Geometric Algebra, Geometric Calculus

[kram1032]

It's some russian server, judging from the url.
Well, have fun working through that 

[kram1032]

another big summary of the topic:
Geometric Algebra: An Introduction with Applications in Euclidean and Conformal Geometry - Richard Alan Miller

[hermann]

Hallo Kram1032,

thanks for the link to the paper of Alan Miller on the introduction on Geometric Algebra.
An exellent introduction into the subject. Easy to read and to understand.

[kram1032]

I have a feeling that, as more and more people understand and use geometric algebra, its grand summaries will become more and more comprehensible, which is great

[Dinkydau]

Hallo Kram1032,

thanks for the link to the paper of Alan Miller on the introduction on Geometric Algebra.
An exellent introduction into the subject. Easy to read and to understand.

Yes, this doesn't look so difficult as all those other scientific documents. Maybe even I could understand some of it.

[kram1032]

Just in case you are not familiar with linear algebra: This is NOT a geometric algebra formulation of space. It's rather a 10-page crash-course-summary of the most important aspects of linear algebra. All that is done here can also be done in Geometric Algebra, though in many cases the Geometric Algebra formulation actually seems cleaner.

A Geometric Review of Linear Algebra

[hermann]

Nice paper, I hope it is also readable for people who are not familiar with this subject.

[kram1032]

http://ryushare.me/dietmar-hildenbrand-foundations-of-geometric-algebra-computing-geometry-and-computing-pdf/#.Uw6Xgfl5PpE

another Springer book - a bit annoying to download though. Enjoy/be quick

[jehovajah]

Thanks Kram1032.

This author or authors have read Grassmann I can tell from the first few opening remarks!

I think I will enjoy this approach because they understand the man and his background!

[kram1032]

Grassmann Algebra in Game Development (slides from a talk at GDC'14)

It's interesting in that most other sources present geometric algebra in a very different way.
Usually, you first see the introduction of the geometric product of two vectors and, from there, a definition of the scalar and the wedge product.
In this one, they start with the wedge product and its dual form, the anti wedge product, and arrive at all the other things from there.

It also doesn't even mention the geometric product. I assume it's probably much closer to the formalisms of Grassmann than the later developments by Hestenes. But I'm not versed in the historical details. Clearly, this forum already has an expert on that front.

[kram1032]

This very much isn't an introductory paper but it extends the ideas of conformal geometric algebra which has, as native objects, lines, planes, circles and spheres (for R³), to include quadratics or conics, e.g. arbitrary elipsoids, hyperboloids, paraboloids, cones and such things.
The idea is fairly simple, especially if you know a thing or two about conics/quadratics, but the presentation in this paper is, perhaps, a bit needlessly complex at times, at least imho.
Still, it's an interesting paper. With it you should be able to define "the" "elliptic"/"hyperbolic"/"parabolic" Mandelbrot set or a higher-dimensional analog.
(That higher-dimensional analog would most definitely just be a rotatory symmetric shape though. Although it might be a bit bendier than just a plain axial rotation, because the linear transformations in their conic Clifford Space do not represent linear transformations in Euklidean space. So things should *probably* distort a bit more...)

Anyway, here it is:
http://arxiv.org/pdf/1403.6665.pdf

[jehovajah]

Once again Kram1032 thanks!

Yes it is a succinct exposition of the main differences between Grassman ,Clifford and Hamilton algebras. In a way it shows the kind of universal thinking of the time, in that Helmholtz attempted to combine the electromagnetic formulae of Maxwell and the Stokes / Navier equations into one by simply introducing an index factor. In this case the index is the choice of the square of terms !

I have, until now, always found Clifford's product distinctions confusing, but my research into Grassmanns concept has opened a can of worms! Well, I like worms and spiders, as you may know!

I have quite a long way to go in fully understanding Hermanns thinking, but I am grateful for small mercies. The recent translation was very revealing to me, because Hegel was unknown to me up to that point!

Thanks again and thank you for your research in this thread!

[jehovajah]

I have digested the slides and recognise them as pure Grassmann, excepting only the cross product. The cross product, despite it's failings, is still widely used and widely understood, so it merits treatment, but as he clearly points out, it is not up to the job! The example of a signed triangle and it's normal orientation is a case in point.

A normal orientation to a plane is still an important line segment, it is just not necessary for rotation!

Every aspect of these slides is exposited in detail by Norman Wildberger in his "Wild linear algebr"a series, except the anti vector, I think. That is unless he treats it under the kernel and basis videos which I have not fully explored.

This is a great Overview of Grassmanns thinking and the thinking of some of his followers in terms of the lineal algebra. The Schwenkung algebra of Grassmann would be interesting to see, but I think this is entirely developed within the Clifford algebras.

[kram1032]

As said, the previously posted paper allows for natively defining all conics within the propsed Geometric Algebra.

This makes reflection along any of the figures rather straight forward:



For circles, this is the commonly known sphere inversion. For ellipses, it's exactly the same, but it preserves the stretching. So I can only assume that an M-Set based on that would simply be a stretched M-set.

For Hyperbolas it becomes a little more interesting with the relationship between a line and its reflection being a bit less obvious.

Though the weirdest, to me, is the reflection of a line in a parabola: Note how for any of those reflections other than for the parabola, all the line-reflections go through the origin.
This is because the origin's inverse is the point at infinity. And of course any line meets at the point at infinity, so the reflection of any line must go through the origin.

Since that is not the case for a parabola, I wonder, for one thing, what significance the origin even has, as far as reflections are concerned, and for another: If we were to use the generalized M-set formula for even just the 2D case on this, would anything interesting happen? How would it look like?

Remember: The generic formula for squaring a number in GA is:

r² := r e r

where r is a vector and e is a unit vector which gives the direction the M-Set should point to. (The positive real axis in the original M-set)

So if you were to do this for a generic parabola and some unit vector, we should get "the" "parabolic" M-Set.
And I'm not even asking about the 3D M-Set resulting from that. It would most likely, once again, simply be a rotation of the 2D-Set. Although the above image suggests that, perhaps, it might have some interesting different behavior.
And even if not, at very least the 2D-M-Set resulting from this should act at least somewhat differently.

[kram1032]

Here is another paper covering this "Quadric Geometric Algebra" which might be a bit easier to understand. It certainly uses more traditional notation.

http://www.gaalop.de/wp-content/uploads/134-1061-Zamora.pdf