modified 1998 Jan 04.
Thoughts on dimensions, dimensionality, ... Some of my thoughts, and a lot of other people's thoughts.
David Cary also maintains related pages:
see also "thought-space"
The geek code http://www.geekcode.com/ uses over 32 dimensions (!) is one of the more humorous ways of categorizing humans using many more than 3 axis (dimensions). [FIXME] A more serious method uses only 4 dimensions.
_Cyberspace: First Steps_ by Benedikt "a good article on N-Dimensional visualization" -- recc. Jerry Isdale/Sysop, firstname.lastname@example.org
"Random Versus Rational Which Is Better for General Compound Screening? " http://www.netsci.org/Science/Screening/feature09.html by S. Stanley Young, Mark Farmen, Andrew Rusinko III asks "What is the inherent dimensionality of chemical space pertinent to drug design?" and gives an estimate of "11 to 18 dimensions". "It would be very useful to have some small number of variables that are pertinent to biological activity. Computational chemists can search for the magic few dimensions or analysis of data might indicate the important few dimensions for a particular type of biological activity. The fact that high dimensional spaces are so large is often termed "the curse of high dimensions." In the drug design situation it means that it is unlikely that a compound good enough to be considered an optimized drug is to be found by screening 10,000 or even 100,000 compounds. Experience supports this assertion." "The many facets of a molecular structure that are pertinent to drug action can only be described in high-dimensional space. Reduction of such space using multi- dimensional scaling ... or principal components ... have been used to simplify the problem of selecting a "diverse" sets of compounds. These methods project points from a high dimensional space into a lower dimensional space, but are most likely an over-simplification."
Date: Mon, 29 Jan 1996 09:18:52 -0700 To: miiko sokka <miiko.sokka at pp.inet.fi> From: cary at agora.rdrop.com (David Cary) Subject: Re: 1-2-3-??? dimension? Cc: Bcc: X-Attachments:
I'm also interested in dimensions other than 3.
There are some digital signal processing (DPS) theorems that are most easily explained in N-dimensional space (where N is the number of basis functions, or the number of data samples).
In particular, the sphere-packing probem, where you want to pack the most "spheres" as possible in a fixed volume, corresponds exactly to finding the best signal/power ratio given a fixed noise energy. We usually use x1 to indicate the distance in the first dimension, .... x25 to indicate the distance in the orthogonal 25th dimension ... xN to indicate the distance in the Nth direction.
This uses "plain" Euclidean geometry, where the distance between 2 points is
d = sqrt( x1^2 + x2^2 + x3^2 + .... + x25^2 + ... + xN^2).
Special Relativity (I don't know about General Relativity) is most easily
explained using 4 dimensions, x,y,z,t,
where t is the "special" fourth dimension known as time.
The "proper spacial distance" between 2 events is
d = sqrt( x^2 + y^2 + z^2 - t^2).
The "proper time" between 2 events is
d = sqrt( t^2 - (x^2 + y^2 + z^2) ).
Fractal theory is concerned with non-integral dimension. For example, in fractal geometry, it makes sense to say Britain's coastline is "1.2 dimensional". Most of the fractals people talk about are embedded in 2D space and have dimensions between 0 and 2, merely because these are the easiest to draw on 2-dimensional computer screens and print out on 2-dimensional paper. There's quite a few people working on fractals embedded in 3D space that have dimension slightly higher than 2.00. However, I know one person who is working on visualizing a fractal that is embedded in 4 dimensions (I forget the exact dimensionality, but it's obviously less than 4.00). I don't know any other work on fractals with dimension greater than 3, but that's more than likely merely due to difficulty in displaying it rather than any mathematical problem.
