4 ms·
Yet another Geometric Algebra introduction claiming GA is better than quaternions, yet ending (section "Problem Solved" near the end) with the exact same formul
by chombier 6y ago
Yet another Geometric Algebra introduction claiming GA is better than quaternions, yet ending (section "Problem Solved" near the end) with the exact same formulas.
Really I like GA and it sure brings a unified treatment for geometric calculations, but in these tutorials I would very much like to see examples of something quaternions can not do, e.g. involving exterior algebra, subspace intersections, duality etc.
- jacobolus 6y agoA quaternion is a kind of multivector consisting of a scalar + a bivector (a bivector is an oriented magnitude with the orientation of a plane). You can get one by taking the quotient of two 3-dimensional vectors (a vector is an oriented magnitude with the orientation of a line). Note that a planar “complex number” is the same kind of object as a quaternion, just with the bivector part always oriennted in the same plane. When people limit themselves to Gibbs-style vectors or to quaternions as a primary formalism what they are doing is pretending that vectors and bivectors are the same kind of object, and this pretense leads to massive amounts of confusion. cf. https://en.wikipedia.org/wiki/Pseudovector https://en.wikipedia.org/wiki/Pseudovector The biggest thing GA adds compared to quaternions is that the same technology generalizes to higher or lower dimensions and to pseudo-Euclidean spaces (and to generalized Möbius transformations, non-metrical contexts, to modeling points and circles as multivectors, etc.), and there are many vector identities which are awkward to express when you are pretending that vectors and bivectors are the same. Another advantage is that you don’t need any arbitrary conventional rules about multiplication of {i, j, k}, since typically the basis is specified in terms of orthonormal unit vectors e.g. e₁, e₂, e₃, so that it becomes very obvious how basic bivectors e₁e₂, e₂e₃, e₃e₁ should multiply: (e₁e₂)(e₂e₃) = e₁e₃ = −e₃e₁, etc.
- creata 6y ago> non-metrical contexts How do you get a geometric algebra without at least a pseudo-Riemannian metric?
- jacobolus 6y agohttp://geocalc.clas.asu.edu/pdf/PGwithCA.pdf http://geocalc.clas.asu.edu/pdf/PGwithCA.pdf http://geocalc.clas.asu.edu/pdf/UGA.pdf http://geocalc.clas.asu.edu/pdf/UGA.pdf
- chombier 6y agoI agree with all of this, and to sibling reply as well. My point is that these advantages are seldom (if at all) discussed in most online introductions to GA I've read so far, yet to me they're the real selling points of GA.
- pjbk 6y agoQuaternions are left handed, which is annoying but not a big deal by itself, but most importantly collapse the scalar with the pseudoscalar. For regular applications of unit quaternions in 3D euclidean space such as rotations, screw displacements and dual quaternions, that is not much of an issue since the scalar part is initially zero anyways (e.g. translation and rotations), it wholly contains the quadrature information after the operation (e.g rotation), or the scalar part is always separate from the vector quaternion (e.g. moments). However if you want to see the quaternions as a vector subspace in problems of higher dimensions or in non-euclidean manifolds, for example path interpolation, finding integral solutions or representations, they fall short and it makes sense to use a Clifford algebra that is compatible with differential operations (e.g Lie) and gives consistent calculations across all dimensions.