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	Rotation (mathematics) - Wikipedia, the free encyclopedia
		In ordinary three dimensional space, a coordinate rotation can be defined by three Euler angles, or by a single angle of rotation and the direction of a vector about which to rotate.
		Rotations about the origin are most easily calculated using a 3 by 3 matrix transformation called a rotation matrix.
		Rotations about another point can be described by a 4 by 4 matrix acting on the heterogeneous coordinates.
	en.wikipedia.org /wiki/Coordinate_rotation (550 words)

	Orthogonal matrix - Wikipedia, the free encyclopedia
		While it is common to describe a 3×3 rotation matrix in terms of an axis and angle, the existence of an axis is an accidental property of this dimension that applies in no other.
		A Jacobi rotation has the same form as a Givens rotation, but is used as a similarity transformation chosen to zero both off-diagonal entries of a 2×2 symmetric submatrix.
		The polar decomposition factors a matrix into a pair, one of which is the unique closest orthogonal matrix to the given matrix, or one of the closest if the given matrix is singular.
	en.wikipedia.org /wiki/Orthogonal_matrix (2811 words)

	Matrix and Quaternion FAQ
		Rotation in X transforms Y and Z Rotation in Y transforms X and Z Rotation in Z transforms X and Y The argument to this goes as follows: Given a vertex V = (x,y,z), rotation angles (A,B and C) and translation (D,E,F).
		For all powers, the matrix must be square, that is orthogonal and the same width and height For example, -1 M is the inverse of the matrix 0 M generates the identity matrix 1 M leaves the matrix unchanged.
		The inverse of an identity matrix is the identity matrix.
	www.j3d.org /matrix_faq/matrfaq_latest.html (7736 words)

	PlanetMath: rotation matrix
		See Also: orthogonal matrices, example of rotation matrix, decomposition of orthogonal operators as rotations and reflections, derivation of rotation matrix using polar coordinates, derivation of 2D reflection matrix
		This is version 14 of rotation matrix, born on 2005-02-19, modified 2006-06-13.
		I think, v should be unit vector or R cannot be rotation matrix.
	planetmath.org /encyclopedia/RotationMatrix.html (147 words)

	Rotation matrix Did You Mean rotation_matrix (Site not responding. Last check: 2007-10-13)
		See also the general formula for a 3 × 3 rotation matrix in terms of the axis and the angle.
		The set of all rotations about a given axis, together with the operation of composition, form a continuous group.
		Any rotation matrix is orthogonal, that is, the inverse of a rotation matrix is its transpose.
	www.did-you-mean.com /Rotation_matrix.html (140 words)

	Matrix and Quaternion FAQ (Site not responding. Last check: 2007-10-13)
		Also, the resulting matrix has an order of AxD Thus, it is possible to multiply a 4xN matrix with a 4x4 matrix but not the other way around.
		have the same width and height For example, -1 M is the inverse of the matrix 0 M generates the identity matrix 1 M leaves the matrix undamaged.
		Using a 4x4 matrix library, the algorithm is as follows: ---------------------------------------------------------------------- for (n = 0; n < 4; n++) m4_to_spherical(mat[n], andv_sph[n]); /* Spherical coordinates / m4_multspline(m_cardinal, v_sph, v_interp); / Interpolation vector */...
	www.flipcode.com /documents/matrfaq.html (7478 words)

	PlanetMath: example of rotation matrix
		You can use rotation matrices to show that if the slope of one line is
		"example of rotation matrix" is owned by swiftset.
		This is version 2 of example of rotation matrix, born on 2005-03-24, modified 2005-10-22.
	planetmath.org /encyclopedia/ExampleOfRotationMatrix.html (94 words)

	Rotation matrix ortho-normalization - GameDev.Net Discussion Forums (Site not responding. Last check: 2007-10-13)
		The way I figured it, all you need to do is take a unit vector, transform it by your current rotation matrix, then inspect the resulting vector and build a some kind of a scaling matrix that makes the resulting vector unit length, then concatenate this matrix with your current one to fix it.
		Rotation of a vector by a quaternion while it is possible it is about 8 time more expensive than rotation by the matrix equivalent.
		The rotation matrix mapped by the product of two quaternion is different that the product of the matrices mapped by the quaternions.
	www.gamedev.net /community/forums/topic.asp?topic_id=278410 (3730 words)