miiko sokka <miiko.sokka at pp.inet.fi> wrote: +I would like to know what different dimensions there are according to +theories. I know something about fibretheory (is it correct?) where is +said to exist at least 10 dimensions. + +Yes, and I want also to know what theories there are. + +And what theory says that time is the fourth dimension? What are the +opinions of modern day physicists? How many agrees with who? + +What you say about imaginary dimension? + +Greetings for all, Miiko J. Sokka (posted and emailed)
Newsgroups: comp.graphics.algorithms From: email@example.com (Jim Paris) Subject: Re: 3D shadow of 4D object ? Sender: firstname.lastname@example.org (news) Organization: The KD3BJ Usenet BBS Date: Tue, 1 Oct 1996 01:28:37 GMT Lines: 26 MORTALIS <email@example.com> writes: >> It is an object that extends into the fourth dimension. > >duh! I was wondering what the 4 dimension was? Is it just an >attribute(that could be anything) or what? Sorry, I was in a bad mood. When he refers to a 4D object, he's referring to an object that extends into the fourth spacial dimension. It's very hard to imagine; only if you can look at the 3D axes (X, Y, and Z) and imagine a line that is perpindicular to every single one of them can you imagine the fourth dimension. It's usually referred to as W, ie, (X,Y,Z,W). The original poster was interested in 4D objects and one of the best ways to study them is to project them into 3D, just like a 3D object can be projected into the 2D computer screen. Usually, the 4D object is projected to 3D, then 2D, for display on a computer screen, which _really_ causes a loss of data. One program I've seen, called HyperGeo, actually let you use 3D glasses so you were only projecting once - it was very effective. -blackbob [firstname.lastname@example.org] -http://www.jtan.com/~jim/
From: email@example.com (Terence Tao) Newsgroups: sci.physics.relativity Subject: Re: The Basis of Special Relativity Date: 3 Mar 1997 19:45:56 GMT Organization: UCLA Mathematics Department Lines: 30 ... Daryl McCullough <firstname.lastname@example.org> writes: > >> 2. Describe what happens within a posited physical space. This is >> the objective approach. > >It is also the approach of special relativity. True, but I think you and Henry are talking at cross purposes as to the meaning of "space". Henry seems to think that the only space in which physics can take place is in the familiar three-dimensional space that we experience at every moment of our existence. That, though, is like Plato's prisoners thinking that the only space which is truly physical is the two-dimensional cave wall on which they see their shadows. Usually in physics the "space" that things live in is quite different from R^3. In Hamiltonian mechanics phase space is used, which is usually 6N dimensional (N = number of particles). In relativity four-dimensional spacetime is the best stage in which to set the theory. In quantum mechanics a Hilbert space is used. In string theory a high-dimensional spacetime manifold is the setting. Nobody really knows what the proper "space" is for quantum gravity yet. And so forth. We see only three dimensions at a time, but the space that things _really_ live in could be much, much bigger. Terry
J.B.S.Haldane: _On Being the Right Size_ (Oxford University Press, London). DAV: if this is the book I think it is .... very good on scaling laws.
see also computer_graphics_tools.html#fractals
Date: Wed, 21 Feb 1996 00:48:14 -0700 To: Sandra Hawkes <hawkes at unixg.ubc.ca> From: cary at agora.rdrop.com (David Cary) Subject: dimensions Well, <blush> I haven't put on any exhibits; only shown my art to a few friends. The images I create are abstract fractals, not in any way representative of real objects. They're just patterns that "look cool" (or in the words of my friends, "They're eye-candy"). I really enjoyed _The Algorithmic Beauty of Plants_ by Prusinkeiwicz and Lindenmayer. I usually skip the math, look at the extremely pretty color pictures. _The Fractal Geometry of Nature_ by Benoit Mandelbrot