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| Keywords: Biholomorphism illustration.svg Illustration of a biholomorphism self-made with MATLAB tweaked in Inkscape 06 40 31 January 2008 UTC Oleg Alexandrov Source code MATLAB <source lang matlab > Illustration of a biholomorphim in this case w exp z function main N 11; num of grid points epsilon 0 1; displacement for each small diffeomorphism num_comp 10; number of times the diffeomorphism is composed with itself Sx linspace -1 1 N ; Sy linspace 0 pi/2 N ; X Y meshgrid Sx Sy ; graphing settings lw 3 5; KSmrq's colors blue 0 129 205/256; green 0 200 70/256; yellow 254 194 0/256; white 0 99 1 1 1; mycolor blue; start plotting figno 1; figure figno ; clf; mode 1; do_plot X Y lw figno mycolor mode saveas gcf 'Biholom1 eps' 'psc2' ; plot2svg 'Biholom1 svg' ; I sqrt -1 ; Z X+I Y; F exp Z ; XF real F ; YF imag F ; figno 2; mode 2; do_plot XF YF lw figno mycolor mode saveas gcf 'Biholom2 eps' 'psc2' ; plot2svg 'Biholom2 svg' ; function do_plot X Y lw figno mycolor mode figure figno ; clf; hold on; axis equal; axis off; minX min min X ; maxX max max X ; minY min min Y ; maxY max max Y ; small 0 2; if mode 1 else small 0 5 exp 1 small; end A minX-small/2; B maxX+small; C minY-small/2; D maxY+small; plot A B 0 0 'linewidth' lw 'color' black ; plot 0 0 C D 'linewidth' lw 'color' black ; M N size X ; for i 1 N plot X i Y i 'linewidth' lw 'color' mycolor ; plot X i Y i 'linewidth' lw 'color' mycolor ; end red 0 867 0 06 0 14; gray 0 2 1 1 1; arrow_size 0 07 max maxX maxY ; sharpness 20; arrow_type 1; tiny 0 01; scale_lw 0 8; arrow A 0 B 0 scale_lw lw arrow_size sharpness arrow_type gray ; arrow 0 C 0 D scale_lw lw arrow_size sharpness arrow_type gray ; fs 10 max maxX maxY ; myrad 0 009 max maxX maxY ; in the two pictures the text to be displayed is different if mode 1 smallx -0 03 max B D ; smally -0 12 max B D ; text minX+smallx smally sprintf ' d' minX 'fontsize' fs 'color' gray ; text maxX+smallx smally sprintf ' d' maxX 'fontsize' fs 'color' gray ; ball minX 0 myrad mycolor ; ball maxX 0 myrad mycolor ; ball 0 maxY myrad mycolor ; else smallx -0 01 max B D ; smally -0 12 max B D ; text minX+smallx smally 'e -1 ' 'fontsize' fs 'color' gray ; text maxX+smallx smally 'e' 'fontsize' fs 'color' gray ; ball exp -1 0 myrad mycolor ; ball exp 1 0 myrad mycolor ; ball 0 exp -1 myrad mycolor ; ball 0 exp 1 myrad mycolor ; end function ball x y radius color draw a ball of given uniform color Theta 0 0 1 2 pi; X radius cos Theta +x; Y radius sin Theta +y; H fill X Y color ; set H 'EdgeColor' color ; function arrow start stop thickness arrow_size sharpness arrow_type color Function arguments start stop start and end coordinates of arrow vectors of size 2 thickness thickness of arrow stick arrow_size the size of the two sides of the angle in this picture -> sharpness angle between the arrow stick and arrow side in degrees arrow_type 1 for filled arrow otherwise the arrow will be just two segments color arrow color a vector of length three with values in 0 1 convert to complex numbers i sqrt -1 ; start start 1 +i start 2 ; stop stop 1 +i stop 2 ; rotate_angle exp i pi sharpness/180 ; points making up the arrow tip besides the stop point point1 stop - arrow_size rotate_angle stop-start /abs stop-start ; point2 stop - arrow_size/rotate_angle stop-start /abs stop-start ; if arrow_type 1 filled arrow plot the stick but not till the end looks bad t 0 5 arrow_size cos pi sharpness/180 /abs stop-start ; stop1 t start+ 1-t stop; plot real start stop1 imag start stop1 'LineWidth' thickness 'Color' color ; fill the arrow H fill real stop point1 point2 imag stop point1 point2 color ; set H 'EdgeColor' 'none' else two-segment arrow plot real start stop imag start stop 'LineWidth' thickness 'Color' color ; plot real stop point1 imag stop point1 'LineWidth' thickness 'Color' color ; plot real stop point2 imag stop point2 'LineWidth' thickness 'Color' color ; end </source> diffeomorphisms Complex analysis Files by User Oleg Alexandrov from en wikipedia Images with Matlab source code | ||||
