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Projective geometric algebra for anyone wondering. A null basis vector is added to the basis vectors of the space you're working in. This allows the algebra to
by at_compile_time 3y ago
Projective geometric algebra for anyone wondering. A null basis vector is added to the basis vectors of the space you're working in. This allows the algebra to represent geometric objects that do not pass through the origin.
- epistasis 3y agoI've been working through a bunch of Geometric Algebra on the web and YouTube lectures in recent weeks. Though I guessed Projective Geometric Algebra, I still wasn't certain as it's the first time I can recall seeing the acronym!
- lupire 3y agoThis called "affine transformation" in linear algebra language. (Linear algebra is stretches and rotations. Affine (affinity?) adds translations) https://people.computing.clemson.edu/~dhouse/courses/401/notes/affines-matrices.pdf https://people.computing.clemson.edu/~dhouse/courses/401/not... In 2 dimensions: Rotation = multiplying by an imaginary unit. Stretches = multiplying by a real number Translation = adding a complex number. In higher dimensions, the analogy to complex numbers breaks down.
- xeonmc 3y agoIt is not affine transforms per se but rather the expansion into homogeneous coordinates that enables translation by treating it as if it's a shear that leaves the reciprocal dimension untouched. > Rotation = multiplying by an imaginary unit. This is also not quite right. Rotation is multiplying by a complex number with a magnitude of 1 (or perhaps you meant to say "raising a number to the power of i"?)
- Joker_vD 3y agoBut... aren't the traditional 4x4 transformation matrices already use projective space, essentially?