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Analytical Methods for Squaring the Disc

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https://www.slideshare.net/chamb3rlain/mappings-for-squaring-the-circular-disc

see accompanying paper in http://arxiv.org/abs/1509.06344
presented in Seoul ICM 2014

Abstract:
We present and analyze several old and new methods for mapping the disc to a square. In particular, we present analytical expressions for mapping each point (u,v) inside a circular disc to a point (x, y) inside a square region. Ideally, we want the mapping to be smooth and invertible. In addition, we put emphasis on mappings with desirable properties. These include conformal, equiareal, and radially-constrained mappings. Finally, we present applications of such mappings to logo design, panoramic photography, and hyperbolic art.

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Analytical Methods for Squaring the Disc

  1. 1. SEOUL ICM 2014 Analytical Methods for seout . . *-. Q’ 58% Squaring the DISC Chamberlain Fong chamberlain@alum. berkeley. edu g - I We present and analyze several old and new methods for mapping the disc to a square. In particular, we present analytical expressions for mapping each point (u, v) inside a circular disc to a point (x, y) inside a square region. Ideally, we want the mapping to be smooth and invertible. In addition, we put emphasis on mappings with desirable properties. These include conformal, equiareal, and radia| |y—constrained mappings. Finally, we present applications of such mappings to cartography, panoramic photography, and hyperbolic art.
  2. 2. How to map every point in the unit disc to a square region? (X, Y) : f (U, V) mapping f: ]R2—> [R232 (u, v)= f_1(x, y) / N T7 LM domain: {(u, V)I u? ‘ + V2 3 1} range: [—1,1]x[-1,1]
  3. 3. fleyfrgzéle properzfey of 122g2,z9,z9f19gZS° some jargon from differential geometry angle_pr, eser. vin9 area—preservIng (no shape distortions) ( "° Size dl51'°'"fi°"5 )
  4. 4. Desirable Properties (continued) / < > ¢ V points only move radially _ _ from the center of the disc Tadlal COT1StI'a1nt during the mapping process i. e. we impose that the angle that point (u, v) makes with the x—axis must be the same angle as that of point (x, y) during the mapping
  5. 5. mathematical details of radial constraint L! 'l. =5""'l3.“"' unit circle inscribed inside _ V square (x, y) "". c"L- , . I» , =~. ‘ 9%. :-! :’~" if 6 is the angle between point (u, v) and the x—axis, these equations must hold u x - V 3’ C056: zj s1nl9=j— , /u2+v2 , /x2+y2 / U2-I-172 — x/ X2+y2 Meanwhile, each point (u, v) can be parameterized by its polar coordinates u= ;-C059 v= rsin9
  6. 6. mapping #1: Naive Stretching x2 just linearly , /x2 + y2 right s°°“°“ stretch from y/ uz + U2 right section 9; y . . u _ rim to rim — ‘/11? + I22 top section (I362 + 312 top Seem“ X = —x2 -, /u2 + 172 left section I: —-u . 7 /1t2 + 172 bottom section ffz + yz left section bottom section forward equations inverse U = Z , /uz + 1,2 right section , /uz + 1,2 top section V 7 / U2 + I72 left section —‘/ u2 + 122 bottom section right section 4., X 2 0 and X 2 | y| top section 4-» y 2 0 and | x| 3 y Ieitsection 4-> X S 0 and | x| 2 | y| bottom section H Y S 0 and IXI S IYI
  7. 7. mapping #3 Squircle Mapping forward equations: x = S‘gn(u) uz + v2 — . /(u2 + v2)(u2 + 172 — 4142172) vx/7 where sgn(x) is the _ Sgn(v) signum function 2+ 2_ 2+2 2+ 2_422 u‘/7 u v , /(u v)(u 1: uv) inverse equations: u_ , /x2+y2 y / x2 _, _y2 _x2y2 v— A
  8. 8. Fernandez-Ci‘uas't1's SC[”LIlT; C1€ a smooth algebraic curve in IRZ 2 x2+y2_S_2x2y2:r2 OSSS1 r . / 1 s= o.95 when s= O, the squircle becomes ' Zr ' 7 a circle with radius r when 5:1 , the squircle becomes a square with side length Zr
  9. 9. key idea: map concentric circles to squircles It Osrsl continuum of concentric continuum of concentric circles inside the disc squircles inside the square u2+v2=r2 x2+y2—x2y2=r2 2 r: x/ u2+v2 = (/ x2+y2—x2y2
