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	Möbius transformation - Wikipedia, the free encyclopedia
		Möbius transformations should not be confused with the Möbius transform or the Möbius function.
		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.
	en.wikipedia.org /wiki/Mobius_transformation (3063 words)

	PlanetMath: Möbius transformation
		A Möbius transformation is a bijection on the extended complex plane
		The geometric interpretation of the Möbius group is that it is the group of automorphisms of the Riemann sphere.
		This is version 13 of Möbius transformation, born on 2002-02-19, modified 2006-09-13.
	planetmath.org /encyclopedia/MobiusTransformation.html (187 words)

	EucliDraw Transformations
		Transformations are particular objects represented on the drawing sheet by their tag, which is a small rectangle containing their kind and name.
		Transformations don't have anchors and to access their functionality it is more convenient to right-click on their tag and select from the popup menu the desired operation.
		And for the Moebius transformation correspondingly its characteristic parallelogram.
	www.euclidraw.com /Eng_fls/EUC_htmls/Transforms.html (5767 words)

	Apollonian gasket - Wikipedia, the free encyclopedia
		Since there is a Möbius transformation which maps any three given points in the plane to any other three points, and since Möbius transformations preserve circles, then there is a Möbius transformation which maps any two Apollonian gaskets to one another.
		Möbius transformations are also isometries of the hyperbolic plane, so in hyperbolic geometry all Apollonian gaskets are congruent.
		The Apollonian gasket is the limit set of a group of Möbius transformations known as a Kleinian group.
	en.wikipedia.org /wiki/Apollonian_gasket (646 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		The set of Moebius transformations also includes other geometric transformations which are compositions of these two types of transformations.
		To be technically accurate, what we are calling Moebius transformations are really elements of a larger class of transformations in the complex plane which are called Moebius transformations.
		In general a Moebius transformation is an invertible transformation of points z=x+iy in the complex plane having the form
	archive.ncsa.uiuc.edu /Classes/MATH198/whubbard/GRUMC/geometryExplorer/help/noneuclid/examples-moebius.html (625 words)

	Statics - GFD - Conformal mapping (Site not responding. Last check: 2007-10-31)
		The Moebius transformation z'=(p0z+p1)/(p2z+p3) is a relatively simple conformal transformation that always maps Circles on Circles, where a Circle is either a usual circle or a straight line, i.e., a circle with infinite radius.
		There are special Moebius transformations that do not change the radius of one of the circles, but these transformations change the radius of the second circle.
		The Moebius transformation is one of the predefined conformal transformations in MaX-1.
	people.ee.ethz.ch /~hafner/1MaXTut/Conformalmapping2.htm (1138 words)

	CARS - Defining Moebius transformations
		We are mostly interested in hyperbolic Möbius transformations mapping the unit disk (or the upper half-plane) onto itself.
		A hyperbolic Möbius transformation g with a given axis moves all points of the unit disk towards one of the fixed-points of g.
		The reddish circular arcs intersecting the axes are the respective isometric circle and its image under the transformation.
	www.math.fsu.edu /~seppala/Symbolic/Software/Cars/Moebius.html (569 words)

	Kleinian Group Fractals Main Page
		Möbius transformations can be as beautiful and varied as fractals created by other better known methods.
		The trace of a transformation is unchanged by conjugation.
		Möbius transformations can be classified by the value of the trace and the number of fixed points (one or two).
	www.hiddendimension.com /KleinianGroup_Fractals_Main.html (293 words)

	Möbius and Lorentz - Advanced Physics Forums (Site not responding. Last check: 2007-10-31)
		Anyway I gather that the group of Möbius transformations is isomorphic to the Lorentz group (minus reflections).
		What I think this means in geometric terms is: take the \"celestial sphere\" (the set of all light-like lines passing through a point, or the sphere where the \"fixed stars\" are), then apply a stereoscopic projection to turn this sphere into a Riemann sphere.
		The result is the same as if you had instead applied the Lorentz transformation corresponding to the Möbius transformation, which is to say the sky has changed as if you turned your head or set off at some velocity.
	www.advancedphysics.org /forum/showthread.php?t=1669 (689 words)

