Math n Stuff (Specifically Geometric Algebra)

So one thing, I've been doing lately instead of building things is studying geometric algebra.

Now you might as why on earth would someone devote them self to some obscure type of math when vector calculus, differential equations and linear algebra satisfy most engineers. Geometric Algebra while not that widely known as far as I can tell (at least I know no mechanical engineers that know of it, but that really isn't saying much), has some pretty snazzy properties. I will babble about it for for a variety of reasons:
1. Its pretty cool (for debatable definitions of cool). 
2. It has applications to a robotics engineering problem I have encountered and would like to solve. 
3. Geometric algebra is like a general enough form of math that it encompasses several other forms of math used in various areas of engineering in a single construct and could be useful in a wide variety of engineering applications.
4. Writing down the important bits will help me remember it.

Honestly as a mechanical engineer it's been a bit of a 'mindfuck' learning mathematics from papers on maths. After reading several papers, maths from a mathematician's perspective seems very different from how I've thought about math up to this point and how I think more applications oriented think about math. As a meche I was looking at maths in what I think was a practical yet backwards thought process that dictated math functions the way it does because that's the way the world works; but I think mathematicians would say math functions the way it does because of the way that people decided to define it is as a logical construct, regardless of how reality works. Which I would say is much more accurate and leaves the door open to interesting ways types of constructs. Math only functions how it does because of how we define it, it just so happens the world was nice enough to allow its self to match up with some of out logical patterns that we like to call 'math'.

Either way, back to geometric algebra.
Some of its useful characteristics: 
Geometric Algebra naturally models geometric things we tend to care about in a concise fashion,  it can be extended to n dimensions, and structures that represent imaginary numbers are naturally formed by the algebra without actually having imaginary numbers. Remember those weird things where \(i^2 = -1 \)? That is an 'imaginary' number. In geometric algebra (what I consider one of the most useful properties) you can represent more than just \(i\), the algebra can be used for quaternion representations where the imaginary numbers are extended to \(ijk\) or extended even further to represent arbitrary set of 'imaginary' constants which would all have the property \(n^2 = -1\) while remaining orthogonal to each other in an whatever dimensional space.
If you're an engineer you might realize the utility of that last bit, it allows you to perform rotations of abitrary dimension using rotors which are kind of like generalized versors.

While I don't know of any applications where you would want to use >6 dimensions but I think it's kind of nice that geometric algebra gives you the option.

For a bit of history on Geometric Algebra

Otherwise I'm going to talk about some of the technical details of geometric algebra. This are defined in terms of the geometric product, which is the fundamental  operation of geometric algebra, it is denoted by a lack of symbol ,similar to what we do with multiplication. It has the properties of multiplicative associativity, and distributivity over addition. In the following example \(a,b\) and \(c\) are multivectors but you can think of them as normal vectors right now. The geometric algebra rules that:

1.Associative: \(a(bc) = (ab)c\)
2.Distributive: \(a(b+c) = ab + ac \)

Note the geometric product is non commutative, I think that this property is best thought of as a feature because it ends up giving the outer product of the geometric some important characteristics. The inner and outer products of the geometric algebra are defined in terms of the geometric product as:

outer product: \((ab-ba)/2 = a \wedge b\)

inner product: \((ab+ba)/2 = a \cdot b\)

geometric product in terms of the inner and outer product: $$ab = a \cdot b + a \wedge b$$

And the outer and inner products are analogous to the cross products and dot products of normal vector math how ever they have some subtle differences that I wont go into right now.

 More on the outer product:
The outer product is the antisymmetric part of the geometric product. Also from now on the outer product will be noted as \(\wedge\) in equations. This property of antisymmetry has some useful effects, and results in something almost the same as a cross product. The cross product of two parallel vectors is 0. similarly the outer product of two vectors that are scalar multiples(a.k.a. parallel) will also result in 0. 

$$a \wedge b = - b \wedge a$$
if  $$ b = a $$
$$ a \wedge a = - a \wedge a $$
$$ \therefore a \wedge a = 0 $$ 

If vectors are orthogonal then the following is true: \(ab = a \wedge b \). The result of vector \(vector2 \wedge vector1\) is not a vector, it is of greater dimension or 'grade', a vector is of grade 1, a plane is of grade 2 and a volume element would be of grade 3. When two things are wedged together their grades add as long as they are not linear multiples of each other.
$$ a , vector, line$$
$$ a \wedge b, grade 2 plane$$
$$ a \wedge b \wedge c, grade 3 volume$$
$$ a \wedge b \wedge a = - a \wedge a \wedge b = - 0 \wedge b, grade 0$$

This is to show some of the basic properties of geometric properties of geometric algebra but I don't have time to write a text book so I'm just going to show a few more things about the outer product right now.  Consider two vectors \(a\) and \(b\) with orthonormal bases (fancy terms for \( \hat x\) and \( \hat y \) ) \( e_{1}, e_{2} \) and scalars \( a_{1} , a_{2}, b_{1}, b_{2} \).

Consider the following: 

$$ a \wedge b = (a_{1} e_{1} + a_{2} e_{2}) \wedge (b_{1} e_{1} + b_{2} e_{2}) $$
Remember in geometrric algebra stuff distributes over addition so:
$$ a_{1} b_{1}  e_{1} \wedge e_{1} + a_{1} b_{2}  e_{1} \wedge e_{2} + a_{2} b_{1}  e_{2} \wedge e_{1} + a_{2} b_{2}  e_{2} \wedge e_{2} = a_{1} b_{2}  e_{1} \wedge e_{2} + a_{2} b{1}  e_{2} \wedge e_{1} $$

and the outer product anti-commutes so:
$$ = ( a_{1} b_{2} - a_{2} b_{1})  e_{1} e_{2}$$ 

If you are familiar with the vector math this may be apparent but the outer product of two vectors results in a planar element with a magnitude of the parallelogram spanned by the vectors \(a\) and \(b\). This is the same as a cross product except that the cross product would produce a third vector perpendicular to the other two rather than a plane spanning the two vectors. This pattern continues with higher dimensions, the outer product of 3 vectors would produce a signed volume element with the magnitude of the spanned parallel piped. In fact this would hold to n-dimensional hyper volumes!

I mentioned before that one of my favorite properties of geometric algebra was how it treated imaginary numbers. Lets take a look at how 'imaginary numbers' occur as a result of the geometric algebra. Consider orthonormal vectors of magnitude 1, \(a\) and \(b\). The inner product is 0 for orthogonal vectors so in this case \(ab = a \wedge b\).

$$ (ab)^{2} = abab = -aabb = - (aa)(bb) = -1 $$

That's right. \( (ab)^{2} = -1\). The imaginary numbers naturally appear out of the geometric algebra. Part of the nifty thing is is that it does not require anything to define this besides the geometric product, and isn't actually dependent on a number of dimensions to exist besides having more than two. In this way you can construct multiple 'imaginary' planes and perform rotations higher dimensional systems in the same manner as a lower dimensional system. Some of you guys might note the possibility of the Euler equation to be applied here which does happen. Infact is allows us to construct the powerful rotor which allows arbitrary rotation across a vector. But that is for another time. 

End of geomtric algebra for now next time I will talk about the inner product and general screw motion.

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