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Topic: Mobius Transformation

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	Möbius transformation - Wikipedia, the free encyclopedia
		In mathematics, a Möbius transformation is a bijective conformal mapping of the extended complex plane (i.e.
		The usefulness of this representation is that the composition of two Möbius transformations corresponds precisely to matrix multiplication of the corresponding matrices.
		The subgroup of parabolic transforms is an example of a Borel subgroup, which generalizes the idea to higher dimensions.
	en.wikipedia.org /wiki/M%F6bius_transformation (2819 words)

	Möbius transformation
		In mathematics, a Möbius transformation, named in honor of August Ferdinand Möbius;, is a conformal mapping that is a bijection on the extended complex plane (that is, the complex plane augmented by the point at infinity, written ∞)
		If this is the case, then the transformation will be an affine transformation (some combination of rotation, dialation, and translation).
		Every transformation is similar to some particular linear transformation having one fixed point at infinity and another at 0.
	www.sciencedaily.com /encyclopedia/moebius_transformation (727 words)

	PlanetMath: Möbius transformation
		A Möbius transformation is a bijection on the extended complex plane
		It can be shown that the inverse, and composition of two mobius transformations are similarly defined, and so the Möbius transformations form a group under composition.
		This is version 12 of Möbius transformation, born on 2002-02-19, modified 2004-02-23.
	www.planetmath.org /encyclopedia/MobiusTransformation.html (185 words)

	Compressor Stalls and Moebius Transformations
		Then we can iterate the Mobius transformation z[k-1] z[k] = --------------------------- (3) id / id\ e + (1 - e) z[k-1] \ / and the real and imaginary parts of z[k] are proportional to the axial and circumferential velocity disturbance components at the angular position kd.
		The figure below shows the velocity disturbance profiles as given by Moore's formulas (2), and then superimposed on those curves are the discrete values generated by iterations of the Mobius transformation (3) with a value of d corresponding to 12 degrees.
		This Mobius transformation has fixed points at 0 and 1, and the simple linear function that transforms the general Mobius transformation to this "bi-polar" LFT maps the real axis to the "Riemann line" 1/2 + yi.
	www.mathpages.com /home/kmath245.htm (506 words)

	Möbius transformation (Site not responding. Last check: 2007-10-15)
		In mathematics, a Möbius transformation, named in honorof August Ferdinand Möbius, is a conformal mapping that is a bijection on the extended complex plane (that is, the complex plane augmented by the point at infinity, written ∞)
		Any Möbius map can be composed from the elementary transformations - dilations, translations and inversions.If we define a line to be a circle passing through infinity, then it can be shown that a Möbius transformation maps circles tocircles, by looking at each elementary transformation.
		Everytransformation is similar to some particular linear transformation having one fixedpoint at infinity and another at 0.
	www.therfcc.org /m%F6bius-transformation-199283.html (705 words)

	Mobius Transformations and The Night Sky
		If we apply a Lorentz transformation of the form (1) to this observer, specified by the four complex coefficients a,b,c,d, the resulting change in the directions of the incoming rays of light is given exactly by applying the LFT (also known as a Mobius transformation)
		This is certainly a proper orthochronous Lorentz transformation, because the determinant is +1 and the coefficient of t is positive.
		We've seen that the general finite transformation of the incoming null rays can be expressed naturally in the form of a finite Mobius transformation of the complex plane (under sterographic projection).
	www.mathpages.com /rr/s2-06/2-06.htm (1843 words)

	ACJ Response: The Mobius Band Meets The Lucas Sequence (Site not responding. Last check: 2007-10-15)
		The actualization of hypertextual space, which is situated in the present act and altered by successive transformations in context, has led theorists to see similarities in the narrative qualities of hypertext and those of oral literature.
		The unitary subject of the modernist era is thus transformed into the nomadic subject contemplating the bard's experience of social action.
		According to Gane's (1991) interpretation, the Mobius band is "a spiral of the reversibility of all signs in the shadow of seduction and death" (p.
	www.acjournal.org /holdings/vol6/iss3/responses/keith.htm (2517 words)

	Moebius - Wikipedia, the free encyclopedia
		Mobius, the planet of the Sonic the Hedgehog universe
		Mobius, a minor planet in the universe of Gene Roddenberry's Andromeda
		In Ace Combat 4 your character's callsign is Mobius 1.
	en.wikipedia.org /wiki/Mobius (179 words)

	hyperbolic, parabolic, elliptic transformations
		If the eigenvectors of the matrix representation of a Möbius transformation are its fixed points, there remains the question of interpreting the eigenvalues.
		, the diagonal form of the transformation executed, and the inverse mapping be used to restore the fixed points.
		Elliptic transformations will shepherd points towards the line joining the fixed points or draw them away into the vast space remote from the fixed points.
	delta.cs.cinvestav.mx /~mcintosh/comun/complex/node14.html (520 words)

