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From: Fridger Schrempp <t00fri@ma...>  20021217 23:46:22

Chris: >A very simple way to implement 'look backwards' is to rotate 180 >degrees about any axis . . . Something like: >sim>setObserverOrientation(sim>getObserverOrientation() * >Quatf::xrotate(PI)); That's essentially what I had implemented so far. > I could use the magnitude of the quaternion and represent angular > velocity, but I don't think I've done that anywhere in Celestia. This is what actually confused me, since a quaternion contains extra info. besides a pure rotation... >I'm interested in hearing more about >how quaternions are used in theoretical physicswhat are Pauli matrices >exactly? Here we go;)... First of all, you sure know that quaternions were originally introduced as generalization of the complex numbers. While the latter represent a mapping of 2 real numbers (x,y) => x+iy, with i^2=1, quaternions are a mapping of 4 real numbers (x,y,z,w) => x+iy+jz+kw with i^2=j^2=k^2=1, involving noncommuting multiplication rules. Now to slightly less trivial issues and physics: (Please forgive me, if I am boring you, but I have no idea about your background in these matters): In theoretical physics, symmetries and their representations are of basic significance (Rotation symmetry, Spin, Isospin, Relativistic symmetry=Lorentz group, Conformal Symmetries, Gauge symmetries,..). Mathematically, symmetries are described by means of "group theory". In particular, socalled Lie groups play the dominant role. The classical Lie groups include the /S/pecial /O/thorgonal groups, SO(N), whose fundamental representation matrices are NxN real, orthogonal matrices, O O^T=O^T O = 1, det(O)=+1, 1. They act on N dimensional /real/ basis vectors, and the most familiar realization are the 3d SO(3)rotation matrices of a real vector (x,y,z). This is what is primarily needed in Celestia;). Besides the SOgroups, the socalled /S/pecial /U/nitary, SU(N), groups are actually the most important ones in particle physics, notably the simplest one: SU(2). The latter appears already in Quantum Mechanics, to describe the rotations of particles with /half/integer intrinsic angular momentum ("Spin"), e.g. the electron or the proton. Also the symmetry of the nuclear forces under proton <=> neutron exchange are associated with a SU(2)"Isospin" group. The fundamental representation matrices of SU(2) are /unitary, complex/ 2x2 matrices, U U^* = U^* U =1, with det U=1 (*=hermitian conjugation= transposition & complex conjugation), acting on /complex/ 2d basis vectors. Such unitary SU(N) matrices may obviously be parametrized as U=exp(i*sum(alpha_k*T_k)), in terms of NxN /traceless, hermitian/ matrices, the socalled /generators/ of the group and N^21 real parameters alpha_k . The generators define the SU(N)/algebra/ in terms of commutation relations [T_i,T_j] = i* f_ijk * T_k, with the numbers f_ijk called "structure constants" and [x,y]=x*yy*x. The fundamental SU(2) generator matrices, are exactly the 2^21= 3 /Pauli matrices/: tau_1=(0 11 0); tau_2=(0 ii 0); tau_3=(1 00 1), first introduced by Nobel prize winner Wolfgang Pauli, when Quantum Mechanics was constructed. They have various important properties. Another basic 2x2 matrix is of course tau_0=(1 00 1), the 2d unit matrix. Now we are ready for quaternions: We can map our 4 real quaternion numbers from above to a complex 2x2 matrix q as follows: q=w*tau_0+x*tau_1+y*tau_2+z*tau_3 With q q^* = q^* q = w^2+x^2+y^2+z^2. A unit quaternion is thus nothing but a general SU(2) matrix. Since the SU(2) group is /locally equivalent=isomorphic/ to the SO(3) rotation group, we have a achieved a representation of arbitrary 3d rotations in terms of unit quaternions (complex 2x2 unitary matrices). Let me stop here for now. If you like, let me know whether these prerequisites were essentially familiar to you or not. The real physics applications of quaternions are starting from here...;). Bye Fridger 
From: Christophe Teyssier <chris@te...>  20021217 22:45:45