is the book that really popularized "fractals", objects with fractional dimensions -- like trees, coastlines, clouds. (I read this years ago, and I've forgotten if it was difficult/easy to read). David Wright has put some extremely abstract fractals on his page http://www.math.okstate.edu/~wrightd/limitsets/limitsets.html (I don't understand any of the math, I just look at the pretty pictures). I've actually met David Wright, and he actually studied under Benoit Mandelbrot. Web browsers are getting really easy to use once they're installed -- just point and click. But sometimes it takes a *long* time to get them installed :-( There are some aspects of fractal geometry that are easy to explain to a 6-year-old, but there's also some hairy math that is still a little beyond my grasp. It's a lot like farming. A child can understand harvesting the wheat, but if you yank the side off the harvester it would be difficult for him to understand all the little gears and gizmos that pull the wheat through the machine. Then there's the concept of wheat hybrids and the gene pool which ... I really don't understand. Hm. "dimensions for children". I'll have to think about that one. There's a book called _Flatland_, by Abbot, I think... it's a fairy tale that explains "1 dimensional", "2 dimensional", etc. Were you able to access the alt.dreams.lucid alt.sci.time-travel newsgroups ? >From: Sandra Hawkes <hawkes at unixg.ubc.ca> >To: David Cary <cary at agora.rdrop.com> ... >I've cut back on my screen time. I just can't sit and read for long. >Can you recommend a easy to understand book? Maybe even a book on how >the dimensions function written for children? That would be most >helpful. I have serious problems understanding physics, even basic >math. I've read children's books on other adult topics and it worked for >me. ... >The stuff about your art sounds really interesting. Had any exhibits? >An exhibit catalogue perhaps that I could look at? I have text only >capabilities in how I read the newsgroups. I'm not complaining. Am >grateful to be able to read them. And I really don't want to attempt to >upgrade my computer abilties. Gotta keep things really simple.
From: email@example.com Newsgroups: sci.physics Subject: REQ: Hyperspace type pages Date: Mon, 16 Sep 1996 20:05:12 GMT I am interested in visiting URLs concerning theoretical physics, specifically centering on the ideas of high dimensions and parallel universes, like those discussed in Michio Kaku's Hyperspace. If you have any such URLs, can you please email me. Usenet is so informal. Thank you for your time
Newsgroups: comp.graphics.algorithms From: firstname.lastname@example.org (Jim Paris) Subject: Re: 3D shadow of 4D object ? Organization: The KD3BJ Usenet BBS Date: Tue, 1 Oct 1996 00:10:21 GMT Lines: 16 Yamaha / XYZZ <scriven@CS.ColoState.edu> writes: >If you want to know more about four physical dimensions, try reading >"The Boy Who Reversed Himself" by William Sleator, or "The Fourth >Dimension" (author unknown), or an older book called (I think) >"FlatLand". The first two are about four physical dimensions in >relation to our three. FlatLand is about three dimensions in relation >to a two-dimensional creature. Or try "Hyperspace". It's a very good book on the subject. The sept/oct issue of Quantum also has some information on 4D space.. -blackbob [email@example.com] -http://www.kd3bj.ampr.org/~jim/
From: Debbie Foch/LIBRARIAN, 74354,1675 To: John Ratcliffe, 70253,3237 Topic: Fun With Tony Msg #2761, reply to #2757 Section: HOT! Roswell Video  Forum: UFO Date: Sat, 1995 Jul 29, 21:56:07
I agree with your comments to Pete. The problem is that I don't think we or scientists in general understand the true nature of the universe and what it means to be hyper or multidimensional. Existence or travel outside of space-time is of course quite foreign to us; however, that doesn't mean it's not possible. Use of non-linear types of communication, outside of space and time, is also foreign to most of us (at least consciously), but is none-the-less valid.