	Mobius Transformations
		This family of transformations is also called fractional linear transformations.
		Composition of Mobius transformations is a Mobius transformation.
		Linear transformations are composed of shifts, rotations and scale multiplications, so they transform lines to lines and circles to circles.
	www.geocities.com /assafwool/Mobius/Mobius.html (614 words)

	Möbius transformation - Information (Site not responding. Last check: 2007-10-15)
		A Möbius $\mathfrak{H}$ transformation is uniqely defined by its two fixed points $\gamma_1, \gamma_2$ and by its characteristic constant $k$ .
		A translation is similar to the identity transform, having $k = 1$ .
		A transform $\mathfrak{H}$ can therefore be specified with two fixed points $\gamma_1, \gamma_2$ and the pole $z_\infty$ .
	www.book-spot.co.uk /index.php/Mobius_transformation (1186 words)

	The Mobius Strip
		The mathematical equation is known as The Mobius Transformation, also known as bilinear transformation or linear fractional transformation.
		The Mobius Strip is an expression of non-duality.
		The Mobius Strip is a spiritually significant symbol of balance and union.
	www.mobiusproductsandservices.com /tms.html (493 words)

	Energy for Knots (Site not responding. Last check: 2007-10-15)
		The key idea behind their results is a proof that the energy of a knot is unchanged by a "Möbius transformation".
		This is a special way to deform space, with the pleasant property that it transforms circles either to circles or to straight lines.
		Applying a Möbius transformation leads to a similar statement about round circles, but now the appropriate energy level turns out to be 4, not zero.
	www.fortunecity.com /emachines/e11/86/knotprob.html (1017 words)

	Talk:Möbius transformation - InformationBlast
		In fact in higher dimensional Euclidean space the Möbius transformations, which are defined by stereographic projection rather than using complex numbers, are the only conformal mappings.
		I'm not convinced that linear fractional transformations in more variables are normally called Möbius transfomations.
		The point about the conformal group in higher dimensions is already mentioned (at conformal geometry?).
	www.informationblast.com /Talk:M%F6bius_transformation.html (313 words)

	Homework from February 3 (Site not responding. Last check: 2007-10-15)
		The `group' can be considered a transformation group on the origin `space', the operation is multiplication of two complex numbers.
		So, in particular, the `point' 1 in the `space' is transformed by multiplication of the `group' element i to the new `point' 1*i = i.
		Garrick described a procedure to factor any Mobius transformation into a composition of the translation T and the double reflection R. Find the factorization of the Mobius transformation M(z) = (2z + 1)/(z + 1).
	www.willamette.edu /~zizza/Courses/SeniorSeminar/HW3.html (322 words)

	Conformal Mappings
		is called a linear transformation and it is a one-to-one mapping of the complex z-plane onto the complex w-plane.
		Since a linear transformation can be considered as a composition of a rotation, a magnification, and a translation, it follows that a linear transformations preserve angles.
		These mappings are conveniently expressed as the quotient of two linear expressions and are commonly known as linear fractional or bilinear transformations.
	mathews.ecs.fullerton.edu /fofz/conformal/c0.htm (444 words)

	[No title] (Site not responding. Last check: 2007-10-15)
		The authors choose the so-called generalized Cayley transform: C(A;\alpha_1,\alpha_2) := (A - \alpha_1 I)^{-1} (A - \alpha_2 I), \alpha_1 < \alpha_2, \alpha_1 not an eigenvalue of A. This is nothing other than the well-known Mobius transform for matrices.
		Obviously the eigenvalues of A are transformed by the corresponding Mobius transformation.
		The authors describe extensively the mapping properties of the Mobius transformation and the corresponding relation to the convergence behavior of the iterative solvers.
	www.math.niu.edu /~rusin/known-math/00_incoming/mobius (381 words)

	nrich.maths.org::Mathematics Enrichment::NRICH
		Briefly a Mobius map is a function f: C* -> C* of the form z -> (pz + q)/(rz + s).
		They call it a Fractional Linear Transformation, but this seems to be the same as a Mobius map.
		ii) a Mobius map with more than two fixed points is the identity (in fact a Mobius map which isn't the identity has exactly one or two fixed points in the extended complex plane).
	www.nrich.maths.org.uk /askedNRICH/edited/3865.html (410 words)