On Monday 16 December 2002 00:03, Christophe Teyssier wrote: > Indeed, I'll add that. > What'd be nice would be to have a URL syntax that represents a location > on/relative to a body. Something like: > cel://SyncOrbit/Body/?long=...&lat=...&alt=... > That way you can even bookmark your home town ;) This is a first try at such relative URLs. Use ALTR or Bookmark>Add Relative Bookmark to create one. As usual the difficulty is to decide what state data to keep/exclude. At the moment, the time and orientation are excluded. It's probably a good idea to ignore the TimeScale and possibly the Mode which should be overwritten as SyncOrbit. Note: the difference in units between getSelectionLongLat() and gotoSelectionLongLat() is a real gotcha!  Christophe 
From: Chris Laurel <claurel@ww...>  20021217 18:13:25

I made a small checkin this morning to fix a couple bugs reported by Rassilon. The change affects only render.cpp, and eliminates shadows on clouds and rings when a planet is marked as emissive. I had the idea to expand the definition of a galaxy to include nebulae and globular clusters. I want to introduce an asyettobenamed superclass, from which Galaxy and Nebula will derive. There won't be too much change in how these objects are handled by the renderer, except that nebulae will use a 3DS mesh instead of the fuzzy blobs of galaxies. I also want to change the format of galaxy catalogs to something flexible enough that it can be used for other deep sky objects. And finally, the limitation that all galaxies must be defined in data/galaxies.dat needs to be removed. This is all 1.2.6 work . . . Chris 
From: Chris Laurel <claurel@ww...>  20021217 02:04:25

On Sun, 15 Dec 2002, Fridger Schrempp wrote: > Chris: > > At present the '*'key does 'sim>setObserverOrientation(Quatf(1));' which is > not exactly what would be the most usefulinteresting 'reverse' operation. > > > I am currently trying to understand your various quaternion > operationsdefinitions, in order to incorporate a true 'reverse Observer > Orientation' ("looking backwards") on the '*'key. For you, this is probably > trivial, but I will catch up;)...A 'looking backwards' function would have a > number of neat applications, like when a planet has just passed the observer or > the observer overtook the planet. Mainly, it is fun for earthbound observations > (after approaching earth via HOME, you want to turn back) and for switching > forth and back between 2 celestial objects, etc. A very simple way to implement 'look backwards' is to rotate 180 degrees about any axis . . . Something like: sim>setObserverOrientation(sim>getObserverOrientation() * Quatf::xrotate(PI)); > I am amazed by the fact, how important the role of quaternions is in > Celestia;). To me as a theoretical physicist, they are of course very > familiar, but in quite different contexts. E.g. I often use quaternions q to > characterize /conformal transformations in 4d/ Euclidean space, by associating > to any 4vector q_mu the quaternion q = q_mu sigma_mu (summed over mu), with > sigma_i = Pauli matrices, i=1..3, and sigma_4=2dunit matrix. > > General conformal trf's then take the form (x, x' are 4d spacetime vectors > written as quaternions;)) > > x => x', with x'=(alpha x + beta) (gamma x + delta)^{1}. > > In Celestia, you apparently use this 4d Rotation group formalism, since you > want to carry along besides the 3x3 rotation matrices3 vectors another piece > of info. What exactly is it. Angular velocity? I could use the magnitude of the quaternion and represent angular velocity, but I don't think I've done that anywhere in Celestia. I find quaternions very convenient because they're more compact than 3x3 rotation matrices and they don't require any complicated reconditioning to keep error from accumulating. All that's required is a normalization to assure that you've still got a rotation. Perhaps most useful of all, it's very straightforward to interpolate between two orientations when they're represented as quaternions. I'm interested in hearing more about how quaternions are used in theoretical physicswhat are Pauli matrices exactly? > In the file quaternion.h you set up various functions involving q's, including > the conversion of a quaternion into an (axis,angle). In case of the > ObserverOrientation, what /exactly/ is the 'axis' and the 'angle'? > Everything is ultimately referred to a coordinate system where the yaxis is normal to the J2000.0 ecliptic, the xaxis points along the J2000.0 equinox, and the zaxis completes the right handed coordinate system. But depending on whether you premultiply or postmultiply, the rotation may be either in the local or the global coordinate system. > All the rest I have coded already, a method > 'Simulation::reverseObserverOrientation()' and its association with the '*'key > in celestiacore.cpp. It also works nicely, but I just want to be sure I exactly > understand all your definitions, involved. > > Quaternions for ever!;) Yes! Chris 