[HN Gopher] Symplectic Geometry in 2D - Points, Lines, Circles
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Symplectic Geometry in 2D - Points, Lines, Circles
Author : fango
Score : 44 points
Date : 2022-02-21 13:54 UTC (9 hours ago)
(HTM) web link (www.researchgate.net)
(TXT) w3m dump (www.researchgate.net)
| chriswarbo wrote:
| > In symplectic geometry an area is the fundamental quantity,
| whereas Euclidean geometry measures lengths and angles.
|
| > 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 = a1xb1
| + a2xb2
|
| Where a = a1x + a2y and b = b1x + b2y. This is just the first
| components multiplied together, plus the second components
| multiplied together; and is easily generalised to N dimensions:
| ab = Sa[?]b[?]
|
| So far, so familiar. We tend to measure vectors using their
| length, which is the square-root of the vector's inner-product
| with itself, e.g. |a| = [?](aa)
|
| However, this is quite restrictive: the inner-product only
| requires + and x, which are well-defined for all sorts of fields
| (real numbers, complex numbers, finite fields, rational numbers,
| etc.); square-roots aren't so easy to define, which restricts
| Euclidean distance to only a few fields (e.g. real numbers and
| complex numbers).
|
| 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 ab (the rules of matrix multiplication require us
| to transpose the 1xn row-vector b into the nx1 column-vector b).
|
| By default, this matrix formulation doesn't alter the inner
| product: it's still Sa[?]b[?]. However, it gives us the
| flexibility to introduce an nxn matrix M in-between the vectors:
| aMb
|
| 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 nxn matrices we get different geometries. In
| particular, the matrix [[1, 0], [0, -1]] gives us the "red"
| inner-product a1xb1 - a2xb2; and [[0, 1], [1, 0]] gives us the
| "green" inner-product a1xb2 + a2xb1. 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 )
| Koshkin wrote:
| A very nice article.
|
| 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.
| q-big wrote:
| ... and don't forget that gyrovector spaces are the best
| mathematical framework for relativistic mechanics - much better
| than Lorentz transformations:
|
| https://en.wikipedia.org/wiki/Gyrovector_space
|
| Just mentioning ... ;-)
| jacobolus wrote:
| Gyrovectors are a generally poor representation for rotations
| compared to quaternions (or the like).
|
| 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.
| Qem wrote:
| Can you recommend good geometric algebra books starting at
| a elementary level, say, not assuming starting knowledge
| much beyond high school mathematics? I did some preliminary
| research, but I had the impression the intended audience
| for most books in this area is people that already master
| the conventional approach but are open to see the subject
| under a new light, so a lot of previous knowledge is
| assumed.
| jacobolus wrote:
| What I have repeatedly found with GA is that I can solve
| some problem I have using some other brute-forceish tools
| with a few pages of tricky error-prone scratch-work that
| balloons out to a complicated mess before simplifying
| back down at the end, and then afterward think about it a
| bit and come up with 2-6 lines of simple GA identities
| showing the same thing in a much higher-level coordinate-
| free way, and with most of the steps geometrically
| interpretable, rather than just opaque calculation. But
| coming up with the simple version at the beginning is
| hard.
|
| The tricky part about it is that there are a _lot_ of
| useful identities that can be written down, and properly
| learning a decent number of them and figuring out which
| ones to apply in which situation takes probably years
| practice, ideally with some guidance /support from
| someone who knows more than you. (I do not feel like I
| have mastered the subject.) The same thing happens using
| whatever other formalism, with the difference that many
| identities that are pretty short to write down in GA are
| much more complicated to write down, so people don't even
| try to use them.
|
| I'm not sure if there's really a good beginner source,
| but I haven't ever really sat down and tried to go
| comprehensively through the exercises in any books
| pitched at a relatively elementary level. You could try
| Alan MacDonald's book _Linear and Geometric Algebra_
| which is designed as an intro undergraduate textbook. If
| you want to also learn some mechanics, you could try
| Hestenes's book _New Foundations for Classical
| Mechanics_.
| blablabla123 wrote:
| Not sure, vector calculus isn't very intuitive either if you
| start with it, with Pauli/gamma matrices it's even worse.
| Having studied Physics myself, I haven't encountered one
| lecture where they were able to give a reasonable geometric
| explanation. (Symplectic Geometry and GA provide it) IMHO if
| the the same amount of effort was used to force vector calculus
| into people's heads, it should be doable with these tools as
| well. Unfortunately there is already a lack of books about the
| topics
| Koshkin wrote:
| > _vector calculus isn 't very intuitive either_
|
| 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
| [deleted]
| ogogmad wrote:
| I've read a bit of this textbook on projective geometry:
| https://www.amazon.co.uk/Perspectives-Projective-Geometry-Th...
|
| 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
| ReleaseCandidat wrote:
| Interesting definition of 'Euclidian geometry'. I am used to call
| 'something' Euclidean, if the parallel postulate of Euclid holds
| (that is, there exists exactly one parallel line to a given line
| through a point (which does not lie on the given line)).
| pfortuny wrote:
| I guess he isnusing it in the metric sense (Euclidean metric).
| Koshkin wrote:
| To be fair, the term they are using is "Euclidean structure,"
| which, strictly speaking, is more algebraic than (purely)
| geometric. The notion of the "angle" (and from there that of
| parallelism) logically follows from it.
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