	Computation of the Rotation Matrix
		In motion analysis, it is often necessary to compute the transformation matrix directly from the coordinates of the markers fixed on a moving body.
		Once the transformation matrix is known, the orientation angles and eventually the location of center of rotation as well may be computed.
		In addition, matrix c' must be orthogonal because all three matrices composing c' are orthogonal.
	kwon3d.com /theory/jkinem/rotmat.html (415 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		Instead of rotating an object through a series of successive rotations, a quaternion allows the programmer to rotate an object through a single arbitary rotation axis.
		Because the rotation axis is specifed as a unit direction vector, it may be calculated through vector mathematics or from spherical coordinates ie (longitude/latitude).
		Using a 4x4 matrix library, the algorithm is as follows: ---------------------------------------------------------------------- for (n = 0; n < 4; n++) m4_to_spherical(mat[n], &v_sph[n]); /* Spherical coordinates / m4_multspline(m_cardinal, v_sph, v_interp); / Interpolation vector */...
	www.eecis.udel.edu /~chandra/640/Fall05/matrixfaq.txt (5761 words)

	rotation matrix (Site not responding. Last check: 2007-10-13)
		In this case, if you rotate 120 degrees about z, x goes to a combination of x and y and y goes to a combination of y and x, and you must consider this.
		Here is a generalized rotation matrix for rotating about the z axis (the angle here is expressed in radians, and n is the order of the rotation, 3 for C
		If you rotated about a line that passed through the origin but was equidistant from x, y and z, x would go to y, y would go to z and z would go to x.
	www.wellesley.edu /Chemistry/chem341/rotationmatrix.html (378 words)

	Maths - Rotation Matrices - Martin Baker
		Rotations can be represented by orthogonal matrices (there is an equivalence with quaternion multiplication as described here)
		First rotation about z axis, assume a rotation of 'a' in an anticlockwise direction, this can be represented by a vector in the positive z direction (out of the page).
		If we want to represent rotation and translation using a single matrix we need to use a 4x4 matrix as explained here.
	www.euclideanspace.com /maths/algebra/matrix/orthogonal/rotation (652 words)

	Rotation Matrix
		Vector rotation is equivalent to the axis rotation in the opposite direction.
		One should not be confused by the axis rotation and the vector rotation.
		In vector transformation, the axis rotation matrices should be used instead of the vector rotation matrices because vector transformation means change in the perspective.
	kwon3d.com /theory/transform/rot.html (205 words)

	Rotation PRINCALS Solutions
		This paper describes how the solution of a non-linear principal components analysis can still be rotated when data were originally intended to serve in an ordinary principal components analysis, but doubts about the measurement level or the linearity of relations have risen.
		The procedure may be used for orthogonal rotation as well as for oblique rotation of the component loadings of the variables.
		When rotating orthogonally, it is also possible to rotate the object scores (component scores) for data reduction purposes.
	oase.uci.ru.nl /~rkonig/rotation_princals (1466 words)

	Rotation about the z-axis... - MSDN Forums
		Rotating about the x and y axis works as expected (the thing spins horizontally and vertically), but when I rotate it about the z-axis, it goes up and down, deforms a bit, and just looks weird.
		If all else fails, you could also try applying only the rotation matrix to verify if the problem is caused by that one or by another matrix.
		For now forget the scaling and rotation and play with translating, then start adjusting the values you pass for translating to find the centre, the bottom centre, etc… It won’t take long and after combine the rotation and scaling.
	forums.microsoft.com /MSDN/ShowPost.aspx?PostID=292764&SiteID=1 (1107 words)

	Forum Thread : Rotation Matrix Problem :. GarageGames
		The 3rd column of the matrix should be the point P from your diagram.
		In three dimensions, this has infinite solutions because the rotation of vectors perpendicular to v and v' (out of the screen in the picture) is not defined..
		Plop their values in the matrix, multiply your vector, and you've re-oriented the vector.
	www.garagegames.com /mg/forums/result.thread.php?qt=19828 (723 words)

	Forum Thread : Rotation Matrix Nightmare's! :. GarageGames
		As soon as I get a little more math and programming under my belt (which may take a while), I hope to give back to the community in the best way I know how...by sharing this knowledge which seems to be scattered o'er the planet.
		All he wants to do is rotate an object about a point, if I understant correctly.
		You can also use the angle to rotate the shape so one side is also facing the other shape.
	www.garagegames.com /mg/forums/result.thread.php?qt=25057 (1262 words)

	SLA_DAV2M - Rotation Matrix from Axial Vector (Site not responding. Last check: 2007-10-13)
		SLA_DAV2M - Rotation Matrix from Axial Vector ;
		Form the rotation matrix corresponding to a given axial vector (double precision).
		The axis is called the Euler axis, and the angle through which the reference frame rotates is called the Euler angle.
	www.hartrao.ac.za /nccsdoc/slalib/sun67.htx/node45.html (117 words)

	SLA_AV2M - Rotation Matrix from Axial Vector (Site not responding. Last check: 2007-10-13)
		SLA_AV2M - Rotation Matrix from Axial Vector ;
		Form the rotation matrix corresponding to a given axial vector (single precision).
		The axial vector supplied to this routine has the same direction as the Euler axis, and its magnitude is the Euler angle in radians.
	www.hartrao.ac.za /nccsdoc/slalib/sun67.htx/node25.html (117 words)

	Rotation matrix or complex translation formula - Beyond3D Forum
		One of my particle effects has particles moving in a circle, and the plane of the circle is rotated around one axis.
		Currently I'm using translation matricies to draw the particles going around the circle, and then a rotation matrix to rotate the circle around the axis.
		It seems like multiplying the translation matrix and rotation matrix together for each particle ends up being a whole lot of multiplications (64 float multiplications per particle per frame, right?).
	www.beyond3d.com /forum/showthread.php?t=2524 (2415 words)

