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Less Weird Quaternions Using Geometric Algebra

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Re: Less Weird Quaternions Using Geometric Algebra

#11

Post by auggierose »

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.

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 three great sources that helped me to understand GA:

1. https://www.amazon.co.uk/Geometric-Algebra-Computer-Science-...

2. https://www.amazon.co.uk/Linear-Geometric-Algebra-Alan-Macdo...

3. https://www.amazon.co.uk/Algebra-Graduate-Texts-Mathematics-... , pages 749-752

The first source gives great motivation and intuition for GA and its various products. Its mostly coordinate free approach is very refreshing and makes the subject feel exciting and magical. This is also the problem of the book, it's easy to end up confused and disoriented after working through it for a while. The second source is great because it grounds GA firmly on LA, and makes everything very clear and precise. The third source gives a short and concise definition of what a Clifford algebra is.

Re: Less Weird Quaternions Using Geometric Algebra

#12

Post by klodolph »

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 multivectors will always give you an even-ordered multivector, and 1 is even-ordered, so the even-ordered multivectors form a subalgebra.

We can also conceive of this subalgebra, the quaternions, as a Clifford algebra. Clifford algebras are generalizations of exterior algebras. Instead of saying v * v = 0, we can put something else on the RHS, and for quaternions we can start with just two basis vectors e_1 and e_2, and then define e_1 * e_1 = e_2 * e_2 = -1. The third basis vector for quaternions is then just e_1 * e_2.

Re: Less Weird Quaternions Using Geometric Algebra

#13

Post by sebastos »

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…

> Redundant information. The latter follows from the former.

It's a blog post, not a paper. If I give you a brand new operator ^ that takes an arbitrary object a and another object b and tell you that it has anti-commutativity, it's not a given that a ^ a = 0. So it's useful to highlight that this IS the case when a and b are the vectors of GA, for intuition's sake. a = -a iff a is the zero vector. What if our objects are rotations or something though? R = -R does not imply R = the zero rotation.

Re: Less Weird Quaternions Using Geometric Algebra

#14

Post by tgb »

Grustaf wrote: post #8
tgb wrote: post #5

Earlier quoted context omitted.

I think that's a good motivation why we would study quaternions, but it's kinda hiding the big difference between 2D and 3D under the rug. In 2D, we have a nice, global coordinate system for the space of all rotations: what we call the angle. (Actually, it's a coordinate system for the "universal cover" of the space of rotations since angle X and angle X + 2pi give the same rotation, which mostly doesn't really matte…

> Meanwhile, in 3D, there is no global coordinate system for the space of rotations There are plenty, it's just that you can't have a 3-dimensional one without singularities. https://en.wikipedia.org/wiki/Hairy_ball_theorem

The term "coordinate system" isn't well-defined without context. I was taking non-singular implicitly as part of that - actually, not so implicit since that's sort of the point of the digression. Of course, other contexts are happy to say that, say, polar coordinates are a coordinate system even including the singular point at radius 0, so I probably should have been more careful and said a "coordinate chart."

Re: Less Weird Quaternions Using Geometric Algebra

#15

Post by catnaroek »

sebastos wrote: post #13
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…

> Redundant information. The latter follows from the former. It's a blog post, not a paper. If I give you a brand new operator ^ that takes an arbitrary object a and another object b and tell you that it has anti-commutativity, it's not a given that a ^ a = 0. So it's useful to highlight that this IS the case when a and b are the vectors of GA, for intuition's sake. a = -a iff a is the zero vector. What if our object…

^ is the multiplication operator of an algebra, though.

Re: Less Weird Quaternions Using Geometric Algebra

#16

Post by yequalsx »

klodolph wrote: post #3
marcv81 wrote: post #2

Quarternions aren't weird. With a 2D angle we use 2 numbers (sin and cos) to calculate vector rotations. It turns out that in 3D we use 4 numbers.

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 to..." was going to be very difficult.

Re: Less Weird Quaternions Using Geometric Algebra

#17

Post by twic »

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…

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.

Re: Less Weird Quaternions Using Geometric Algebra

#19

Post by defen »

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

Re: Less Weird Quaternions Using Geometric Algebra

#20

Post by abainbridge »

defen wrote: post #19
twic wrote: post #17

Earlier quoted context omitted.

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

That hasn't helped!
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