	Class Plan
		Such a transformation is called a Möbius transformation.
		Show that a Möbius transformation with c nonzero can be written as a composition of translation followed by an inversion, followed by a dilation, and again by a translation.
		Show that a Möbius transformation with three fixed points must be the identity.
	www.msci.memphis.edu /~botelhof/X.html (183 words)

	IFS Fractals Main Page
		The transformation can scale, stretch, skew and rotate the rectangle, still keeping it within the confines of the graphic plane of width W and height H. The image produced is a fractal.
		With each round of iteration one of the transformations is chosen randomly, using the probability as factor in the choice, and the transformed point is plotted on the graphic plane.
		A single transformation is used for each iteration, with the transform used selected randomly by the same method used for affine transformations described above.
	www.hiddendimension.com /IFS_Fractals_Main.html (445 words)

	[No title]
		Intuitively, in the Euclidean plane, a motion is a transformation that "moves things around", without changing their size or shape, or the relationships among them.
		Now, a line is part of a Möbius circle, so perhaps we should be looking at transformations which preserve these circles.
		Furthermore, it seems likely that these transformations will also preserve distances and angles, since both are calculated in terms of cross ratios: this part, we'll have to check.
	www.cecm.sfu.ca /~jalester/DEMO/POINCARE/BookPages/motions-1.html (246 words)

	Energy for Knots
		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 (1035 words)

	Equations (Site not responding. Last check: 2007-10-31)
		All isometries in the Poincaré Disk are given by Moebius Transformations of the form
		Given a Moebius transformation of the form T(z) = (az + b) / (cz + d), we define the trace of the transformation to be tr(T) = a + d.
		Note: Moebius transformations in the plane are determined by images of three points.
	www.geom.uiuc.edu /~crobles/hyperbolic/hypr/isom/pncr/eq.html (77 words)

	Analytic solutions - Conformal mapping (Site not responding. Last check: 2007-10-31)
		MaX-1 transforms the grid points only and it draws straight lines from each grid point to its neighbors if the 3D Grid box in the Field representation group of the Field dialog is checked.
		You can use this transformation to transform the two concentric circles of the default coaxial cable into non-concentric circles or into a circle in front of a straight line.
		If you apply the Moebius transformation with these parameters, all grid points will be transformed in one and the same point w=1.
	people.ee.ethz.ch /~hafner/1MaXTut/Conformalmapping1.htm (1390 words)

	Summary of facts about Möbius maps (Site not responding. Last check: 2007-10-31)
		A Möbius transformation is a transformation of the Riemann sphere defined by one of the two formulas
		The former kind of Möbius transformations are angle-preserving, or alternatively conformal, while the latter kind (with
		All Möbius transformations preserve the size of angles between curves; the difference lies in whether or not they also preserve the sense.
	www.math.okstate.edu /~wrightd/INDRA/MobiusonCircles/node1.html (104 words)

	Mobius Transformations
		The Möbius transformation, Green function and the degenerate elliptic equation.
		Some properties of linear-fractional transformations and the harmonic mean of matrix functions.
		Enhancing the convergence region of a sequence of bilinear transformations.
	math.fullerton.edu /mathews/c2003/MobiusTranformationBib/Links/MobiusTranformationBib_lnk_2.html (280 words)

	The action of on
		As a trivial consequence of this exercise we have:
		Besides sending circles to circles, Moebius transformations have the important property that they preserve angles and orientation.
		Using that Moebius transformations preserve angles, we can give a simple proof that the stereographic projection preserves angles.
	www.math.poly.edu /courses/projective_geometry/chapter_three/node2.html (953 words)

