- You get one unified simple framework which replaces or incorporates vector, matrix, complex, quaternion, tensor and spin algebra (all of which would otherwise need their own incompatible notations). [1]
- Objects and transformations of objects can be expressed by the same multivectors. In the computer graphics of today you would have let's say: Vectors for points and translations, quaternions for interpolatable rotations, matrices for chaining up transformations and you would have separate objects for rays, planes, etc. But in geometric algebra all of that can be expressed by one class: The multivector. [2] [3]
- It generalizes the same way in all dimensions (which is not true for e.g. vector algebra and the cross product).
- One can easily derive geometric calculus from geometric algebra, thus have derivatives and integrals (which is a lot harder to do when you have 6 different frameworks and notation systems)
- Bonus: Because it is unified, it has less edge cases and you need less workarounds, possibly making it also more stable / robust.
It would be interesting if anyone could contribute if there are serious downsides except for not being widely used (mostly for historic reasons I guess).
Maybe it simpler that quaternions (debatable) or simpler than learning all of the concepts you've mentioned separately, but I wouldn't really call it a "simple framework".
I think that simple should be understood here as easy to use. As a crude analogy, the mechanical principles underlying a car with an automatic shift aren't simple, but the abstraction it provides makes driving much easier to learn.
A more mathematically grounded example could be the complex plane compared to the real line: when encountered for the first time, it is not simple in any way, but it offers a clean solution to many (originally real valued) problems (perhaps by the fact that it is algebraically closed?).
NB: the author of this library also made a very interesting presentation of geometric algebra (using this library as vehicle) for SIGGRAPH 2019: https://www.youtube.com/watch?v=tX4H_ctggYo
I agree that if you only need one specific part, than your needs are better met by using just one of the other frameworks. But in my experience that is rarely ever the case. Usually I have to work with at least 3 or 4 different ones. You don't even have to use all of the possibilities for a unified framework to be simpler already. So the word "simple" is meant in relation to the fragmented alternatives.
[1]: https://en.wikipedia.org/wiki/Geometric_calculus#/media/File... [2]: http://projectivegeometricalgebra.org/projgeomalg.pdf [3]: https://bivector.net/3DPGA.pdf