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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…”
Lasenby: Computation is no longer a bottleneck for geometric algebra
“I don't think it's not computational anymore, because we're building up more and more tools.”
Lasenby: Geometric algebra provides huge simplification in electromagnetism
“Electromagnetism is also, I should have mentioned this, is a field whereby you really get huge simplification.”
Lasenby: Geometric algebra requires unlearning standard commutative mathematics
“It's anti-commutative. It's not a commutative algebra. So immediately you throw away everything you've learned as a kid and through school and through university. So it makes perfect sense once you're into it, but there's a little, you know, there's a little h…”
Lasenby: Geometric algebra empowers practitioners with geometric intuition over advanced math
“A lot of people have a lot of geometric insight But maybe not the mathematical sophistication. And I think this will certainly sort of give them a big advantage because it seems to me to be the way the world works. This, if it's a unifying language, it's what …”
Lasenby: Engineers are developing geometric algebra instruction sets for FPGAs and chipsets
“We have a community, not a massive community, but there are lots of people who are very interested in, in you know, potentially getting instructions for chip sets, et cetera. FPGA is, it is. So, and of course we have quite a lot of Programs that people have be…”
Lasenby: The vector cross product only works in three dimensions
“Now, that's all very nice, but it only works in three D. It doesn't work in any other dimension. Because in a plane, I've got no perpendicular. You're stuck. In four dimensions, there's no concept of a perpendicular to a plane.”
Lasenby: Quaternions prevent singularity issues and parameterize rotations smoothly
“Quaternions have been particularly nice because they are minimally parameterized. They have three components. They are smooth. They don't suffer from singularity problems.”
Lasenby: Conformal geometric algebra turns geometric shapes directly into algebraic objects
“Well, what this gets you is that points, lines, planes, circles, and spheres become objects in the algebra. They're objects. You give me a, it's a C. This big C is a circle. It's a trivector in my five dimensional space. And rotors, Which are these, this class…”
Lasenby: Conformal geometric algebra handles non-Euclidean geometries by altering invariants
“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 o…”