Fractals/Computer graphic techniques/2D/transformation

From Wikibooks, open books for an open world
Jump to navigation Jump to search

Plane transformations

Names[edit | edit source]

  • transformations
  • projections
    • Map projection[1] ( in case of 3D to 2D graphic)
  • mappings

Theory[edit | edit source]

Mapping in 2D raster graphic is a complex function that maps from complex plane to complex plane:


It is implemented as point transformations. For instance, Mercator:[2]

function Spherical_mercator(x, y) {
  return [x, Math.log(Math.tan(Math.PI / 4 + y / 2))];
}

function transverseMercatorRaw(lambda, phi) {
  return [log(tan((halfPi + phi) / 2)), -lambda];
}

transverseMercatorRaw.invert = function(x, y) {
  return [-y, 2 * atan(exp(x)) - halfPi];
};

If one work with discrete geometry ( polygons and polylines) then the projecting is much harder to imlement. One have to balance accuracy and performance.[3]

classifications[edit | edit source]

Maps clasifications

  • object-based mappings ( Image objects are sets of connected pixels having the same integer value )
  • pixel based mappings


Transformations ( maps) in 2-D graphics

  • basic types
  • Composite Transformation [4]

Implementation[edit | edit source]

The matrix operations in homogeneous coordinates[edit | edit source]

types[edit | edit source]


Interesting maps[edit | edit source]


Examples:[5][6]

Plane inversion[edit | edit source]

Parabola inversion[edit | edit source]

Inversion of a parabola = cardioid

Mandelbrot set inversion[edit | edit source]

Inversion of lambda Mandelbrot set with different translations

Julia set inversion[edit | edit source]

  • f(z) = z^2+c convert to f(z) = z^2 + 1/c
  • inverting the polynomial formula Ax(1-x) in the circle of radius 1, resulting in the formula f(x)= x^2/A(1-x).[19]

basilica[edit | edit source]

  • normal basilica f(z) = z^2 - 1
  • inverted basilica
    • rational with degree = 2 : reversed basilica[20] : f(z) = z^2/(z^2 -1)


Conformal map[edit | edit source]

  • "Conformal maps preserve angles and take infinitesimal circles to infinitesimal circles. Non-conformal maps take infinitesimal circles to infinitesimal ellipses (or worse)." Claude Heiland-Allen


Conformal maps


Dictionary of Conformal Mapping by John H. Mathews, Russell W. Howell ( 2008)

cylindrical projection[edit | edit source]

The geometry for the normal (equatorial) tangent projections of the sphere to the cylinder.

Cylindrical projection ( or cylindric p. ) maps from sphere ( without poles) to cylinder [21][22]

Joukowsky transformation (map)[edit | edit source]

Description :

domain coloring or complex phase plots[edit | edit source]

Visualisation of complex functions

modular forms[edit | edit source]

References[edit | edit source]

  1. Map projection in wikipedia
  2. Transverse_Mercator_projection in wikipedia
  3. : Geographic projections, spherical shapes and spherical trigonometry in JavaScript
  4. geeksforgeeks : composite-transformation-in-2-d-graphics
  5. opentextbc.ca: nature of geographic information
  6. wolfram : ComplexFunctionsAppliedToASquare
  7. scikit-image.org docs: swirl
  8. Log-Polar Mapping by Alexandre Bernardino
  9. Weisstein, Eric W. "Parabolic Cylindrical Coordinates." From MathWorld--A Wolfram Web Resource
  10. Weisstein, Eric W. "Parabolic Coordinates." From MathWorld--A Wolfram Web Resource.
  11. Numerical approximation of conformal mappings by Bjørnar Steinnes Luteberget
  12. processing : transform2d
  13. Squares that Look Round: Transforming Spherical Images by Saul Schleimer Henry Segerman
  14. fractalforums inflection-mappings
  15. theinnerframe : playing-with-circular-images
  16. theinnerframe :squaring-the-circle
  17. riemman-for-anti-dummies
  18. fractalforums: fractal-on-sphere
  19. Video "Curling Tentacles" by Valannorton
  20. Iterated Monodromy Groups of Rational Mappings by Kuang, Zheng
  21. Spherical Projections (Stereographic and Cylindrical) Written by Paul Bourke
  22. Weisstein, Eric W. "Cylindrical Projection." From MathWorld--A Wolfram Web Resource.
  23. johndcook : joukowsky-transformation