Fractals/Computer graphic techniques/2D/transformation

Plane transformations Names transformations Geometric Image Transformations Geometrical transformations coordinate transformations projections Cartographic Map projectionMap projection in wikipedia ( in case of 3D to 2D graphic)observable: plot-projections mappings Theory Mapping in 2D raster graphic is a complex function that maps from complex plane to complex plane: It is implemented as point transformations. It is usually denoted as where is a point of complex plane ( input) is a point of image complex plane ( output = result) Example code For instance, Mercator:Transverse_Mercator_projection in wikipedia 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.: Geographic projections, spherical shapes and spherical trigonometry in JavaScript // https://github.com/adammaj1/Mandelbrot-Sets-Alternate-Parameter-Planes/blob/main/src/lcm/d.c // projection from p to c complex double map_parameter(const ParameterTypeT ParameterType, const complex double parameter){ complex double p; // plane transformation switch(ParameterType){ case c_identity :{p = parameter; break;} case c_inverted :{p = 1.0/parameter; break;} case c_parabola :{p = 0.25+ 1.0/parameter; break;} case c_Myrberg_type :{p = cf - 1.0/parameter; break;} case c_inverted_2_type :{p = -2.0 + 1.0/parameter; break;} case c_exp :{p = cf + cexp(parameter) ; break;} // here one can change cf to get different image case lambda_identity :{ p = parameter; break;} case lambda_inverted_type :{p = 1.0/parameter; break;} case lambda_inverted_1_type :{p =1.0+ 1.0/parameter; break;} default: {p = parameter;} } classifications Maps classifications 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 geeksforgeeks : composite-transformation-in-2-d-graphics Implementations Use of matrices: without matrices ( using functions) with matrices Coordinate with Cartesian coordinate with homogenous coordinate Matrix "Matrix multiplication is not commutative. The order of the transformations is vital– Rotation followed by translation isvery different from translation followed by rotation"2D transformations and homogeneous coordinates by Dr Nicolas Holzschuch University of Cape Town Implementations Imath is a C++ and python library of 2D and 3D vector, matrix, and math operations for computer graphics types zoom inversion unrolling (a closed curve ) Conformal maps applied to pictures Moebius transformation Exponential and Mercator map Affine transformations To represent affine transformations with matrices, we can use homogeneous coordinates. This means representing a 2-vector (x, y) as a 3-vector (x, y, 1), and similarly for higher dimensions. thumb|right|2D affine transformation matrix Transformation name Affine matrix Example Identity (transform to original image) File:Checkerboard identity.svg Translation File:Checkerboard identity.svg Reflection File:Checkerboard reflection.svg Scale File:Checkerboard scale.svg Rotate File:Checkerboard rotate.svg where θ π 6 30° Shear File:Checkerboard shear.svg The affine transforms are applicable to the registration process where two or more images are aligned (registered). An example of image registration is the generation of panoramic images that are the product of multiple images stitched together. scale or resize Scaling of object about origin by by a scale factor becomes: Scaling Uniform (maintains the object’s proportions as it scales): sx = sy non-uniform: sx != sy Resizing objects while maintaining a fixed centre point = Scaling object about their own center width' = width * sx height' = height * sy compute coordinate of corners from the center translation Complex transtlation is a mapTerr, David. "Complex Translation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. where Using this system, translation can be expressed with matrix multiplication. The functional form becomes: rotation For rotation by an angle θ counterclockwise (positive direction) about the origin the functional form is . Written in matrix form, this becomes:http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamics-fall-2009/lecture-notes/MIT16_07F09_Lec03.pdf March 2022 Similarly, for a rotation clockwise (negative direction) about the origin, the functional form is the matrix form is: These formulae assume that the x axis points right and the y axis points up. the counter-clockwise rotation matrix in normalised coordinate: C code without matrix /* C program to rotate an object by a given angle about a given point gcc r.c -Wall -Wextra -lm ./a.out */ #include <stdio.h> #include <math.h> #include <complex.h> // complex numbers : https://stackoverflow.com/questions/6418807/how-to-work-with-complex-numbers-in-c //The sin() function returns the value in the range of [-1, 1] /* https://stackoverflow.com/questions/2259476/rotating-a-point-about-another-point-2d First subtract the pivot point (cx,cy), then rotate it (counter clock-wise), then add the point again. */ complex double rotate_point(const complex double center, const double angle_in_radians, const complex double point ) { // translate point to center complex double translated_point = creal(point) - creal(center) + (cimag(point) - cimag(center))*I; // rotate point counter clock-wise by a given angle about a given pivot point ( center) double s = sin(angle_in_radians); double c = cos(angle_in_radians); complex double new_point = creal(translated_point) * c - cimag(translated_point) * s + (creal(translated_point) * s + cimag(translated_point) * c)*I; // translate point back new_point = creal(new_point) + creal(center) +(cimag(new_point) + cimag(center))*I; return new_point; } #define kMax 6 // center, angle_rad, point double examples[kMax][5] = { {50,-50, -0.7853982, 100,100 }, {50,-50, -0.7853982, 100,200 }, {50,-50, -0.7853982, 200,200 }, {0,0, 1.570796, 100,100}, {0,0, 1.570796, 150,200}, {0,0, 1.570796, 