Take care, Debbie
From: "Timothy J. Ebben" <firstname.lastname@example.org> Newsgroups: sci.physics Subject: Re: EPR Solution 4D-Space! Date: 18 Jan 1997 15:49:37 GMT Organization: Ascension Enterprises, Inc. Lines: 165 X-Newsreader: Microsoft Internet News 4.70.1160 ... George Penney <email@example.com> wrote ... > A Solution to the Einstien-Podolsky Paradox. This > is a solution based on 4D Space.If a Flatlander lived in his 2D universe > which consisted of a flat plane and we intersected it with a circular ring > which say was spring loaded so we > could make it bigger or smaller at will.Now if we laid this rind flat on > his plane,he would see a circle,if we now lift the ring out of his plane so > it is inclined 90 degrees to his plane he would see two points (or > dots),seperated by the dia of the ring > ..We then rotate the ring(say clockwise),perpendicular to the plane,he would > observe them moving in unsion in his space,but not conected.He would > conclude they obayed some law(such as a force between them that made them > move in unsion).He then positons him > self on one of the dots(lets say one is blue and the other red,he's on the > blue dot).He can't visalize why they move together as he is in 2D-space.We > then make the dia of the ring larger so the dots are further apart,he still > can see both and that they ar > e moving in unsion.Everytime the blue dot moves so dose the red dot.Then we > slowly increase the dia so it's dia is 6x10^8M in dia(The speed of light in > his universe is 3x10^8M/S,also we can rotate the ring as slowly as we > like). We now reverse the rotatio > n of the ring CCW and although he cant observe both the blue dot (which he > is on),and the red dot simaltanesly he just knows that the red dot has also > reversed it's direction.But he reasons how can this be since information > can't travel between them at gr > eater than the speed of light?? Note now that the flatlander now has a > paradox the same as the Einstien-Podolsky paradox.Two pairs are > communicating information(as QM would predict),but faster than the speed of > light)??.Of course what he dont know is that > the pairs are connected in 3D-Space.If he did he can conclude that > Reletivity and Quantum Mech are not in violation of each other!!!.Thus we > have a resolution of this paradox.The same would apply if we had this > paradox in our 3D-Space and concluded that > the pairs are connected in 4D Space. > In working out this solution I also noticed a peculiar property of > partical spin in 4 or higher dimensional spaces.It goes as follows:--- > First lets discuss some aspects > of an n-dimensional object intersecting an (N-1)-dimensional space. I'll do > this by going back to the flatlander and our 3-space.In the flatlander's > universe his circle is the same to him as our sphere(Keep this in mind) in > that he can't enter his circle > without breaking it's circumference(lets assume his circle is not solid > inside like we would have if we shaded the circle inside).To us in 3-space > we could step inside his circle without breaking the circumference due to > the fact that we have access to 1 > more dimension than he has.His circle is the same as our 3D hollow > sphere,we woudn't be able to enter our sphere without breaking it's surface > but it could easily be done from 4D-space because in 4-space our 3D-sphere > would be equivalent to their 4D circ > le.Now if we inter- sect the 2d circle with out sphere perpendicular to and > in the center of his circle passing the sphere down through the plane of > the circle,if he were inside he would see a dot that would become a small > circle that would get bigger in > dia as we continued to pass the sphere through the 2D circle--- (let the > dia of OUR sphere = the dia of his circle),midway through our sphere would > form a concetric circle with his own,then start to decrease in size back to > a point and finaly dissapear c > ompletly!!.This would seem very odd to him as all he is observing are cross > sectional pieces of our sphere at any one instant in time.It would be the > same as if we saw an object suddenly appear in our 3-Space,continualy > change shape and then dissapear.We > can get even stranger effects if we intersect irregular shaped geometric > objects from higher spaces into lower spaces.Of course the flatlander can't > visualize our sphere due to his restriction of being confined to his 2D > universe,however he can construct > the laws of 3D-Space Geometry based on the cross sections that he seen of > our 3D objects Similerly we can construct the laws of 4D-Space and > N-Dimensional Spaces and their Geometries. > Keepin > g