Lasenby: Conformal geometric algebra handles non-Euclidean geometries by altering invariants
Joan Lasenby · Joan Lasenby on Applications of Geometric Algebra in Engineering · Y Combinator · Sep 17, 2018 · at 27:03
Cambridge University engineering reader Joan Lasenby discusses how conformal geometric algebra generalizes beyond Euclidean space to hyperbolic and spherical geometries.
0:00 / 0:33exact quote · 33.9s
▶ Watch the full episode on YouTube → 720p mp4 · rendered on demand · StarZero watermark
“Now, if you have a different underlying geometry, so if you have hyperbolic or spherical geometry. Then in this algebra, you have to change, you, in conformal algebra, Euclidean geometry is the thing that keeps the point at infinity invariant. Then if I keep other things invariant, I get these other geometries. And I can just use my standard apparatus and do exactly the same things.”
quote is from the automated transcript, cleaned for reading: filler sounds and stutters are removed, nothing is rephrased. names can be misheard (the analysis reads context, assessments check outside sources). how →
More from Joan Lasenby
Lasenby: Geometric algebra unifies classical mechanics, linear algebra, and quantum physics
“So if you know this language, you can not only do rigid body dynamics in engineering, classical mechanics. You can do linear algebra without matrices and without tensors. And you can do complex things in quantum space-time physics with the same mathematical sy…”
Lasenby: Rotation matrices are numerically inconvenient due to manifold constraints
“Rotation matrices are not numerically nice to deal with because you have to keep them on the manifold. You have to make sure if you change, if you update rotation matrices, you have to make sure it obeys the constraints.”
Lasenby: Geometric algebra enables easy calculus on coordinate-free objects
“So I've said that I have these geometric objects, my rotations are objects, so I can write down coordinate free expressions. But not only can I write them down, I can differentiate with respect to them, easily. So, because I have this algebra of objects. I can…”
Lasenby: Geometric algebra requires more underlying compute but simplifies code
“Underlying, you've got an algebra of A much bigger algebra than three-dimensional space where I've got three vectors. So actually, computationally, there's more going on. There's more going on. Yeah, but at a higher level, you know, I can get code to do all th…”
Lasenby: Geometric algebra adds no new capabilities, only intuitive clarity
“Geometric algebra won't really give you anything that you can't do conventionally. What it might enable you to do is to see how, how to do that thing.”
Lasenby: Geometric algebra eliminates trial-and-error matrix hacks in computer vision
“If people have worked with computer vision, they will know that often things don't work. So instead of a rotation matrix R, they try R transpose. And instead of a translation vector T, they try R transpose T. And they mess around until it works because it's ki…”