	Materials : Rotation Matrix transformation using misorientation
		So if N(the number of final orientations) is say 10, then theres going to be 10 orientations from initial to final that rotate in such a way that their misorientation is linear in incrementation.
		DG is the required rotation matrix to move Gi to Gf.
		But I think theres a lack of information in trying to determine a Rotation Matrix from a misorientation angle, because there is no unique solution.
	www.physicsforums.com /showthread.php?threadid=60496 (441 words)

	Implementing Rotation Matrix Constraints in Analog VLSI (ResearchIndex)
		Abstract: We describe an algorithm for continuously producing a 3x3 rotation matrix from 9 changing input values that form an approximate rotation matrix, and we describe the implementation of that constraint in analog VLSI circuits.
		This constraint is useful when some source (e.g., sensors, a modeling system, other analog VLSI circuits), produces a potentially "imperfect" matrix, to be used as a rotation.
		The9 values are continuously adjustedover time to find the "nearest" true rotation matrix, based on...
	citeseer.ist.psu.edu /92158.html (381 words)

	Rotation Matrix?
		Let R(a) be the rotation matrix about the z-axis through angle a.
		Let R(v) be the rotation matrix that sends the vector v to the z axis.
		Then R(v) is a rotation about the z axis through the angle -c, followed by a rotation about the y-axis through angle -b.
	www.physicsforums.com /showthread.php?p=916564#post916564 (646 words)

	[chimera-dev] Accessing rotation matrix of molecule models (Site not responding. Last check: 2007-10-13)
		The xform first applies the rotation matrix and then a translation.
		You probably see the translation changing when you rotate the molecule because the center of rotation when using the mouse is not at the origin (0,0,0).
		(The center of rotation for mouse rotations can be controlled -- see the Rotation tab of the Side View dialog.) When you get atom coordinates with atom.coord() you get the original xyz values from say a PDB file.
	www.cgl.ucsf.edu /pipermail/chimera-dev/2002/000020.html (224 words)

	[chimera-dev] Accessing rotation matrix of molecule models (Site not responding. Last check: 2007-10-13)
		I'm interested in comparing/aligning two structures and I would like the user to rotate the models until they think it looks good.
		Then I want to compute the distance between the atoms, but I'd need to know what they're coordinates are in the same space.
		Along with that, is there then a way to set the rotation matrix for a model?
	www.cgl.ucsf.edu /pipermail/chimera-dev/2002/000017.html (133 words)

	Rotation Matrix - GameDev.Net Discussion Forums (Site not responding. Last check: 2007-10-13)
		You could calculate all possible bases produced by the algorithm, you'd have to do it 6 times though (once for each ordering of the input base vectors).
		Then pick the matrix which, when subtracted from the original, has least (modulus of the) determinant, i.e.
		You may need to check that the final matrix has determinant +1 rather than -1 though, otherwise you get a reflection as well (so you need to reverse one of the output base vectors).
	gamedev.net /community/forums/topic.asp?topic_id=137042&whichpage=1 (260 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		Energy Citations Database (ECD) Document #4584634 - REDUCED ROTATION MATRIX: PLOTS AND ZEROS.
		Availability information may be found in the Availability, Publisher, Research Organization, Resource Relation and/or Author (affiliation information) fields and/or via the "Full-text Availability" link.
		DATA TABULATIONS/on plots and zeros for reduced rotation matrix;NUCLEAR THEORY/rotation matrix in, table of plots and zeros for reduced
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=4584634 (105 words)

	[ODE] Getting X,y,Z rotation from rotation matrix (Site not responding. Last check: 2007-10-13)
		Previous message: [ODE] Getting X,y,Z rotation from rotation matrix
		for > How can I get the X,Y,Z rotation radians from the Rotation matrix I > get from dBodyGetRotation?
		> > James > > ----- Original Message ----- > From: Alessandro Monopoli > Date: Fri, 21 May 2004 03:46:17 +0200 > Subject: [ODE] Getting X,y,Z rotation from rotation matrix > To: ode at q12.org > > Hi all!
	www.q12.org /pipermail/ode/2004-May/012946.html (184 words)

	[ODE] Getting X,y,Z rotation from rotation matrix (Site not responding. Last check: 2007-10-13)
		Next message: [ODE] Getting X,y,Z rotation from rotation matrix
		> > for > > > How can I get the X,Y,Z rotation radians from the Rotation matrix I > > get from dBodyGetRotation?
		> > > > James > > > > ----- Original Message ----- > > From: Alessandro Monopoli > > Date: Fri, 21 May 2004 03:46:17 +0200 > > Subject: [ODE] Getting X,y,Z rotation from rotation matrix > > To: ode at q12.org > > > > Hi all!
	www.q12.org /pipermail/ode/2004-May/012947.html (208 words)

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