To some extent, the book justifies Arthur Cayley (the inventor of matrix algebra)'s adage that "Projective geometry is all geometry". Towards the end of the book, models of non-Euclidean geometries are built within CP^2. I've written up an overview in this Wikipedia sandbox: https://en.wikipedia.org/wiki/User:Svennik/sandbox
Symplectic geometry feels different once area and volume diverge.
On the other hand, I have a feeling that symplectic geometry (in 3D) is being pushed by its proponents onto the unsuspecting public as the best framework for understanding Hamiltonian mechanics, similar to how geometric algebra people claim that theirs is the best mathematical framework for physics.
Personally, I find both largely unintuitive and, at deeper levels, too complicated to be useful.
That's, by the way, why we have the calculus of differential forms which, unlike vectors with all their flavors (free; polar; axial/pseudo), have a clear geometric meaning, and with which many statements about fields acquire an especially simple form. There are many excellent guides; for the motivation, see, for example, https://www.jpier.org/PIER/pier148/09.14063009.pdf
https://en.wikipedia.org/wiki/Gyrovector_space
Just mentioning ... ;-)
In a spherical context, a “spherical gyrovector” can represent any rotation of the sphere whose axis is on the equator, with the representation being the point where the north pole gets sent. This gets you 2 out of 3 degrees of freedom for spherical rotations. Then you can represent an arbitrary rotation of the sphere as the composition of a “gyrovector” and a rotation about the north pole. But the details here are tricky and unintuitive and a lot of the symmetries of spherical rotation are not reflected in the representation.
The deficits of this system are a bit less obvious in a context (hyperbolic space) that students are less familiar with. But if you represent the hyperbolic plane as a paraboloid in pseudo-Euclidean space (akin to representing a sphere as a surface embedded in Euclidean space), a tool similar to unit quaternions is also a more convenient and natural representation for hyperbolic rotation.
* * *
Geometric algebra as a language makes it easy and natural to understand and describe the meaning and relationships between various rotation representations, and is much better for this purpose than e.g. matrices.
> yield the results in a coordinate, matrix and trigonometry-free manner
Some related ideas, for simplifying and generalising geometry:
Euclidean geometry is characterised by inner-product/symmetric-bilinear-form, shown in Section 2.1:
𝐚𝐛 = a₁×b₁ + a₂×b₂
𝐚𝐛 = Σaₙbₙ
|𝐚| = √(𝐚𝐚)
Remarkably, we can do a lot of geometry without using length at all, hence not requiring square roots, and therefore generalising our results to many more fields. Instead, we just work with quantities like 𝐚𝐚 directly, which can be interpreted as the area of a square with side-length |𝐚| (AKA a "quadrance"). An obvious example is Pythagoras' theorem, which relates the quadrances of a right-triangle's sides.
This use of area is probably connected to symplectic geometry, but I haven't looked into that yet.
The approach described above is called Rational Trigonometry; which also avoids transcendental functions like cos/sin, by replacing angles with "spreads" (equivalent to the sin^2 of an angle), which range from 0 = parallel to 1 = perpendicular.
Looking again at the inner-product 𝐚𝐛, there's another degree of freedom lurking in there if we interpret it as matrix multiplication 𝐚𝐛ᵀ (the rules of matrix multiplication require us to transpose the 1×n row-vector 𝐛 into the n×1 column-vector 𝐛ᵀ).
By default, this matrix formulation doesn't alter the inner product: it's still Σaₙbₙ. However, it gives us the flexibility to introduce an n×n matrix 𝐌 in-between the vectors: 𝐚𝐌𝐛ᵀ
If 𝐌 is the identity matrix [[1, 0], [0, 1]] (denoted 𝐈 in the article), then we again keep the original behaviour. In this sense, Euclidean geometry is characterised by 𝐈 (encoding its symmetric bilinear form).
If we use other n×n matrices we get different geometries. In particular, the matrix [[1, 0], [0, -1]] gives us the "red" inner-product a₁×b₁ - a₂×b₂; and [[0, 1], [1, 0]] gives us the "green" inner-product a₁×b₂ + a₂×b₁. These are closely related to each other (one is a rotation of the other; both are 2D analogues of special-relativity), and to the "blue" Euclidean geometry. This colour-coding come from Chromogeometry, which studies their relations.
These are explained more in An Introduction to Rational Trigonometry and Chromogeometry (which I just submitted at https://news.ycombinator.com/item?id=30418194 )