  10. 10. mapping #3: SCHWARZ-CHRISTOFFEL CONFORMAL sou using Complex Analysis to map the disc to a polygon conformally Elwin Christoffel Hermann Schwarz 1367 {C E LC: In1( > 0} 1869 general “0 = /C polygon 2:}: ): C dw : /517 / square (C win’? -1) ( H ' / )
  11. 11. fundamental mapping in the complex plane / w'1:u_1+V1i z1=x1+y1i (0:1) (L0) (0: 0) (‘W (-2Ke.0) (‘W (‘Ker‘Ke) Cemered “nil Circle off—centered & tilted square 0" Ihe C°mP"’-X Plane on the negative side of the real axis of the complex plane 1 W1 = Cn(Z1»—) ‘/2 unaligned mapping equation Ke 3 1-854
  12. 12. ful| y—aligned mapping equations forward equafions: ( in situ canonical mapping space ) i; I= , %,l11 :1 F(C°s*<, %2li 1:Ii: I> er Lai inverse equations: ii = e is. :1 w< %i: : ii iii + it] is where K3 is the complete elliptic integral of the 1st kind % 1 Ke_£ ‘“ z 1.854 F is the Legendre elliptic integral of the 1st kind cn is a Jacobi elliptic function
  13. 13. mapping #4: Elliptical Arc Mapping forward equations: 1 1 x: E l2+2/ §u+u2—i22 — E l2—2/ §u+u2—v2 1 1 y: E l2+2/ §v—u2-+122 -5 l2—2/ §v—u2+i22 inverse ecIuations= also squircle—based but not radially-
  14. 14. key idea; map elliptical arcs inside the disc to axis—aligned lines inside the square ellipses get more eccentric as x or y A . approaches zero . u2 122 vertical line of constant x gets mapped from an arc in F + 2 — x2 = 1 ellipses get more i C : circular as x or y approaches -i-1 2 horizontal line of constant y gets mapped from an arc in 2 u 2 V2 — 3’ 3’
  15. 15. I’ i i ‘fl l I , J I , ‘C “J 4’ ,7‘ K I i i ‘-e , . ' - l how many corporatem logos can you identify below? I ‘ ‘ -1 l ' ‘Z‘'«': :L’‘{$‘i I 1 I is ‘(i"*'') E Y l 1 1 C is L ‘m l I I
  16. 16. ci; >pllcCcg: lloI'I5:C lcii"lo, <35i‘Cc»_pl*iLi: ,- Lambert Azimuthal 2 mapped to a 22 Equal Area Projection square of the world
  17. 17. applications: Panoramic Photography 4. 2. —, , . ..' / /"/ ’ . ,_ .34»/ ‘- ' » ‘ _ . _" 3 -:7 r -—- There is a « sphere of light that surrounds us that we can capture photographically into a spherical panorama widest possible 1* photograph from a single viewpoint project the spherical panorama to a circular image then convert this disc to a square photograph
  18. 18. example } spherical R I ; i.2ill: C:'f"i? 'lll ii Pam mm“ ‘ , ;»‘~”: I ” (equirectangular ‘L, /,4 projection) azimuthal projection
  19. 19. - . :'—-I applicaiions. ;, :i, l-i, _. «, : .)0 ic; . _=, i the Poincare d is. ’-< model of non-Euclidean hyperbolic qeometry M. C. Escher's circle Limit Iv Circle Limit IV
  20. 20. i. i[ ii ’ 1) naive stretching 2) squircle mapping ' '“0dl0”Y — radially C°“5l'"0l“2d constrained — circular circular contours contours map to map to squares squircles 3) 5chwarz—Christoffel 4) €”iPTlCOl 0|"C m0PPll’|9 - conformal - not radially 2/ requires constrained C0mPl€>< - circular GnGlYSlS <3 contours elliptic map to functions squircles
  21. 21. SEOUL $47‘ ICM ° , .. 2014 References R. Bedard, 2009. Squaring the Thumbsticks (blog). http: //theinstructionlimit. com/ squaring-the-thumbsticks M. Fernandez-Guasti, Analytic Geometry of Some Rectilinear Figures, International Journal of Mathematical Education in Science and Technology 23, pp. 895-901. 1992. C. Fong, Revolvable Indoor Panoramas Using a Rectified Azimuthal Projection. http: //arxiv. org/ abs/12062068 C. Fong, An Indoor Alternative to Stereographic Spherical Panoramas. Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture. P. Nowell, Mapping a Square to a Circle httpI/ /mathproofs. blogspot. com/2005/07/mapping-square—to—circle. html P. Shirley, K. Chiu, A Low Distortion Map Between Disk and Square. Journal of Graphics Tools, volume 2 number 3. pp.45—52. 1997. Wikipedia authors, Schwarz-Christoffel Mapping http: //en. wikipedia. org/ wiki/ Schwarz-Christoffel_mapping SEOUL ICM 2014

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