Fractals/moebius

thumbnail|right|Riemann sphere Möbius transformation is an example of plane transformation definition A Möbius transformation Möbius transformation in wikipediaMoebius transformation animated GIFs by Fritz MuellerMöbius Transforms App by (c) Robert Woodley, 2016-2017.Transformations of the projective line of extended complex plane is a rational function f of the form of one complex variable z. Here the coefficients a, b, c, d and the result w are complex numbers satisfying Representation or form function matrix Function Matrix In matrix form by using homogeneous coordinates:oeis.org : Moebius transformation Matrix M is a square 2x2 invertible matrixMatrix in wikipedia Examples Transforms using Complex Numbers Moebius transformationSquares that Look Round: Transforming Spherical Images by Saul Schleimer and Henry Segerman Paper: Squares that Look Round: Transforming Spherical Images by Saul Schleimer Henry Segerman and Shadertoy : Spherical Images by soma_arc and python code Henry Segerman Moebius transformation in mandelbrot-graphics library Mathematics of Kleinian Group Fractals by hiddendimension Generalized circles and Möbius transformations by Stéphane Laurent Projective Transformation and Mobius Transformation by Tadao Ito simple The following simple transformations are also Möbius transformations: is an identity is a translation is a combination of a homothety and a rotation. If then it is a rotation if then it is a homothety inversion and reflection with respect to the real axis) How to ...? eigenvalue and eigenvector A number and a non-zero vector satisfying are called an eigenvalue and an eigenvector of matrix M, respectively. For dimensions 2 formulas involving radicals exist that can be used to find the eigenvalues. The eigenvalues can be found by using the quadratic formula: diagonalization a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. In other words all off-diagonal elements are zero in a diagonal matrix. the main diagonal of a matrix is the list of entries where , here the diagonalization of a matrix M gives a pair of matrices: D, P such that:How to diagonalize a matrix ? in wikipedia D is diagonal (all elements not on the diagonal are 0) For 2x2 matrices there is a simple closed form solutioninterpolating moebius transformations Product with a scalar If A is a matrix and c a scalar, then the matrices and are obtained by left or right multiplying all entries of A by c. trace The trace of a square 2x2 matrix is the sum of its diagonal entries So determinant determinant of matrix inverse thumb|right|Mapping of generalised circles by inversion w=Sz=1/z (primitive Möbius-transform). Straight lines are mapped to circles/lines. Inverse Möbius transformationinverse of a matrix by Bruce Simmons . interpolation How to smootly interpolate between möbius transformations?mathoverflow question: ow-to-smootly-interpolate-between-moebius-transformationsinterpolating moebius transformations by Claude Heiland-Allen If you have two Möbius transformations represented as: where coefficients are complex numbers Is it possible to derive a third function , where and , which "smoothly" interpolates between the transformations represented by and ? The solution: Specifying a transformation by three points Given a set of three distinct points z1, z2, z3 on the one Riemann sphere ( let's call it z-sphere) and a second set of distinct points w1, w2, w3 on the second sphere ( w-sphere) , there exists precisely one Möbius transformation f(z) with : for i=1,2,3 Mapping to 0, 1, infinity The Möbius transformation with an explicit formula :Triple transitivity by David J Wright 2004-12-04 maps : z1 to w1= 0 z2 to w2= 1 z3 to w3= ∞ the unit circle to the real axis - first method Let's choose 3 z points on a circle : z1= -1 z2= i z3= 1 then the Möbius transformation will be : Knowing that :math.stackexchange questions : how-to-do-this-transformation-of-complex-rational-function one can simplify this to : In Maxima CAS one can do it : (%i1) rectform((z+1)*(%i-1)/((z-1)*(%i+1))); (%o1) (%i*(z+1))/(z−1) where coefficients of the general form are : so inverse function can be computed using general form : Lets check it using Maxima CAS : (%i3) fi(w):=(-%i-w)/(%i-w); (%o3) fi(w):=−%i−w/%i−w (%i4) fi(0); (%o4) −1 (%i5) fi(1); (%o5) −%i−1/%i−1 (%i6) rectform(%); (%o6) %i Find how to compute it without symbolic computation program (CAS) : (%i3) fi(w):=(-%i-w)/(%i-w); (%o3) fi(w):=−%i−w/%i−w (%i8) z:x+y*%i; (%o8) %i*y+x (%i9) z1:fi(w); (%o9) (−%i*y−x−%i)/(−%i*y−x+%i) (%i10) realpart(z1); (%o10) ((−y−1)*(1−y))/((1−y)^2+x^2)+x^2/((1−y)^2+x^2) (%i11) imagpart(z1); (%o11) (x*(1−y))/((1−y)^2+x^2)−(x*(−y−1))/((1−y)^2+x^2) (%i13) ratsimp(realpart(z1)); (%o13) (y^2+x^2−1)/(y^2−2*y+x^2+1) (%i14) ratsimp(imagpart(z1)); (%o14) (2*x)/(y^2−2*y+x^2+1) thumbnail|right|2 steps of unrolling the main cardioid of Mandelbrot set: Moebius map and conformal map right|thumb|Unrolled main cardioid of Mandelbrot set for periods 7-13 So using notation : one gets : It can be used for unrolling the Mandelbrot set components Stretching cusps by Claude Heiland-Allen the unit circle to the real axis - second method Function : sends the unit circle to the real axis : z=1 to w=0 z=i to w=1 z=-1 to Mapping to the imaginary axis Function sends the unit circle to the imaginary axis.math.stackexchange questions: what-mobius-transformation-maps-the-unit-circle-z-z-1-to-the-real-axis/335061#335061 visualisations mobius revealed by Mark McClure live demo by Tim Hutton Iterating the Möbius transformation by Tim Hutton References moebius-transformation by Claude Heiland-Allen