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Less Weird Quaternions Using Geometric Algebra

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Re: Less Weird Quaternions Using Geometric Algebra

#21

Post by defen »

abainbridge wrote: post #20
defen wrote: post #19

Earlier quoted context omitted.

A bivector is a plane spanned by two vectors, with an associated orientation.

That hasn't helped!

In a geometric setting - if you have two vectors, you can position them so that both have one end at the origin. This spans a plane (test it out yourself in 2D or 3D space with two pencils, put the eraser at the origin for each; it's a parallelogram). The area of the plane will depend on the length of the pencils. You can assign an orientation to the plane by imagining a rotor embedded in the plane that spins either clockwise or counterclockwise.

Re: Less Weird Quaternions Using Geometric Algebra

#22

Post by imh »

twic wrote: post #17
yequalsx wrote: post #16

Earlier quoted context omitted.

I think there is a lot of unintentional irony in what you wrote. You start out saying, "There's a lot of hand waving in that phrase..." and then go on to write: "Start by using bivectors to represent reflections, then take the closure of your bivectors and you get the even-ordered subalgebra." It reminds me of the running joke we had in graduate school. Any book whose title starts off with "An Elementary Introduction…

There's definitely something a bit "just a monoid in the category of endofunctors" about that description. Which is not to say that it's not both true and helpful - it's just not very accessible. Perhaps if there was a one-sentence explanation of what a bivector was, it would be a lot clearer.

A scalar is just a magnitude (call it amount). A vector is a direction with magnitude (call it length). A bivector is a pair of directions with magnitude (call it area). You can keep going and make something with volume and even more into higher dimensions.

Imagine putting a 2d rectangle into 3d space with some orientation. Starting from some corner of the rectangle, you have two sides coming out from it. In 3d space, those sides make 3d vectors. You can generalize it to 3 vectors making a cube with magnitude (volume). And these don't have to be perfect rectangles and cubes, they can be parallelograms and parallelopipeds (3d parallelograms) and higher dimensional analogues.

Re: Less Weird Quaternions Using Geometric Algebra

#23

Post by logfromblammo »

klodolph wrote: post #12

A couple things to add. For notation, we would often see the basis vectors named (e_1, e_2, e_3) instead of (x, y, z). The quaternions are the even-ordered subalgebra of the 3D exterior algebra. The exterior algebra has scalars (1), vectors (x, y, z), bivectors (xy, yz, zx), and pseudoscalars (xyz). The even-ordered subalgebra is scalars and bivectors (1, xy, yz, zx). Adding or multiplying two even-ordered multivecto…

The odd-ordered subalgebra (x, y, z, xyz) is symmetrical to the even-ordered subalgebra (1, xy, yz, zx), and can also represent quaternions.

Re: Less Weird Quaternions Using Geometric Algebra

#24

Post by madhadron »

klodolph wrote: post #7
catnaroek wrote: post #6

> OK so what is this Geometric Algebra? It’s an alternative to linear algebra. No. Geometric algebra is a use case of linear algebra. How can it be an alternative? > Before I tell you how to actually evaluate the wedge product, I first have to tell you the properties that it has: > 1. It’s anti-commutative: a \wedge b = -b \wedge a > 2. The wedge product of a vector with itself is 0: a \wedge a = 0 Redundant informat…

It's definitely an alternative in the sense that it gives you an alternative framework for concepts that are taught under the banner of linear algebra in school. For example, it gives an alternative construction for quaternions as a subalgebra, and it gives the exterior product as an alternative to the cross product.

No, it's not an alternative. You're conflating the grab bag of topics in an undergraduate linear algebra class with the subject of linear algebra.

Linear algebra is the study of linear operators on vector spaces over fields (a special case of modules over rings). Some vector spaces are inner product spaces, but most are not.

Exterior algebra is an example of multilinear algebra. Clifford (or geometric) algebras are constructed as an algebra over a vector space. Quaternions come via a different route: instead of constructing a multilinear algebra, they are one of a handful of very special algebras like C and the octonions.

Re: Less Weird Quaternions Using Geometric Algebra

#25

Post by klodolph »

logfromblammo wrote: post #23
klodolph wrote: post #12

A couple things to add. For notation, we would often see the basis vectors named (e_1, e_2, e_3) instead of (x, y, z). The quaternions are the even-ordered subalgebra of the 3D exterior algebra. The exterior algebra has scalars (1), vectors (x, y, z), bivectors (xy, yz, zx), and pseudoscalars (xyz). The even-ordered subalgebra is scalars and bivectors (1, xy, yz, zx). Adding or multiplying two even-ordered multivecto…

The odd-ordered subalgebra (x, y, z, xyz) is symmetrical to the even-ordered subalgebra (1, xy, yz, zx), and can also represent quaternions.

The odd-ordered subspace is a subspace, not a subalgebra, because multiplication is not closed.

Re: Less Weird Quaternions Using Geometric Algebra

#26

Post by klodolph »

auggierose wrote: post #11
klodolph wrote: post #7

Earlier quoted context omitted.

It's definitely an alternative in the sense that it gives you an alternative framework for concepts that are taught under the banner of linear algebra in school. For example, it gives an alternative construction for quaternions as a subalgebra, and it gives the exterior product as an alternative to the cross product.