this in mind let's get to rotation or spin properties of 2D space and 3D > space,then apply this to higher spaces.If we rotate the Flander's circle or > if HE rotates what he considers to be his SPHERE it can only rotate in TWO > directions(CC or CCW) or if y > ou like it can rotate one way or the other[for the terms CC & CCW can > interchange depending on where you view the rotation from in space].Now if > we in 3-Space take the circle(----- his sphere)and rotate it down- > ward(around it's dia) into the plane of his > 2-Space and perpendicular to it we can rotate it in two more directions in > our 3-space!!He would observe two points seperated by the dia of his sphere > that would be stationary.Again he would not be able to visualize that the > circle(his sphere) had TWO mo > re modes of rotation or spin in 3D-Space!From this it follows that a sphere > in our 3D-Space which has only 2 directions of spin would have MORE than 2 > directions of spin in 4Space and even more in higher dimensional > spaces.Like the Flander this goes again > st our common sense(common sense being that layer of predudice layed down > prior to the age of 16---I could'nt resist getting that in).Also we would > not be able to mentaly visualize this.(unless we were Charles Hinton who > claimed he could visualize 4D obj > ects such as Hypercubes and so on). > > George Penney > > > >
one particular theory with more than 4 physical dimensions
From: Mitchell Porter <firstname.lastname@example.org> Subject: >H designer universes, spin networks, and string theory Transhuman Mailing List [Eugene Leitl] > if we can tweak our spacetime to cause > precipitation of an edge-of-chaos high-dimensional CA universe and > stabilize the access channel long enough to bootstrap, then we can > fabricate our own Eden. I've thought a bit about what this would require. Assume that you're doing this by causing inflation to occur in a very small region of space, where (somehow) you've controlled the state of the Higgs and other fundamental fields so as to determine the effective physics of the new universe. Since we're used to thinking in terms of atoms - relatively pointlike nuclei surrounded by fuzzed-out electrons - let's suppose that the quanta of the new universe will form atom-like structures, although the electron orbitals will be hyperspheres rather than 3D. (I'll keep speaking of electrons, nuclei and atoms, although it should be understood that these are particles in the new universe which behave somewhat like those in existing matter.) Hyper-atoms might form as they did in our universe, either shortly after inflation, or in stars. Either way, what you will want to do is to tweak their properties so that they have tendencies to form a very regular crystalline lattice (of d dimensions). The hyper-atoms in this crystal then form the cells of your cellular automaton, and the spin states of some electrons can form the cell states. The most difficult thing is to keep the "access channel" (which is, I presume, an umbilical wormhole connecting the baby universe to ours) open, since the baby universe is, by hypothesis, of a different effective dimensionality. I guess it just takes more of that "exotic matter" which you build wormhole rigging out of. As you pass down the wormhole throat, some of the space dimensions which are Planck-size in our universe would acquire a wider and wider radius, until they were on an equal footing with our three large dimensions. What sort of intermediate structures this would entail, I have no idea yet. There's actually a preprint out there on d-dimensional atoms: The Periodic Table in Flatland It focuses on the two-dimensional case. I think there's also something in the appendices to A.K. Dewdney's _Planivese_. [text lost, but Eugene was saying that in some current string theories space has "27 (?) dimensions"] > dimension, but most are curled up real small so that we can't see > them at our scale. The original string theory (before supersymmetry) had 26 dimensions - 25 space, 1 time. It's not really current (post-1984 superstring theory has ten spacetime dimensions, and there are some lower-dimensional models), although it's been suggested that you can get superstrings out of 26-dimensional string theory. [Anders Sandberg] > On Wed, 17 Apr 1996, Eugene Leitl wrote: > > If fact, if we can tweak our spacetime to cause > > precipitation of an edge-of-chaos high-dimensional CA universe and > > stabilize the access channel long enough to bootstrap, then we can > > fabricate our own Eden. > > I had a similar idea for how life near the Omega Point could create a > state in a Penrose spin lattice to