scholarly journals On the Images of Ellipses Under Similarities

2013 ◽  
Vol 21 (2) ◽  
pp. 189-194
Author(s):  
Nihal Yilmaz Özgür
Keyword(s):  
Complex Plane ◽  
Möbius Transformations ◽  
Mobius Transformations

AbstractWe consider ellipses corresponding to any norm function on the complex plane and determine their images under the similarities which are special Möbius transformations.

Convex fundamental domains of Fuchsian groups

1974 ◽  
Vol 76 (3) ◽  
pp. 511-513 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Complex Plane ◽  
Unit Disc ◽  
Fuchsian Group ◽  
Discrete Group ◽  
Fuchsian Groups ◽  
Finitely Generated ◽  
Möbius Transformations ◽  
Fundamental Domains ◽  
Mobius Transformations

In this paper a Fuchsian group G shall be a discrete group of Möbius transformations each of which maps the unit disc △ in the complex plane onto itself. We shall also assume throughout this paper that G is both finitely generated and of the first kind.

scholarly journals Hyperbolic Motions

1967 ◽  
Vol 29 ◽  
pp. 163-166 ◽  
Author(s):  
Lars V. Ahlfors
Keyword(s):  
Half Space ◽  
Hyperbolic Space ◽  
Complex Plane ◽  
Two Dimensions ◽  
Möbius Transformations ◽  
Do So ◽  
Mobius Transformations

The observation by Poincaré that Möbius transformations in the complex plane can be lifted to a half-space raises the need to be able to handle motions in hyperbolic space of more than two dimensions by means of an analytic apparatus of not too forbidding complexity. In my experience the best way to do so is to be guided by analogies with the familiar twodimensional case. The purpose of this little paper is to collect a few formulas that the writer has found useful when working with certain hyperbolically invariant operators.

scholarly journals Hermite Interpolation Using Möbius Transformations of Planar Pythagorean-Hodograph Cubics

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Sunhong Lee ◽  
Hyun Chol Lee ◽  
Mi Ran Lee ◽  
Seungpil Jeong ◽  
Gwang-Il Kim
Keyword(s):  
Complex Plane ◽  
Hermite Interpolation ◽  
Möbius Transformation ◽  
Möbius Transformations ◽  
Interpolation Problems ◽  
Pythagorean Hodograph ◽  
Improved Stability ◽  
Mobius Transformation ◽  
Extra Parameter ◽  
Mobius Transformations

We present an algorithm forC1Hermite interpolation using Möbius transformations of planar polynomial Pythagoreanhodograph (PH) cubics. In general, with PH cubics, we cannot solveC1Hermite interpolation problems, since their lack of parameters makes the problems overdetermined. In this paper, we show that, for each Möbius transformation, we can introduce anextra parameterdetermined by the transformation, with which we can reduce them to the problems determining PH cubics in the complex planeℂ. Möbius transformations preserve the PH property of PH curves and are biholomorphic. Thus the interpolants obtained by this algorithm are also PH and preserve the topology of PH cubics. We present a condition to be met by a Hermite dataset, in order for the corresponding interpolant to be simple or to be a loop. We demonstrate the improved stability of these new interpolants compared with PH quintics.

scholarly journals Sharp Lipschitz Constants for the Distance Ratio Metric

2015 ◽  
Vol 116 (1) ◽  
pp. 86 ◽  
Author(s):  
Slavko Simić ◽  
Matti Vuorinen ◽  
Gendi Wang
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Unit Disk ◽  
Complex Plane ◽  
Distance Ratio ◽  
Möbius Transformations ◽  
Lipschitz Constants ◽  
The Distance Ratio Metric ◽  
Mobius Transformations

We study expansion/contraction properties of some common classes of mappings of the Euclidean space $\mathsf{R}^n$, $n\ge 2$, with respect to the distance ratio metric. The first main case is the behavior of Möbius transformations of the unit ball in $\mathsf{R}^n$ onto itself. In the second main case we study the polynomials of the unit disk onto a subdomain of the complex plane. In both cases sharp Lipschitz constants are obtained.

scholarly journals Möbius geometry for hypersurfaces in S4

1995 ◽  
Vol 139 ◽  
pp. 1-20 ◽  
Author(s):  
Changping Wang
Keyword(s):  
Invariant System ◽  
Principal Curvatures ◽  
Möbius Transformations ◽  
Möbius Geometry ◽  
Mobius Transformations

Our purpose in this paper is to study Möbius geometry for those hypersurfaces in S4 which have different principal curvatures at each point. We will give a complete local Möbius invariant system for such hypersurface in S4 which determines the hypersurface up to Möbius transformations. And we will classify the so-called Möbius homogeneous hypersurfaces in S4.

MÖBIUS TRANSFORMATIONS IN QUANTUM AMPLITUDE AMPLIFICATION WITH GENERALIZED PHASES

2010 ◽  
Vol 08 (06) ◽  
pp. 923-935 ◽  
Author(s):  
CÉSAR BAUTISTA-RAMOS ◽  
NORA CASTILLO-TÉPOX
Keyword(s):  
Error Probability ◽  
Quantum Algorithm ◽  
Linear Transformations ◽  
Möbius Transformations ◽  
Good State ◽  
Quantum Amplitude ◽  
Initial States ◽  
Phase Angles ◽  
Amplitude Amplification ◽  
Mobius Transformations

The iteration of the operators employed in quantum amplitude amplification with generalized phases is analyzed by using elementary properties (geometric and algebraic) of the Möbius transformations (fractional linear transformations). It is shown that, for a given quantum algorithm without measurement, which produces a good state with probability a of success, if the phase angles φ and ϕ which mark the good and initial states respectively satisfy φ = ϕ with a small enough, then, for a number n of iterations with [Formula: see text] we get an error probability that is at most O(aϕ2).

scholarly journals Ellipses, near ellipses, and harmonic Möbius transformations

2005 ◽  
Vol 133 (9) ◽  
pp. 2705-2710 ◽  
Author(s):  
Martin Chuaqui ◽  
Peter Duren ◽  
Brad Osgood
Keyword(s):  
Möbius Transformations ◽  
Mobius Transformations
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