## General Multidimensional Rotations

Higher-dimensional geometry (previously "Polyshapes").

### General Multidimensional Rotations

Hi !. I'm new in this site, in first place, excuse me by my basic english

Actually i'm working on my thesis work in the university, about computer graphics and the title is the same of this post.

Ok, the intention of my message is to make you know what i traying to do in my thesis and i hope you can tell me some recomendations or suggestions.

Specially i hope, you can tell me any direct aplication where n-dimensional rotations can be used.

Thanks a lot.

If you understand spanish, here is the Formal Proposal of my thesis.

And here, i'll write a small abstract:

Problem definition
There isn’t a generalization of the transformation equations that permit rotating n-dimensional objects, not only around the principal hyperplanes, but around a arbitrary hyperplane.

If we want to visualize the phenomena in nD (n>3), it’s necessary also, to generalize the projections equations (parallels and perspectives) that permit carrying the information in nD to 2D, by means of successive projections, in order to visualize it on a computer monitor.

The wire model of polytopes can be drawn from their vertexes, and we can know how to obtain that information, though, the order in which the vertexes should be drawn, is a process still not automatized, for example, to draw a 4D hypercube, we need to draw its 32 edges, the question is: In what order? This is another point to analyze, because it’ll be useful for automatically generating and drawing nD polytopes and placing them in the hyperspace.

General objective
The main goal of this research is to find the principal rotations in nD, to simulate the rotations of nD polytops around a general hyperplane, and finally show the result through projections in a computer monitor.

Specific objectives
• Definition of the projection equations to 2D, which permit visualizing the rotations from different points in the 4D space.
• Analysis and study nD,(n-1)D, …. 3D, 2D projections.
• Generalization of rotations and projections for nD polytopes.
• Definition of the general matrixes, called GO and BACK (necessary matrixes to carry an nD object to the origin, make the rotations, and take back the object to its original position.)
• Automatic generation of coordinates and draw of edges for nD regular polytops.
• Development of a multidimensional navigator.
Last edited by irving on Sat Nov 18, 2006 11:54 pm, edited 1 time in total.

<b><i>You're not thinking 4th dimensionally !</b>

irving
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The phase space of four dimensional rotations is a seven dimensional object hight "biglomohedic pyramid". The actual angular displacements by rotations forms an eight-dimensional object, which i have not fully explored.

In five dimensions, the space of great arrows is ten-dimensional, but the composition of higher spaces is yet unknown, and prehaps 15 dimensions might be the final yield.

One might search for groups like SO4 or SO5 (special orthogonal groups), because these represent the great arrows (great circle + direction), in 4D and 5D. But this represents a sphere rotating around a fixed n-2 space, in each direction.

In four and higher dimensions, one might suppose that there is rotation in this axis too, so SO4 and SO5 are not the complete group. One adds further, double rotations, of fixed or free rotations.

In four dimensions, one might, when both rotations are the same, lead to clifford parallels (and rotation). These exist for 'left' and 'right' handed form, and each corresponds to a set of points on the surface of a 3-sphere. All 4d rotations are formed by a pair of these together, so one might note that this is where the seven-dimensional figure comes from. Pairing off opposite points, and replacing these with a circle, one gets 8d.

In five dimensions, one might have for any axis, a L-clifford-rotation. This corresponds to an eight-dimensional figure in the first instance. But the sum of great arrows is a ten dimensional figure, and one supposes that by adding three (for the right), and subtracting one (for the variations of possibility), one has ten dimensions.

Rotations in higher dimensions map onto things like sine-waves (consider the complex equation x = y + z . cis(v.t) as used in physics &c, and Lojase(?) figures, which are of the form x = a.cis(v1.t+c1), y = b.cis(v2.t+c2). This is much used in electromagnetism.

Four dimensional rotations can be implemented by way of things called 'quarterions', and in Eight, by 'octonions', but i don't know too much about these.

Wendy
The dream you dream alone is only a dream
the dream we dream together is reality.

wendy
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### Re: General Multidimensional Rotations

irving wrote:Problem definition
There isn’t a generalization of the transformation equations that permit rotating n-dimensional objects, not only around the principal hyperplanes, but around a arbitrary hyperplane.
General objective
The main goal of this research is to find the principal rotations in nD, to simulate the rotations of nD polytops around a general hyperplane

Thats not really a problem. The rotation in an arbitrary plane in n-dim space can performed like I described it already here. If you wish you can write it down in matrix notation but the description is already suitable for programming (and indeed I used it already in my 4d building blocks game).
bo198214
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### Re: General Multidimensional Rotations

irving wrote:to simulate the rotations of nD polytops around a general hyperplane

In addition to what bo said, it should also be noted that an n-dimensional object can be rotated simultaneously parallel up to floor(n/2) orthogonal planes.

Another point, more interesting research, in my opinion, would be working on ways to visualize the (n-1)D facets.... wireframes are somewhat unsatisfying. I'm thinking at the moment, that it might be nice if you took a standard wireframe representation, and then while it is rotating, cycle through the facets one at a time, shading in the facet with a semi-transparent color. So, you'd end up with frames like this...
pat
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wendy:
Great information, i will read more about that you wrote.

bo198214:
Ok, I see that this kind of work has been made, and maybe it could be realtively easy to implement, however I think, this generalization should be probed mathematically, and that's another problem that i have to attack ... or is a work about it exist ?

pat:
That you said is a good idea, i will take into acount.

Irving

<b><i>You're not thinking 4th dimensionally !</b>

irving
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You may be interested in the following formulation of mine that describes the rotation of a vector in any arbitrary 2-plane (specified as two orthogonal basis unit vectors) in R<sup>n</sup>, using only vector operations (no Cartesian coordinates needed):

http://eusebeia.dyndns.org/~hsteoh/4d/genrot.pdf
quickfur
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quickfur wrote:You may be interested in the following formulation of mine that describes the rotation of a vector in any arbitrary 2-plane (specified as two orthogonal basis unit vectors) in R<sup>n</sup>, using only vector operations (no Cartesian coordinates needed):

http://eusebeia.dyndns.org/~hsteoh/4d/genrot.pdf

Very interesting paper . ..

Thanks.

<b><i>You're not thinking 4th dimensionally !</b>

irving
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