evolve towards the Omega Point, even > if the time direction would become ambigious. I'd like to hear more about this. [Eugene Leitl] > Can one understand Penrose spin lattice, I mean is it physics fit for a > mere human? Have you a ref handy? week 46 week 55 abstract gr-qc/9505006 The first two should be expository posts by John Baez from sci.physics.research, touching on spin networks, while the latter is a paper in which gravity theorists Carlo Rovelli and Lee Smolin actually calculate something that might be real using spin networks. [Anders Sandberg] > I think Kaufmann wrote about them in a book about knot theory, they > didn't look that absurd to me. I'm not sure it is physics yet, and most > people seems to have moved to string theory instead (a pity, I really > like the idea of the universe as a kind of random graph with a cellular > automaton rule). That would be in _Knots and Physics_ (1993). The paper by R&S, cited above, might be the first time that an actual physical quantity was computed using spin networks. (The quantity in question was the quantum of surface area, in their theory of quantum gravity.) [Eugene Leitl] > > > Can one understand Penrose spin lattice, I mean is it physics fit for a > > > mere human? Have you a ref handy? > > > > Well, theoretical physicists seem to handle them, but if they are human > > is still an open question... :-) > > I also, wonder. I once bought a string theory book from Kaku, just for > the fun of it -- I didn't really expected to understand anything. But > I didn't understand not a single line of it: not contents, of course > not the chapters, nothing. Only the conclusion, because there was no tech > speak in it. This was a really scary experience, this book could > have been written by an alien. I wish I could do physics properly, <sigh>. Is this _Introduction to Superstrings_? I tried to teach myself string theory from that. I didn't quite succeed, but I learnt a lot. You'd need some idea of quantum field theory to start with though (recomendation: Feynman's _QED_). > String theory is not very aesthetical, > unless one happens to be a musician, may be ;) Or a mathematician... The sum over topologies, and the way that strings in 10 dimensions can be described as 10 fields in 2 dimensions, are the two ideas which stick in my mind as elegant. One of the hot topics right now is membrane theory (or, as people insist on calling them, p-branes, p-dimensional membranes) - a membrane being an entity of 2 or more dimensions (point particles are zero-branes, strings are one-branes). Some people think that the superstring theories are all approximations to an 11-D membrane theory, dubbed "M theory". -mitch http://desire.apana.org.au/~qix
see also computer_graphics_tools.html#writing
"Quaternions are not a well known mathematical concept; but ... are the primary choice for representing spacecraft orientation. " "the NPS aeronautical model uses quaternions." -- unknown
the Quaternians can be defined without too much trouble. There are three Quaternians which, together with the Real numbers, generate all of the others. They are given the names i, j, and k. A typical example of a Quaternian is 1 + 3.3i + 9j - 1.5k.
Addition is defined as you would define addition on a polynomial with variables i, j, and k. For example, the sum of 1 + 5j and 2 + 4j + 13k is 3 + 9j + 13k.
Multiplication is also defined just like multiplication on polynomials except that we are allowed to make certain simplifications: i^2 = j^2 = k^2 = ijk = -1. We can derive other relations, such as i = jk, from the ones given above. Thus (1 + 5j)(2 + 4j + 13k) equals (2 + 4j + 13k) + (10j + 20j^2 + 65jk) = -18 + 65i + 10j + 13k.
Newsgroups: comp.graphics.algorithms From: email@example.com (Wayne Cochran - CS) Subject: Re: 4d line Sender: firstname.lastname@example.org (Wayne Cochran - CS) Date: Wed, 19 Mar 1997 21:44:45 GMT Organization: School of EECS, Washington State University In article <E79HwB.xx.B.email@example.com>, firstname.lastname@example.org (D J Hampson) writes: |> > It turns out we that can represent vectors as "points at infinity" = [x y z 0] as well |> > using this model. Also, perspective transformations (which are non-linear in 3D) |> > can be represented with a 4D transformation. |> |> Perspective transformations in 4D sounds quite useful - could you say |> some more about this? |> |> -- |> David Hampson |> e-mail : email@example.com |> Uni Page : http://www.bath.ac.uk/~ma6djh/ If d is the distance of the projection plane to the eye, then here is one of several possible perspective transformations: | 1 0 0 0 | | x | | x | | x*d/z | | 0 1 0 0 | | y | = | y | => | y*d/z | | 0 0 0 1 | | z | | 1 | | d/z | | 0 0 1/d 0 | | 1 | | z/d | | 1 | The resulting homogeneous 4D point contains the info for the perspective division that causes the perspective warp. Note also that this transformation retains (inverted) depth info for a z-buffer or other hidden surface algorithms. What is really neat in 4-D is clipping. We can warp the truncated viewing frustrum to a hypercube called the canonical view volume and do the clipping in 4-D! This is efficient since the faces are parallel to the principal axes. Also, with this warp, we can use a single clipping algorithm that works for both parallel and perspective projections! The moral of the story is that a lot of transformations that are non-linear in 3-D are linear in 4-D which makes life easier. -- wayne Wayne O. Cochran firstname.lastname@example.org http://www.eecs.wsu.edu/~wcochran Ecclesiastes 3:11
Newsgroups: comp.graphics.algorithms Subject: Re: Camera System Algo. like Decent From: Simon Fenney Date: Thu, 09 Jan 1997 05:31:04 -0600 In article <32CDAF88.66BD@chollian.dacom.co.kr>, Kee-Jeong Lee wrote: > > Hi, there! > > I've been tried some camera system but it was not flexile to view any > angle. > My 3D engine flow is following : > > Local coord. -> World coord. -> Camera coord. > > and transfering world to camera is > > 1. translate -CameraX, -CameraY, -CameraZ > 2. rotate -CameraAngleX, -CameraAngleY, -CameraAngleZ > > I compute offset when camera is moving > > camera->Move( SIN(-roll)*czm, > SIN(yaw)*czm, > COS(roll)*czm ); > camera->Angle.Set( yaw, roll, pitch ); > > czm is offset of moved toward. > > This method is doing very well. > but it have some problem when yaw is ANGLE90( when I see ground ) > then it can not be twisted left of right(rotate with roll) > > So, I am looking for Camera system like Decent what has very flexile > view angle. > > Thank you read this and reply me! ;) > > ------------------- > Kee-Jeong > email@example.com What you have is the dreaded "Gimbal Lock" problem which can happen with the gimbal mechanisms supporting gyroscopes etc., and you suddenly lose degrees of freedom. Probably the simplest solution is to ditch your Euler rotations entirely, and use quaternions to specify your angles. These can be combined arbitrarily and you always get a valid rotation. The resulting final Quaternions can then be converted to a rotation matrix quite trivially. There is probably stuff on the net describing them, but, failing that, I can recommend "Advanced Animation and Rendering Techniques" by Watt & Watt Addison- Wesley ISBN:0-201-54412-1 It describes the maths and uses in detail. Simon Fenney -------------------==== Posted via Deja News ====----------------------- http://www.dejanews.com/ Search, Read, Post to Usenet ###
From: firstname.lastname@example.org (Andrew Cooke) Newsgroups: sci.physics Subject: Re: Decomposing rotation matrices?? Date: 30 Nov 1995 16:37:18 GMT Organization: Institute for Astronomy, Royal Observatory Edinburgh Reply-To: A.Cooke@roe.ac.uk In article <email@example.com>, Joe Heafner - Astronomer wrote: >Greetings. > >I'm aware that two or more rotation matrices may be combined to form a >single rotation matrix equivalent to the combination of the others. I >need to do the reverse of this. I need to know if there's a way to look >at a matrix and deetermine which of the three elementary rotations have >been used and how to "decompose" the matrix into a product of these >elementary matrices. FYI, I'm assuming that the three elementary rotation >matrices are for the x, y, and z axes. hi, the decomposition depends on the order of rotations. if you write down the various matrices and take the product you'll see that it's quite easy to work backwards. if the operation is given by XYZ where X doesn't alter the x coordinate, etc., and X = ( 1 0 0 ) ( 0 x22 x23 ) ( 0 x32 x33 ) and Y and Z are like you expect, then, if A = XYZ, you'll find that a13 = y13 a23 = y33 x23 a12 = y11 z12 and since all the elements in x are related (same for y and z) you have got the solution (they are +/- cos/sin of the angle). two things more: 1 - that's not very `beautiful'. sorry! 2 - check what've said by writing them matrices out. hope that makes sense (and works! :-), andrew -- work phone/fax: 0131 668 8356, office: 0131 668 8357 institute for astronomy, royal observatory, blackford hill, edinburgh http://www.roe.ac.uk/ajcwww
