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Comment by blt | Hacker News Reader

if the author is reading this: please define the acronym PGA when first using it!

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.

I'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!

This 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/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.

It 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"?)

Sorry I meant "complex unit".

"complex number with a magnitude of 1" is the definition of a "complex unit".

I like "roots of unity" (not necessarily rational) or "unit phasor" or "non-integer powers of -1"

But... aren't the traditional 4x4 transformation matrices already use projective space, essentially?

Yes! The use of FPGA for "Fast PGA" was particularly confusing.

Apologies. Had that joke sitting around for waaaay to long. Not that great in retrospect :D

I got a giggle out of it!

done. mea maxima culpa.

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