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.

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 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.

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 The Geometrical Basis of 𝒫𝒯 Symmetry

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 494 ◽  
Author(s):  
Luis Sánchez-Soto ◽  
Juan Monzón
Keyword(s):  
Transfer Matrix ◽  
Complex Plane ◽  
Physical Meaning ◽  
Pt Symmetry ◽  
Möbius Transformation ◽  
Invariant System ◽  
Complex Conjugation ◽  
Basic Properties ◽  
Mobius Transformation

We reelaborate on the basic properties of PT symmetry from a geometrical perspective. The transfer matrix associated with these systems induces a Möbius transformation in the complex plane. The trace of this matrix classifies the actions into three types that represent rotations, translations, and parallel displacements. We find that a PT invariant system can be pictured as a complex conjugation followed by an inversion in a circle. We elucidate the physical meaning of these geometrical operations and link them with measurable properties of the system.

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 Visualization of Mandelbrot and Julia Sets of Möbius Transformations

2021 ◽  
Vol 5 (3) ◽  
pp. 73
Author(s):  
Leah K. Mork ◽  
Darin J. Ulness
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Mandelbrot Set ◽  
Iteration Step ◽  
Möbius Transformation ◽  
Möbius Transformations ◽  
Fractal Sets ◽  
Mandelbrot Sets ◽  
Möbius Transforms ◽  
Mobius Transformations

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.

Power Dependent Impedance Measurement Exploiting an Oscilloscope and Möbius Transformation

2017 ◽  
Vol E100.C (10) ◽  
pp. 918-923
Author(s):  
Sonshu SAKIHARA ◽  
Masaru TAKANA ◽  
Naoki SAKAI ◽  
Takashi OHIRA
Keyword(s):  
Impedance Measurement ◽  
Möbius Transformation ◽  
Mobius Transformation
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