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The main problem with GA is if you want a unified transformation hiearchy, geometric algebra could quickly get very complicated, as it requires projective GA to

The main problem with GA is if you want a unified transformation hiearchy, geometric algebra could quickly get very complicated, as it requires projective GA to handle translation and dual-quaternion to handle non-uniform scaling. Geometry algebra appears beautiful to simple rotation cases and explains very well concepts that feel incomplete in simple vector math, but in real application for genric game engine, you can't really code up a scene graph with unified transforms in like 2 hours. Not to mention that even the GA people do not all agree with each other how to do the more advanced GA stuff(strange inversions and ext). Surprisingly very few people talk about this on Youtube with all those GA tutorial videos, and you can find scant information on the bivector site forum. In comparison, despite matrix represetntion's weaker mappng to geometric concepts, it handles everything in a more unified interface without too much complications. For that matter the GA math feels much to be desired. There probably exists a undiscovered better version of GA that handles translation and scaling better and everybody could instantly agree on. Until that day GA probably won't see much general usage.

enkimute 6y ago
Ok, standard 4x4 matrices also implement a projective (aka d+1) model. (the 'w' coordinate is just the projective coordinate). So no difference with GA in that respect. Setting up a unified transformation hierarchy is actually very easy, and again not really different from how you would approach it with matrices. (plus, its more performant). Simply swap the matrix with the appropriate versor. Which (versors or matrices) are appropriate depends on the symmetry group you are interested in : Orthogonal Group (just rotations : distance + origin preserving) in d dimensions -> use the geometric algebra R_d. (classically : complex numbers, quaternions) Lorentz Group (rotations + boosts : spacetime distance + origin preserving) in d space dimensions and 1 time dimension -> use the geometric algebra R_{d,1}. (classically : Lorentz transformations) Euclidean Group (translations + rotations : distance preserving) in d dimensions -> use the geometric algebra R_{d,0,1}. (classically : planar quaternions, dual quaternions) Conformal Group (translations + rotations + dilations : angle preserving) -> use the geometric algebra R_{d+1,1}. (classically : linear fractional transformations) General Linear Group (translations + rotations + sheering + ... : preserves parallelism/incidence) : use d+1 x d+1 matrices. Working in a symmetry group that is 'to big' comes at a cost - both in algorithmic complexity as well as numerical precision. If you only want translations/rotations, but are using matrices you'll have to resort to things like Gramm-Shmidt or SVD to re-orthogonalize your matrices after doing numerical calculations. (you have to project it back to the solution manifold in math terms - this is almost never trivial and often impossible). (for those interested, I explain this in-depth in my GAME2020 talk : https://www.youtube.com/watch?v=ichOiuBoBoQ&ab_channel=Bivector https://www.youtube.com/watch?v=ichOiuBoBoQ&ab_channel=Bivec... )
- hohohmm 6y ago
  Thanks for the information. It's nice of you to point out that 4x4 matrix is projective in nature, and I understand that GA could potentially be more performant for its more compact usage of numbers. 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 thoery concepts, even better, make it in the context of a scene graph hiearchy, with a unified operator like "multiply". 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". 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. Just explaining my experience digging in GA for a couple of weeks.
  - enkimute 6y ago
    > 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.
creata 6y ago
Like enkimute said, your options re translation are exactly the same with matrices and with geometric algebra: either carry a vector offset with you, or move to projective space (homogeneous coordinates).
- hohohmm 6y ago
  The problem with carrying a vector with you is, you can't really concatenate transform. Maybe I am missing something?
  - pfortuny 6y ago
    You work with 4x4 (in 3D space) matrices, with a specific form (a row/columb of 1’s). The affine part of a projective transformation. All this GA stuff is OK but what is the problem with matrices of affine transformations?
    - enkimute 6y ago
      a selection: - no closed form exponential/logarithm (i.e. cant interpolate) - no numerical stability for subgroups (i.e. need Gramm-Shmidt, SVD) - no covariant transformations (i.e. need adjugate to transform axial vectors) - no geometric construction. (e.g. in GA product of two elements is (square of) versor between them) - costly inverses (just some sign swaps in GA)
  - deleted 6y ago
    [deleted]
  - creata 6y ago
    I'm not sure what you mean. Let Ax mean rotating x by A (which is a rotation matrix, quaternion, rotor, whatever). Then, the composition of Ax + b and Cx + d is C(Ax + b) + d = (CA)x + (Cb + d).
    - hohohmm 6y ago
      yes but what about non-uniform scaling. Is it possible to handle it in this case?

