enkimute

6 ms·

1.
by enkimute 2y ago
fixed it. mea culpa.
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by enkimute 3y ago
Thank you, appreciate that!
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by enkimute 3y ago
Apologies. Had that joke sitting around for waaaay to long. Not that great in retrospect :D
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by enkimute 3y ago
don't foget the fantastic Sudgy 'A swift introduction to projective geometric algebra' : https://www.youtube.com/watch?v=0i3ocLhbxJ4 and ofcourse the go-to reference https://bivector.net or join 1
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by enkimute 3y ago
For specifying animations, you should work in the quaternion lie algebra, not in the group as you suggest. There you can represent 1620 degrees without any problem. Furthermore, in the quaternion Lie algebra (pure imaginary quaternions), an
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by enkimute 3y ago
done. mea maxima culpa.
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by enkimute 4y ago
There's no smoke. But there surely are mirrors. (Group theory joke)
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by enkimute 4y ago
These are also called the planar quaternions and are the even subalgebra of 2D PGA. I use and explain these (and their nD generalisations) in my SIBGRAPI 2021 talk on kinematics and dynamics in PGA. https://youtu.be/pq9YfdPH
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by enkimute 5y ago
Not the same concrete example, but one where I do find the Geometric Algebra version substantially more insightful, is the treatment of rigid body mechanics in the geometric algebra of the Euclidean group (R_{n,0,1}). It has the dual quater
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by enkimute 5y ago
Every Lie algebra is a bivector algebra (see 'Lie groups as Spin groups' http://geocalc.clas.asu.edu/pdf/LGasSG.pdf ). Additionally the GA formalism enables closed form solutions for the exponential map for al
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by enkimute 6y ago
They were designed with touchscreens in mind - I agree tho that it didn't work out exactly as I wanted. Someone did make a playthrough video of it : https://www.youtube.com/watch?v=UzEytSmwEpk&ab_channel=probi...
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by enkimute 6y ago
Thanks for this background info. I also agree with your assessment - so far its just gotten some giggles and maybe even some extra attention from students who are not always easy to motivate when it comes to math. Its also racked up quite a
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by enkimute 6y ago
> You're generating code that relates to a certain algebra, likely for providing API for doing calculations based on that algebra (so an algebra library). Why would you call that 'algebra generated by ganja.js'? The '
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by enkimute 6y ago
A bias towards 'vectors must be points' crops in as an unwritten legacy axiom, leading to a broken correspondence with Group theory (which again cannot be turned around, discrete symmetries compose into continuous ones, but not th
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by enkimute 6y ago
The grade 1 element is not the normal vector of the plane, it is the plane itself (as a homogeneous linear form, it includes the distance from the origin as an extra coefficient, which the normal vector does not). Linear equations form
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by enkimute 6y ago
Chris Doran, while not officially in academia, is still very much actively researching these topics and regularly publishes. (he is also a member of the bivector community, very approachable and happy to help out with questions). Anthony La
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by enkimute 6y ago
close? https://enkimute.github.io/ganja.js/examples/coffeeshop.html
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by enkimute 6y ago
Geometric Algebra for Physicists : https://www.amazon.com/Geometric-Algebra-Physicists-Chris-Do...
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by enkimute 6y ago
Absolutely : Algebraically we can always use the Hodge dual and its inverse to move to/from k-vectors from/to n-k vectors. Geometrically we can always say two points define a line or two lines define a point. Group Theor
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by enkimute 6y ago
That is correct - matrices are the natural companion of the general linear group. The GA representation exists, and is called 'the mother algebra', for general 4x4 matrices the GA equivalent is R4,4. (where general linear transfor
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by enkimute 6y ago
> But to really make it popular and understandable, a "simple" version of GA that handles translation, rotation & non-uniform scaling would really help, without the group theory concepts, even better, make it in the context
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by enkimute 6y ago
Take a look at Klein : https://www.jeremyong.com/klein/ (arm implementation is coming) Its geometric product between motors is the equivalent of the 4x4 matrix product. Although the basic computational complexity tells
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by enkimute 6y ago
a selection: - no closed form exponential/logarithm (i.e. cant interpolate) - no numerical stability for subgroups (i.e. need Gramm-Shmidt, SVD) - no covariant transformations (i.e. need adjugate to transform axial vectors) - no geomet
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by enkimute 6y ago
I fully agree, for practical examples (no-install, web-browsable, interactive), check out the dozens of examples in the ganja.js coffeeshop: https://enkimute.github.io/ganja.js/examples/coffeeshop.html
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by enkimute 6y ago
Ok, standard 4x4 matrices also implement a projective (aka d+1) model. (the 'w' coordinate is just the projective coordinate). So no difference with GA in that respect. Setting up a unified transformation hierarchy is actually ver
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Dual Quaternions Demystified. (GAME2020 PGA:The Algebra of the Euclidean Group)(youtube.com)
2 points by enkimute 6y ago|0 comments
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by enkimute 6y ago
The GAME2020 event (Geometric Algebra Mini Event) showcased applications in graphics, physics, robotics, neural networks, mathematics. The lectures (from Cambridge, UVA, UPEM, TU Delft, ..) are all available here : https://www.yo
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Prof. Anthony Lasenby (Cambridge) Geometric Algebra as New Language for Physics(youtube.com)
4 points by enkimute 6y ago|0 comments
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by enkimute 6y ago
This is one of the ideas presented in this Chapter. (i.e. to identify the geometry with the invariant of the transformation). In 3D if x represents a reflection, it also represents the plane left invariant by that reflection. If x represent
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A Guided Tour of the Plane-Based Geometric Algebra PGA(bivector.net)
45 points by enkimute 6y ago|4 comments

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