	PlanetMath: Möbius transformation cross-ratio preservation theorem (Site not responding. Last check: 2007-10-31)
		"Möbius transformation cross-ratio preservation theorem" is owned by rspuzio.
		This is version 4 of Möbius transformation cross-ratio preservation theorem, born on 2003-04-28, modified 2006-05-31.
		value at a of the mobius transformation that takes b,c,d, to
	planetmath.org /encyclopedia/MobiusTransformationCrossRatioPreservationTheorem.html (116 words)

	Twistor Theory
		Hence circles on the sphere will be mapped to circles on the plane (except those containing the south pole, which are mapped to straight lines on the plane).
		Möbius transformations of the plane correspond to Lorentz transformations of the celestial sphere.
		It is clear that Lorentz transformation cannot change the shape of the circle.
	users.ox.ac.uk /~tweb/00006/index.shtml (947 words)

	Can You See the Lorentz-Fitzgerald Contraction?
		The above article on Penrose-Terrell rotations mentions in passing the fact that every Lorentz transformations act on the celestial sphere the same way that the corresponding Moebius transformation acts on the Riemann sphere, but it is not very explicit.
		A Moebius transformation, or linear fractional transformation, of the complex numbers, or more properly of the Riemann sphere (C augmented by a "point at infinity") is given by
		Lorentz transformations may be classified into four types according to their geometric effect on the night sky:
	www.math.ucr.edu /home/baez/physics/Relativity/SR/penrose.html (1653 words)

	Triple transitivity
		Thus, it is not surprising that prescribing the images of a Möbius transformation at three points determines the transformation completely.
		For instance, suppose we wish to have a map that transforms the unit circle to the real line.
		Since three points determine a circle, and Möbius transformations carry circles to circles, this map does indeed map the unit circle to the real axis.
	www.math.okstate.edu /~wrightd/INDRA/MobiusonCircles/node3.html (208 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)

	Math I'm working on
		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)

	Page 2 for Paul McCreary's Home Page (Site not responding. Last check: 2007-10-31)
		Poincare described the conditions necessary and sufficient for a hyperbolic polygon to represent a Reimann surface.
		There are four types of Möbius transformations, three of which can be isometries of the hyperbolic plane.
		The one- and two-dimensional manifolds that are stabalized by the extended Möbius transformations make spectacular introductions to the actions of hyperbolic isometries.
	new.math.uiuc.edu /~paulmcc/KleinGrps.html (234 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		There is an iteration for producing these surfaces that involves Moebius transformations and a Darboux-Baecklund tranformation (DBT) of the Gauss map.
		The first level of UP-iterates of Enneper's surface are obtained by applying the Moebius transformation (az+b)/(cz+d) to the Gauss map of Enneper's surface, g(z)=z, and then performing the DBT.
		The second level of UP-iterates of Enneper's surface are obtained by applying a second Moebius transformation to the first iterates, and again performing the DBT on the resulting Gauss map.
	www-sfb288.math.tu-berlin.de /~cat/rational.html (444 words)

	Table of Contents
		Illuminating, widely praised book on analytic geometry of circles, the Moebius transformation, and two-dimensional non-Euclidean geometries.
		Moebius transformations and anti-homographies as products of inversions f.
		The group R of rotational Moebius transformations c.
	www.doverpublications.com /cgi-bin/toc.pl/0486638308 (133 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		51 (Hint: If s and t are Moebius transformations as in in part (b) and in part (c), consider transformations of the form tst^{-1}).
		Solve the Dirichlet problem for this region, with boundary conditions g(z)=1 on the upper arc, and g(z)=0 on the lower arc.
		(Hint: Use a Moebius transformation which maps the two circles to two straight lines [this forces one of the intersecion points to be mapped to infinity; make a convenient choice for one of the lines]; then solve the problem for the resulting region).
	www.math.ucsd.edu /~wenzl/210A.html (590 words)

	Class Plan
		Let T be a Möbius transformation with only one fixed point (provide some examples).
		Consider a Möbius transformation T with a single fixed point 3+i.
		Solve the same problem if T has a single fixed point, denoted by a.
	www.msci.memphis.edu /~botelhof/XV.html (82 words)

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