enkimute 6y ago

Ok, standard 4x4 matrices also implement a projective (aka d+1) model. (the 'w' coordinate is just the projective coordinate). So no difference with GA in that respect. Setting up a unified transformation hierarchy is actually very easy, and again not really different from how you would approach it with matrices. (plus, its more performant). Simply swap the matrix with the appropriate versor. Which (versors or matrices) are appropriate depends on the symmetry group you are interested in : Orthogonal Group (just rotations : distance + origin preserving) in d dimensions -> use the geometric algebra R_d. (classically : complex numbers, quaternions) Lorentz Group (rotations + boosts : spacetime distance + origin preserving) in d space dimensions and 1 time dimension -> use the geometric algebra R_{d,1}. (classically : Lorentz transformations) Euclidean Group (translations + rotations : distance preserving) in d dimensions -> use the geometric algebra R_{d,0,1}. (classically : planar quaternions, dual quaternions) Conformal Group (translations + rotations + dilations : angle preserving) -> use the geometric algebra R_{d+1,1}. (classically : linear fractional transformations) General Linear Group (translations + rotations + sheering + ... : preserves parallelism/incidence) : use d+1 x d+1 matrices. Working in a symmetry group that is 'to big' comes at a cost - both in algorithmic complexity as well as numerical precision. If you only want translations/rotations, but are using matrices you'll have to resort to things like Gramm-Shmidt or SVD to re-orthogonalize your matrices after doing numerical calculations. (you have to project it back to the solution manifold in math terms - this is almost never trivial and often impossible). (for those interested, I explain this in-depth in my GAME2020 talk : https://www.youtube.com/watch?v=ichOiuBoBoQ&ab_channel=Bivector https://www.youtube.com/watch?v=ichOiuBoBoQ&ab_channel=Bivec... )

hohohmm 6y ago
Thanks for the information. It's nice of you to point out that 4x4 matrix is projective in nature, and I understand that GA could potentially be more performant for its more compact usage of numbers. 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 thoery concepts, even better, make it in the context of a scene graph hiearchy, with a unified operator like "multiply". 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". 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. Just explaining my experience digging in GA for a couple of weeks.
- enkimute 6y ago
  > 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.

creata 6y ago

Like enkimute said, your options re translation are exactly the same with matrices and with geometric algebra: either carry a vector offset with you, or move to projective space (homogeneous coordinates).

hohohmm 6y ago
The problem with carrying a vector with you is, you can't really concatenate transform. Maybe I am missing something?
- pfortuny 6y ago
  You work with 4x4 (in 3D space) matrices, with a specific form (a row/columb of 1’s). The affine part of a projective transformation. All this GA stuff is OK but what is the problem with matrices of affine transformations?
  - enkimute 6y ago
    a selection: - no closed form exponential/logarithm (i.e. cant interpolate) - no numerical stability for subgroups (i.e. need Gramm-Shmidt, SVD) - no covariant transformations (i.e. need adjugate to transform axial vectors) - no geometric construction. (e.g. in GA product of two elements is (square of) versor between them) - costly inverses (just some sign swaps in GA)
- deleted 6y ago
  [deleted]
- creata 6y ago
  I'm not sure what you mean. Let Ax mean rotating x by A (which is a rotation matrix, quaternion, rotor, whatever). Then, the composition of Ax + b and Cx + d is C(Ax + b) + d = (CA)x + (Cb + d).
  - hohohmm 6y ago
    yes but what about non-uniform scaling. Is it possible to handle it in this case?