	RRFB Nova Scotia- MOBY (Site not responding. Last check: 2007-10-15)
		Moby is named after the universal symbol for recycling a mobius loop.
		The mobius loop, a fascinating shape, expresses transformation.
		Discovered in 1858 by German mathematician and astronomer August Ferdinand Mobius, the shape is a continuous loop with one surface and one edge formed by twisting one end of a long, thin rectangular strip 180 degrees (1/2 twist) and attaching this end to the other.
	www.rrfb.com /pages/Secondary%20pages/moby.html (224 words)

	Finite Subgroups of the Mobius Group (Site not responding. Last check: 2007-10-15)
		Every finite group of Mobius transformations is isomorphic (i.e., conjugate) to a group of rotations of the extended complex plane.
		For any positive integer g let n(g) denote the number of conjugacy classes of Mobius transformations each member of which generates a cyclic group of order g.
		This is closely related to the roots of polynomials with coefficients taken from a diagonal of Pascal's triangle.
	www.mathpages.com /home/kmath027.htm (199 words)

	Notes on using Fracitonal-Linear Transforms applet
		For a simple shift (a transformation when the objects are moved by a certain distance in a certain direction), use the matrices with alpha=1, gamma=0, delta=0 and beta determines the magnitude and the direction of the shift.
		That is, the first transformation is translation by two units to the right and the second transformation is rotation through 60 degrees.
		Then, the agreement when both the transformations and their entries are denoted by the same letters (like a or b - they are used to denote Mobius transformations and elements of a Mobius matrix) is very confusing.
	www.wiu.edu /users/fa101/java/Mobius2/Instructions2.htm (3687 words)

	Math 132 Applet 3
		A Möbius transformation (also called a fractional linear transformation, projective linear transformation, or a bilinear transformation by some authors) is any map of the form
		The purpose of the rest of the buttons on the applet is to change this equation to a different Möbius transform.
		grid is the "pole" (or "singularity") of the Möbius transformation.
	www.math.ucla.edu /~tao/java/Mobius.html (892 words)

	PlanetMath: unit disk upper half plane conformal equivalence theorem
		See Also: unit disk, upper half plane, Möbius transformation, Möbius circle transformation theorem
		Cross-references: unit circle, real axis, Mobius circle transformation theorem, Mobius transformation, upper half plane, unit disk, map, conformal, theorem
		This is version 3 of unit disk upper half plane conformal equivalence theorem, born on 2003-05-12, modified 2003-05-13.
	www.planetmath.org /encyclopedia/UnitDiskUpperHalfPlaneConformalEquivalenceTheorem.html (110 words)

	Möbius transformation - InformationBlast
		If both points are at infinity, then the transformation is a translation a = λ,b = λΔ,c = 0,d = λ.
		A translation is similar to the identity transform, having k = 1.
		A java applet allowing you to specify a transformation via its fixed points and so on may be found
	www.informationblast.com /M%F6bius_group.html (742 words)

	MAGOD: a Mobius transformation (Site not responding. Last check: 2007-10-15)
		From the site: "Möbius transformations are widely studied as useful examples of conformal maps.
		This animation shows the identity map (a Möbius transformation!) being deformed into a less trivial Möbius transformation.
		The homotopy is chosen so that all the intermediate steps are Möbius transformations as well.
	socrates.nhc.rtp.nc.us /~magod/archives/000013.html (172 words)

	Mobius Venture Capital :: In the News (Site not responding. Last check: 2007-10-15)
		A leader in its field, the Feld Group has successfully transformed the IT departments of several of the world's most prominent companies including Burlington Northern Santa Fe and PricewaterhouseCoopers and has been working with EDS as a consultant for the past several months.
		Under terms of the transaction, $37 million in cash was paid to Mobius Venture Capital, which held a 40 percent stake in the Feld Group.
		In addition, Mobius received warrants to purchase 898,921 shares of EDS common stock at a strike price of $23.95 per share, valued at approximately $7 million.
	www.mobiusvc.com /pages.php?pn=overview&sub=inthenews&id=952&id=952 (1043 words)

	Math I'm working on (Site not responding. Last check: 2007-10-15)
		One of the big problems of with using the definition is that it is not clear how to retrieve a set of simple closed curves that satisfy the requirements if you start with the transformations.
		Since we are dealing with Möbius transformation it seem reasonable to just consider circles to be the curves in question unfortunately there are Schottky groups that can not be represented by circles--even for two-generator kleinian groups there are examples that need curves other than circles.
		One plausibly way to retrieve the curves would be to compute the limits and draw in two curves and throw them at the generators of the groups and see if they satisfy the properties of the Schottky group.
	www.math.ubc.ca /~morey/math.html (327 words)

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