I'd say alternative is an unlucky choice of words. I'd rather say geometric algebra (GA) is an extension of linear algebra (LA). In order to really understand GA you need first to firmly understand LA. Then it becomes clear that all that GA does is to turn a Hilbert space into an algebra called a Clifford algebra, and to examine the geometric semantics of the various operations that pop up in the process. Here are th…

My personal recommendation for a book on geometric algebra is the one by Hestenes, New Foundations for Classical Mechanics (https://www.amazon.com/dp/0792355148/). I was disappointed by Geometric Algebra for Computer Science and I recently got rid of my copy when I moved to a new apartment, but I have a mathematics background and tend to prefer denser books.

I would say that "alternative" is a viable word here. Yes, you'll need a foundation in linear algebra to understand geometric algebra, but our classes and books on linear algebra go beyond what is necessary for understanding geometric algebra and introduce concepts (like the cross product) which have more natural equivalents in geometric algebra. I'm not even convinced that it's necessary to have a good understanding of matrixes in order to work with geometric algebra.

Re: Less Weird Quaternions Using Geometric Algebra

#27

Post by klodolph »

yequalsx wrote: post #16
klodolph wrote: post #3

Earlier quoted context omitted.

There's a lot of hand waving in that phrase, "it turns out". Sure, "it turns out" that 3D uses four numbers. Why? Geometric algebra explains that in a succinct way that also appeals to our intuition about geometry. Start by using bivectors to represent reflections, then take the closure of your bivectors and you get the even-ordered subalgebra. This will have dimension 2^(N-1)... so 2 for 2D, 4 for 3D, and 8 for 4D.…

I think there is a lot of unintentional irony in what you wrote. You start out saying, "There's a lot of hand waving in that phrase..." and then go on to write: "Start by using bivectors to represent reflections, then take the closure of your bivectors and you get the even-ordered subalgebra." It reminds me of the running joke we had in graduate school. Any book whose title starts off with "An Elementary Introduction…

I aiming that explanation at people who had read and understood the article. The article explains bivectors and how they can be used to represent reflections, and "closure" is a fairly common concept, so once you put those two together you should get a mathematical object which I've called "the even-ordered subalgebra". I haven't explained why it's even-ordered or what a subalgebra is, but I used those terms so you could at least have the vocabulary to talk about it or do a Google search.

Mathematics education is hard. In my experience, you start out with no understanding of a subject and can't understand it when people explain it to you, and at some point it clicks and you can't understand why it was ever difficult. I could be intentionally obtuse and, for example, describe a vector space as an "abelian group, field, and homomorphism from the field to group endomorphisms", but I feel that's the only people who would use that definition already have a good understanding of vector spaces.

The reason that I consider the non-GA approach to quaternions as rotations "hand wavy" is because it's not constructive, or perhaps just because I personally don't understand it. Using GA, I can construct a representation for rotations in any Euclidean space, not just 3D space, but 2D, 4D, 5D, whatever. However, without GA at my disposal, the fact that unit quaternions are a double cover for SO(3) seems like some kind of black magic that came from the void.

I have a few drafts of an introductory article I was writing on geometric algebra sitting on my hard drive, but I've never been able to get the article into a state I'd consider publishable. So instead, I'm trying to inject what I know into HN discussions.

Re: Less Weird Quaternions Using Geometric Algebra

#28

Post by klodolph »

abainbridge wrote: post #20
defen wrote: post #19

Earlier quoted context omitted.

A bivector is a plane spanned by two vectors, with an associated orientation.

That hasn't helped!

Think in terms of magnitude (which can be positive or negative) and direction.

Think of a scalar. It has a magnitude but it doesn't have a direction. It's 0-dimensional.

Think of a vector. It has a magnitude (the size of the vector) and it also has a direction, which points in a straight line through the origin. It's 1-dimensional. For example, the vector (2,0,0) has magnitude 2 and points along the X-axis. You could write that as 2 * x, if x is the vector (1,0,0).

A bivector also has a magnitude, but instead of being 0-dimensional (like a scalar) or 1-dimensional (like a vector, it's 2-dimensional. So you could have a bivector that "points" along the entire XY-plane (remember: two-dimensional) and has some magnitude, say, 5. You could write that as 5 * x * y, if x is (1,0,0) and y is (0,1,0).

If you attach physical units to these things, then you might have units of meters for vectors, and square meters for bivectors.

Having an understanding of subspaces in linear algebra is helpful.

Re: Less Weird Quaternions Using Geometric Algebra

#29

Post by Retra »

catnaroek wrote: post #10
klodolph wrote: post #7

Earlier quoted context omitted.

It's definitely an alternative in the sense that it gives you an alternative framework for concepts that are taught under the banner of linear algebra in school. For example, it gives an alternative construction for quaternions as a subalgebra, and it gives the exterior product as an alternative to the cross product.

Exterior algebras themselves are defined in terms of standard linear algebra concepts (tensor algebras, ideals, quotients). I don't see how you can claim something is an alternative to linear algebra if you need to know linear algebra to define it, nevermind use it.

Left is an alternative to right. Right is the opposite of left. They are alternatives, yet defined in terms of each other. You can't know one without the other.

Doesn't seem contradictory to me...

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