From: "Alexander E. Meier" Newsgroups: sci.physics Subject: Re: Decomposing rotation matrices?? Date: Fri, 1 Dec 1995 10:17:50 -0500 Organization: Chemistry, Carnegie Mellon, Pittsburgh, PA Hi The best way to solve rotation matrix problems is to use quantum mechanical angular momenta ( I know it may seem strange but there is a close symmetrical connection between rotation and angular momentum eigenvectors) I assume the rotations are in normal space..that means that each rotation is a three by three matrix. Those matrix also represent the transformation properties of an angular momentum 1 under rotations...combining two of these matrices to form another one is like combining two angular momentums one to get another angular momentum one...in a mathematical sense it is an unitarian transformation...the coefficients of this transformation are known as the Clebsch-Gordan coefficients or vector-coupling coefficient...If you know the coefficient for the transformation in one way you can just take the inverse transformation and decompose your matrix Anyway I suuggest you to get a book about angular momentum and group theory which would certainly explain things better than I'm able to do Hope it helps :-) Alex
[xnew] [ ] [xold] [ynew] = [ rotation matrix ] [yold] [znew] [ ] [zold]
rotating points an angle T counterclockwise as viewed from a camera +x: (perhaps y to the right, z up)
[ 1 0 0 ] [ 0 cosT -sinT ] [ 0 +sinT cosT ]
With a right-handed coordinate system,
rotating points an angle T counterclockwise as viewed from a camera +y: (perhaps z to the right, x up)
[ cosT 0 +sinT ] [ 0 1 0 ] [ -sinT 0 cosT ]
rotating points an angle T counterclockwise as viewed from a camera +z: (perhaps x to the right, y up)
[ cosT -sinT 0 ] [ +sinT cosT 0 ] [ 0 0 1 ]
the Covariant Theory of physics http://infoweb.magi.com/~jgc/ tries to carefully explain some 4D concepts.
conference on spatial information theory http://www.sis.pitt.edu/~cosit97/ "Spatial information theory is the basis for the construction of Geographic Information Systems (GIS), but also necessary for other uses of geographic information and useful for information system design in general."
Non-Linearity of Thought http://www.catalog.com/sft/bobf/nonlinear.html some practical advice for those who think non-linearly.
On the dimensionality of spacetime Max Tegmark's library: dimensions http://www.sns.ias.edu/~max/dimensions.html
You ask a very interesting question when you ask about other symmetrical solids in 4 and 5 dimensions. The well-known mathematician Donald Coxeter, part of the mathematics department at the University of Toronto, spent many years of his life precisely on this question, worked out the answer, and wrote extensively about it. ...
The reason the question is so interesting is that the theory of regular shapes changes as you change dimension. In 2 dimensions, you can have a regular shape (a polygon whose lengths and angles are all equal) with any number of sides. So, there are an infinite number of regular polygons: the equilateral triangle, the square, the regular pentagon, the regular hexagon, and so on.
However, in three dimensions, there are only five different kinds of regular solids, instead of an infinite number! They are the tetrahedron (a 4-sided solid whose faces are all equilateral triangles), the cube (a 6-sided solid whose faces are all squares), the octahedron (an 8-sided solid whose faces are all equilateral triangles), the dodecahedron (a 12-sided solid whose faces are all regular pentagons), and the icosahedron (a 20-sided solid whose faces are all equilateral triangles). There aren't any others.
The reason is quite interesting. ...
Now, what about 4 and higher dimensions? You can do a similar kind of analysis using solid angles instead of angles, and you find that there is only a limited, finite number of possibilities for the number of hyperfaces meeting at each vertex, depending on what the hyperfaces are (and, since the hyperfaces have to be regular solids, there are only five possibilities for those).
An axiomatic system like this is not the usual method for studying geometry in three and higher dimensions. One would normally employ the language of vector spaces, linear independence, bases, and so on. That gives you a much cleaner theory that is dimension-independent.
The well-known compass and straight-edge construction of the circum-centre and in-centre of a plane-triangle, including the corresponding radii, generalizes to n-dimensional space. In the n-dimensional space, we are given a set of (n + 1) linearly-independent points. We want to find the centres of their circum-scribed and inscribed spheres, including the corresponding radii. We provide a brief background in Linear Algebra.
Started: 1997 ? before Sep 16.
Original Author: David Cary.
Current maintainer: David Cary.
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