that cannot be handwaved away or "too big".

Versors combine just like matrices (using just the ordinary product). (doing this for translations/rotations _are_ the dual quaternions, but you don't have to (and imho shouldn't) call them that.). Non-uniform scaling along your scenegraph (as opposed to in the beginning (object space) or at the end (view space)) is usually frowned upon in professional game development. (it makes it impossible to correct matrices using Gramm-Shmidt, and adds a lot of complexity to things like tracing hit rays etc).

> Also, why the need o a dual(e12, e02, e01) to represent a point when in vector it's just a (e0, e1, e2). This is just counterinuitive. This is what I mean by "quickly gets complicated" and it feels nearly as opaque as cross product in vector math.

This is because geometry and group theory are intricately connected. When you use matrices, you represent elements with vectors and transformations with matrices - they're separate things. In Geometric Algebra, every element also _is_ a transformation. (a plane represents a reflection in that plane, a line represents a 180 degree rotation around that line, a point represents a point reflection in that point). So now there is a strong link. Whatever you use to represent reflections should also represent planes, same for rotations/translations and lines, or point reflections and points.

It is in fact very intuitive and simple, its just different from what you're used to. For example, in 2D, given a point at euclidean position (3,4), here are the two mindsets:

* classic : it is a sum of three times the 'x' vector and 4 times the 'y' vector. (and actually than add in '1' homogeneous vector). '3x + 4y + w' (in memory : 3,4,1 )

* GA : (3,4) is a system of equations. Namely 'x=3' and 'y=4', or homogeneously : 'x-3=0' and 'y-4=0'. Such homogeneous linear equations are lines (in 2D), and represented by vectors : 'e1-3e0' and 'e2-4e0', solving such a system of equations is just the outer product: '(e1-3e0) ^ (e2 - 4e0) = 3e20 + 4e01 + e12'. (in memory : 3,4,1)

so because it is on the bivector basis, this element (3e20 + 4e01 + e12) now represents both the point at (3,4) as well as a rotation of 180 degrees around that point. Just like the line (e1-3*e0) represents both the line `x=3` as well as a reflection w.r.t. that line. For the same reason the product of two lines will give you the rotation or translation between them and the product of two points will always give you the translation.

So I'd argue its a lot more intuitive, don't factor out the time it took you to find the linear algebra approach intuitive.

[hohohmm]

Thanks for the explanations. Just want to elaborate on this point:

>> Also is it possible to collapse a series of such transforms in a single versor like you can do with matrices without going into dual quaternion stuff? In generic game developemnt, translation, rotation, and non-uniform scaling are all extremely basic things that cannot be handwaved away or "too big".

>Versors combine just like matrices (using just the ordinary product). (doing this for translations/rotations _are_ the dual quaternions, but you don't have to (and imho shouldn't) call them that.). Non-uniform scaling along your scenegraph (as opposed to in the beginning (object space) or at the end (view space)) is usually frowned upon in professional game development. (it makes it impossible to correct matrices using Gramm-Shmidt, and adds a lot of complexity to things like tracing hit rays etc).

Though frowned upon, it's important to retain the ability to do non-linear scales, as that's part of tuning things in-engine fast. It could be later corrected but without it, it becomes very cumbersome to do quick tunings of object scales or maybe simply doing quick scene mockups. I work in games in a professional capacity and find this usage very common and almost indispensible.

[cygx]

> this element (3e20 + 4e01 + e12) now represents [...] the point at (3,4)

Note that this is just the Hodge dual of (3e1 + 4e2 + e0), or, as you labelled it previously, (3x + 4y + w)...

[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.