200,200} }; int main(void){ int k; complex double center ; double angle_r ; complex double point ; complex double rotated_point; for (k=0; k<kMax; ++k){ center = examples[k][0] + examples[k][1] * I ; angle_r = examples[k][2]; point = examples[k][3] + examples[k][4] * I ; rotated_point = rotate_point(center, angle_r, point ); fprintf(stdout, "point %f%+f*I rotated about %f%+f*I by %f radians is %f%+f*I \n", creal(point), cimag(point), creal(center), cimag(center), angle_r, creal(rotated_point), cimag(rotated_point)); } return 0; } Output: point 100.000000+100.000000*I rotated about 50.000000-50.000000*I by -0.785398 radians is 191.421359+20.710673*I point 100.000000+200.000000*I rotated about 50.000000-50.000000*I by -0.785398 radians is 262.132040+91.421348*I point 200.000000+200.000000*I rotated about 50.000000-50.000000*I by -0.785398 radians is 332.842715+20.710668*I point 100.000000+100.000000*I rotated about 0.000000+0.000000*I by 1.570796 radians is -99.999967+100.000033*I point 150.000000+200.000000*I rotated about 0.000000+0.000000*I by 1.570796 radians is -199.999951+150.000065*I point 200.000000+200.000000*I rotated about 0.000000+0.000000*I by 1.570796 radians is -199.999935+200.000065*I Other Interesting maps Examples:opentextbc.ca: nature of geographic informationwolfram : ComplexFunctionsAppliedToASquare Jason Davies: maps barrel-distortion swirlscikit-image.org docs: swirl log-polar map Log-Polar Mapping by Alexandre Bernardino Parabolic Cylindrical Coordinates Weisstein, Eric W. "Parabolic Cylindrical Coordinates." From MathWorld--A Wolfram Web Resource Parabolic CoordinatesWeisstein, Eric W. "Parabolic Coordinates." From MathWorld--A Wolfram Web Resource. Numerical approximation of conformal mappings Numerical approximation of conformal mappings by Bjørnar Steinnes Luteberget descriptionprocessing : transform2dSquares that Look Round: Transforming Spherical Images by Saul Schleimer Henry Segerman inflection fractalforums inflection-mappings Altmetric: 25Citations: 1More detail Article | OPEN Exponential Sensitivity and its Cost in Quantum Physics by András Gilyén, Tamás Kiss & Igor Jex Blaschke product theinnerframe : playing-with-circular-images squaring the circletheinnerframe :squaring-the-circle elliptic by Claude Heiland-Allen by Paul Bourke A cartographic projection is a nonlinear transformation in two dimensions map projections and coordinate transformations to spherefractalforums: fractal-on-sphere Mandelbrot set projected on a shrinking Riemann-sphere by Arneauxtje Mandelbrot's Elephant Valley (Short Version) Timothy Chase Mandelbrot Buds and Branches Timothy Chase Timothy Chase Mandelbrot Zoom on a Sphere video by craftvid : "This a 300-trillion time zoom-in on the Mandelbrot set. The images are set on a Spherical "mobius" projection, meant to be wrapped onto a spherical surface. The image zooms in on the front center of the Sphere, while fading away on the back of the sphere." Cat Eye : Inverted Mandelbrot Set by denis archambaud Inverse & Potens Mandelbrot Set by Jens-Peter Christensen Z² + Sin(Cˉᵐ+phase) by Jens-Peter Christensen snibgo's ImageMagick pages: Barrels and pincushions snibgo's ImageMagick pages: Scale, rotate and translate in 3D see Cube earth Conformal map "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 "A complex function is conformal at infinity if the function is conformal at 0. This equivalent to the usual definition of conformal in terms of leaving angles unchanged in size and sense (clockwise/anticlockwise) if you think of as being conformal at infinity if it leaves angles unchanged at the "North Pole" of the Riemann sphere."math.stackexchange question: conformal-mappings-of-riemann-sphere Conformal maps henry seg: spherical_image_editing in python newconformist: Conformally map the hyperbolic plane to arbitrary shapes by Zeno Rogue examples of conformal maps applied to pictures Moebius transformation An Overview of Conformal Geometry Processing (2017) by Keenan Crane Discrete Conformal Deformation: Algorithm and Experiments by Jian Sun, Tianqi Wu, Xianfeng Gu, Feng Luo Conformal map Poincaré Hyperbolic Disk Conformal Mapping Recipe by bugman123 Visualizing Conformal Maps in Python by MJ Gruber Conformal Animation by denis archambaud ConformalMaps is a Julia package for approximating the Riemann map from a simply connected planar domain to a disk. It uses the zipper algorithm as described in Convergence of the zipper algorithm for conformal mapping by Don Marshall and Steffen Rohde. sswatson: ConformalMaps Dictionary of Conformal Mapping by John H. Mathews, Russell W. Howell ( 2008) part 1 part 2 part 3 part 4 Cartographic Map projection cylindrical projection thumbnail|right|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 Spherical Projections (Stereographic and Cylindrical) Written by Paul Bourke Weisstein, Eric W. "Cylindrical Projection." From MathWorld--A Wolfram Web Resource. Joukowsky transformation (map) Description : John D Cookjohndcook : joukowsky-transformation domain coloring or complex phase plots Visualisation of complex functions The 5 ways to visualize complex functions commons : Category:Complex_functions Complex Mappings by Rand Asswad Poincaré Hyperbolic Disk Conformal Mapping Recipe by Paul Nylander david lowry-duda : phase_mag_plot/ modular forms MODULAR FORMS by David Lowry-Duda The L-functions and modular forms database (LMFDB) Riemann mappings The Riemann Mapping Theorem by Christopher J. Bishop Riemman for anti-dummies by ©2015 LYM Canada Maping component to the unit disk ( Riemann map ): Multiplier map and internal ray on the parameter plane on the dynamic plane Boettcher map, complex potential and external ray on the parameter plane parameter ray = field lines complex potential , external angle References immersive linear algebra by J. Ström, K. Åström, and T. Akenine-Möller