qx = ax * sin(angle/2)
qy = ay * sin(angle/2)
qz = az * sin(angle/2)
qw = cos(angle/2)
http://www.euclideanspace.com/maths/geometry/rotations/conve...
http://www.euclideanspace.com/maths/geometry/rotations/conve...
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.
>So don’t think of quaternions as a 4 dimensional hypersphere of radius 1
This is also a bit weird:
>But nobody would ever suggest that we should think of a rotation matrix as a 9 dimensional hyper-cube with rounded edges of radius 3.
Even weirder when they claim that the axis-angle interpretation of (unit) quaternions "breaks down".
Anyway, back in high-school when I first became fascinated with quaternions I certainly didn't expect to be working with them on a daily basis two decades later. The moral of this is that anything you learn can become crucial to your career...
If you write out a multiplication table, it seems that it's isomorphic. But... Octonions aren't associtive. Does the even subalgebra of G4 somehow lose associativity? Is it equivalent to Octonions with a cannonical multiplication order?
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.
This, to me, takes the mystery out of why quaternions can represent rotations, and it places quaternions in a coherent theory of geometry that works in any number of dimensions, not just 3D. Alternatively, we could accept that the math just happens to work out that way, or we could even show that quaternions are a double cover of SO(3), but all that does is analyze why something works, whereas the geometric algebra version is a bit less of a leap and builds quaternions from the ground up.
"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.
This is analogous to the problem of finding a coordinate system for the globe: specifying latitude and longitude tells you were you are, but there's a degeneracy at the poles. And no possible coordinate system can solve this problem entirely. Contrast this to the situation of giving a coordinate system for the circle, which we do with it's angle. This isn't quite a coordinate system, due to the problem we already encountered that X and X + 2pi are the same, but that's OK because the these two points are separated from each other. On the sphere, the latitude/longitude pair (pi/2, x) gives the north pole for any value of x, even ones that are arbitrarily close together. That maps not even locally invertible!
You suggest we think of points on the circle as point in 2D space that happen to lie on the circle (i.e. cos and sin of the angle corresponding to that point). Analogously, we can think of points on the sphere as points in 3D space that happen to lie on the sphere (like some point (x,y,z) with x^2 + y^2 + z^2 = 1). And analogously, we can think of rotations of 3D space as a point in 4D space (that happens to satisfy some conditions), and the quaternions give that 4D point. This is fantastic and convenient in both 2D and 3D! But in 2D we didn't need to do this, but could if we wanted to. For 3D rotations, we do need to, or else we have this terrible degeneracy that never rears its head in 2D. In that sense, 2D and 3D are very different!
There are plenty, it's just that you can't have a 3-dimensional one without singularities.
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 information. The latter follows from the former.
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.
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.
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.