From a brief skim, it looks like this introduction has the same problem as every other introduction to geometric algebra that I've seen: a complete lack of practical examples.
Where is there an introduction that shows you how to use geometric algebra to write better code to perform calculations? Or maybe to do word problems, like when we teach kids algebra or calculus?
Even in a supposedly practical book like Foundations of Game Engine Development, Volume 1, there are code examples throughout the book, except for the chapter about geometric algebra.
I share your pain, as someone who messes around with game engine dev as a hobby and has also read through that book. He sure made geometric algebra sound like it _should_ be useful, but man is it clear as mud to me.
Of course in general I'm not surprised; for instance I'm not surprised that this paper doesn't have many practical examples. It's higher level math, of course, and for people in that space, showing a small example where you use the geometric product to solve an abstract linear algebra problem is about as close to 'practical' as you are going to get.
But I found it much more aggravating in that book. It's specifically a practical book after all, meant to help lay foundations for renderer and physics engine dev -- but as you said, that section is completely devoid of any real examples of how you could integrate it to make Problem A or B easier, or how it simplifies your transform code, or literally anything. The complete lack of code examples or any such real-world applications make the chapter feel a bit out of place, maybe even tacked on or gratuitous. (Which is a shame because he kind of builds up to that chapter, dropping hints here and there about how it will fundamentally shift our understanding.)
From my math undergrad, I feel like generally there are two feelings on this kind of thing:
- No practical application is needed, since studying math is an end in itself
- If there is a practical application, it should be fairly obvious and the author should not have to stoop to the level of trying to enumerate them
I'm not saying I agree, but I think there is part of the math community that doesn't care to enumerate practical applications. Expecting this kind of thing from this area of research may be an uphill battle.
Sure, but geometric algebra is specifically marketed as a unifying framework that makes geometric problems easier to reason about. It's not something you learn to solve previously unsolvable problems. So why is there a distinct lack of down-to-earth problem solving?
I think one would have to have a background in modern algebra and physics to see where the "unification" is useful. Then it becomes a lot more obvious, for example Maxwell's equations can be compressed down to one equation with this type of algebra.
In any case, check section 10 for another application for physics.
Where is there an introduction that shows you how to use geometric algebra to write better code to perform calculations? Or maybe to do word problems, like when we teach kids algebra or calculus?
Even in a supposedly practical book like Foundations of Game Engine Development, Volume 1, there are code examples throughout the book, except for the chapter about geometric algebra.