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Phase Portraits of Hyperbolic Geometry

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Phase Portraits of Hyperbolic Geometry Phase Plotting Differential Geometry Hyperbolic Geometry Conclusion Phase Portraits of Hyperbolic Geometry Scott B. Lindstrom and Paul Vrbik CARMA University of Newcastle Jonathan M. Borwein Commemorative Conference, September 2016 Last Revised September 26, 2017 https://carma.newcastle.edu.au/scott/ Phase Plotting Differential Geometry Hyperbolic Geometry Conclusion OutlineI 1 Phase Plotting The Basic Idea Some Examples Modulus Axis History 2 Differential Geometry The Basics What Has Been Done 3 Hyperbolic Geometry Pseudosphere Poincar´eDisc Beltrami Upper Half Sphere and Klein Disc Dini's Surface 4 Conclusion Phase Plotting Differential Geometry Hyperbolic Geometry Conclusion OutlineII Dedication References Phase Plotting The Basic Idea Differential Geometry Some Examples Hyperbolic Geometry Modulus Axis Conclusion History Complex Phase Portraits (·) |·| f (z) f 0 z Figure: Construction of a complex phase portrait Phase Plotting The Basic Idea Differential Geometry Some Examples Hyperbolic Geometry Modulus Axis Conclusion History The Basics Phase plotting is a way of visualizing complex functions f : C ! C: iθ iθ Where f (r1e 1 ) = r2e 2 , we plot the domain space, coloring points according to argument of image θ2 Top right: z ! z. Bottom right: z ! z3. Phase Plotting The Basic Idea Differential Geometry Some Examples Hyperbolic Geometry Modulus Axis Conclusion History Some Examples Figure: Left to right: z · ez ; W (z); ζ(z). Phase Plotting The Basic Idea Differential Geometry Some Examples Hyperbolic Geometry Modulus Axis Conclusion History Recapturing the Modulus We can also plot in 3d to recapture the modulus information. iθ iθ Let f (r1e 1 ) = r2e 2 Again we plot over the domain space, coloring points according to argument of image θ2 We also give them vertical height corresponding to Figure: Phase portrait with modulus their modulus r . 2 included: z ! z. Phase Plotting The Basic Idea Differential Geometry Some Examples Hyperbolic Geometry Modulus Axis Conclusion History Recapturing the Modulus Figure: 3d phase portraitp for Figure: 3d phase portrait for z ! z2. inversion in circle radius 2 centered at −i Phase Plotting The Basic Idea Differential Geometry Some Examples Hyperbolic Geometry Modulus Axis Conclusion History History Phase plotting is a relatively new tool. Recent attention Elias Wegert's \Visual Complex Functions" published in 2013 [7] \Complex Beauties" annual calendar (of which Jonathan Borwein was quite fond) [4] Figure: Left: Elias Wegert's \Visual Complex Wegert's Matlab code is Functions." Right: Right: \Moment function of a available for download on 4-step Pplanar random walk" by Jonathan M. his site. Borwein and Armin Straub from 2016 Complex Beauties calendar. Phase Plotting Differential Geometry The Basics Hyperbolic Geometry What Has Been Done Conclusion Differential Geometry Conformal Mappings are mappings which preserve the angles at which lines meet (and signs thereof) Direct Motions are mappings such that the distance between points is equal to the distance between their images. Parallel axiom: for a line L and point p there exists exactly one line through p which doesn't intersect L. Geometries which do not obey the parallel axiom: Spherical Geometry (no lines through p) Hyperbolic Geometry (more than one line through p) Both have constant curvature (intrinsic property) The type of geometry determines how many types of direct motions there are. This is because conformal maps can be expressed as compositions of reflections across lines. Phase Plotting Differential Geometry The Basics Hyperbolic Geometry What Has Been Done Conclusion What Has Been Done Phase plotting on the Riemann sphere has already been employed by Wegert. Figure: Left: The construction of the Riemann Sphere with stereographic projection. Right: phase plotting for a M¨obiustransformation (direct motion) on the Riemann Sphere. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Pseudosphere Figure: The conformal map from the pseudosphere to the hyperbolic upper half plane. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Pseudosphere The map is between the pseudosphere and a small area of the upper half plane If we colored according to the planar phase plotting rules, problems: Fewer colors for visualization Coloring would be tied to Euclidean geometry rather than Hyperbolic geometry, warping perspective. Figure: Colors change along Unable to tell if points mapped tractrices rather than out of visible region. Euclidean subspaces. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface What Has Been Done Figure: Left: naive plotting on pseudosphere. Right: assigning unique colors to tractrices. Solution: defined a new coloring scheme unique to hyperbolic space. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface But what to do about points mapped out of viewable region? Solution: color them black CP : P ! [0; 2π] [ fblackg ( Re ◦ f ◦ T (z) Re ◦ f ◦ T (z) 2 [0; 2π] z 7! P P : black otherwise Figure: An h-rotation plotted on a pseudosphere. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Figure: A rotation on hyperbolic space plotted in the Poincar´eupper half plane and on the pseudosphere. We can use the same plot in the upper half plane for comparison. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Poincar´eDisc −1 0 −i K Figure: Anticonformal map IK Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Poincar´eDisc Asymptotic T −1 D f Re 0 0 TD Figure: Construction of the coloring map CD;1 for uniquely coloring asymptotic parallel lines. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Poincar´eDisc Asymptotic T −1 D f 0 Re 0 TD Theorem 1 Let f : C+ ! C+ represent a function on hyperbolic space. The function CD;1 : D ! [0; 2π) z 7! (arg ◦T ◦ Re ◦ f ◦ T −1)(z) D D assigns a unique color to the preimage of each tractrix line. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Poincar´eAsymptotic A true phase portrait! Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Poincar´eAsymptotic Figure: A limit rotation on hyperbolic space. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Poincar´eUltra Parallel T −1 −1 D T f D M 0 ·i 0 TD Figure: Construction of a phase plot CP;2 uniquely coloring ultra-parallel lines on the Poincar´eDisc. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Poincar´eUltra Parallel T −1 T −1 D D f M 0 ·i 0 TD Theorem 2 Let f : C+ ! C+ represent a function on hyperbolic space. The function CD;2 : D ! [0; 2π) z 7! (2 · arg ◦ i · T ◦ M ◦ f ◦ T −1)(z) D D assigns a unique color to the preimage of each h-line corresponding to a circle in C+ centered at 0. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Poincar´eDisc Ultra Parallel Figure: The phase plot CP;2 is also a true phase plot. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Multiple views: motivation Figure: A translation on hyperbolic space. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Multiple views: motivation Figure: A translation on hyperbolic space. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Multiple views: motivation Figure: Translations on hyperbolic space. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Combining Views Figure: Colored landscapes on D. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Combining Views Figure: Colored contour plots on D. Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Beltrami Upper Half Sphere and Klein Disc Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion Dini's Surface Klein Disc Figure: Phase plotting on K Phase Plotting Pseudosphere Differential Geometry Poincar´eDisc Hyperbolic Geometry Beltrami Upper Half Sphere and Klein Disc Conclusion
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