Fractals/Computer graphic techniques/2D/exp
Disambiguation[edit  edit source]
Exponential mapping of the plane is:
 is exponential transformation of the plane
Exponential mapping of the plane is not
 the Julia set or Mandelbrot set of the exponential function ( exponentia map)
 Logarithmic scale on one axis^{[1]}
 Logarithmic scale on both axes: Loglog scale plot^{[2]}
 polar azimuthal equidistant projection ( Exponential map in Riemannian geometry)^{[3]}
 The exponential operator, anamorphosis operator which can be applied to grayscale images.^{[4]}
Compare exponential function by different input
 single number ( 0D space) gives natural exponent of the number
 number line ( 1D space). Exponential scale is not used. Logarithmic scale is used for exponential data. It gives a linear function.^{[5]}
 plane ( 2D space) gives exponential mapping
number[edit  edit source]
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number x to the base b is the exponent to which b must be raised, to produce x
is equivalent to
if b is a positive real number. (If b is not a positive real number, both exponentiation and logarithm can be defined but may take several values, which makes definitions much more complicated.)^{[6]}

a logarithm function and an exponential function

Plots of logarithm functions, with three commonly used bases. The special points log_{b} b = 1 are indicated by dotted lines, and all curves intersect in log_{b} 1 = 0

The graph of the logarithm base 2 crosses the xaxis at x = 1 and passes through the points (2, 1), (4, 2), and (8, 3), depicting, e.g., log_{2}(8) = 3 and 2^{3} = 8. The graph gets arbitrarily close to the yaxis, but does not meet it
scale[edit  edit source]
A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a scale is nonlinear: the numbers 10 and 20, and 60 and 70, are not the same distance apart on a log scale. Rather, the numbers 10 and 100, and 60 and 600 are equally spaced. Thus moving a unit of distance along the scale means the number has been multiplied by 10 (or some other fixed factor). Often exponential growth curves are displayed on a log scale, otherwise they would increase too quickly to fit within a small graph. Another way to think about it is that the number of digits of the data grows at a constant rate. For example, the numbers 10, 100, 1000, and 10000 are equally spaced on a log scale, because their numbers of digits is going up by 1 each time: 2, 3, 4, and 5 digits. In this way, adding two digits multiplies the quantity measured on the log scale by a factor of 100.
Logarithmic scale
 on 1 axis = semilog plot^{[7]}
 linearlog plot
 loglinear plot
 on both axes = loglog plot^{[8]}

LinLin scale

LogLin scale

LinLog scale

LogLog scale
Logarithmic scale is used for exponential data. It gives a linear function.^{[9]}^{[10]}
mapping[edit  edit source]
 GRAPHICS FOR COMPLEX ANALYSIS by Douglas N. Arnold:
 "the complex exponential maps the infinite open strip bounded by the horizontal lines through +/ pi i onetoone onto the plane minus the negative real axis.
 The lines of constant real part are mapped to circles, and lines of constant imaginary part to rays from the origin.
 In the animation we view a rectangle in the strip rather than the entire strip, so the region covered is an annulus minus the negative real axis.
 The inner boundary of the annulus is so close to the origin as to be barely visible.
 We also make the strip a bit thinner than 2 pi, so that the annulus does not quite close up."
 the complex exponential map takes the rectangle Re(z) ≤ a, Im(z) ≤ π onto the annulus
 Composition of complex Mappings Viewer by Paul Falstad  the composition of selected mapping functions: g(f(z)). One have to increase grid size to have tha same effect ( rectangle to annulus mapping)
 Complex exponential map by Siamak on geogebra:
 vertical axis is transformed to the circle on the wplane
 horizontal axis is is transformed to the radii on the wplane
 Images of a square grid of size [3,3]x[3,3] under the (conformal) exponential map givers almost full annulus^{[11]}
 The complex exponential function mapping biholomorphically a rectangle [1,1]x[0 ,pi/2] to a quarterannulus ^{[12]}
 virtual math museum: ConformalMaps, complex exponential function:"a Cartesian Grid is mapped “conformally” (i.e., preserving angles) to a Polar Grid: the parallels to the real axis are mapped to radial lines, and segments of length 2π that are parallel to the imaginary axis are mapped to circles around 0."
exp(x + i · y) = exp(x) · exp(i · y) = exp(x) · cos(y) + i · exp(x) · sin(y)

real part

imaginary part

modulus = cabs

main argument = carg

real part

imaginary part



mercator projection vs rotation

the exponential mapping from the tangent space to the sphere
Name[edit  edit source]
 exponential mapping
 the exponential map^{[13]}^{[14]}
 logpolar mapping ( Logpolar coordinates )
 Mercator ( it is not good name^{[15]} )
 Mercator map
 Mercator projection ^{[16]}^{[17]}^{[18]}
 polar projection
 logarithmic ^{[19]}
 logarithmic projection around a point c0: z> (log(zc0)
 log(z)MandelbrotZooms^{[20]} = side scrolling ( side scrolling fractal zoom )^{[21]}^{[22]}^{[23]}
General names:
 graphical projection
 geometric transformation
Description[edit  edit source]
Informal description[edit  edit source]
 The exponential mapping transforms the entire complex plane into a strip that has unlimited length along the real axis, and a width of 2π along the imaginary axis. Exponential Map from the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872022
 " The idea is to focus on a point in or near the Mandelbrot set, and create an image where one direction is the logarithm of the distance and the other one the angle. The result is very much like the astroph/031571 map^{[24]} and theXKCD cartoon versions.^{[25]} This map projection is conformal, so it does not distort local angles, and objects are usually recognizable on all scales." Anders Sandberg^{[26]}
 Images of the Mandelbrot set with a logarithmic projection around a point c0: z> (log(zc0), arg(zc0)). Anders Sandberg ^{[27]}
 "it is a log map toward the target point (or, as some might say, a Mercator projection with the target point as South pole and complex ∞ as North pole); horizontally it is periodic and I have placed two periods side to side, whereas vertically it extends to infinity at the top and at the bottom, which corresponds to zooming infinitely far out or in, at a factor of exp(2π)≈535.5 for every size of a horizontal period. Horizontal lines (“parallels”) on the log map correspond to concentric circles around the target point, and vertical lines to radii emanating from it; and the anamorphosis preserves angles." David Madore ^{[28]}
 "The coordinate system is such that the angular component goes to the yaxis, and the radius goes to the xaxis of the resulting image. In addition, the xaxis (radius) is normalized with expfunction so that angles are preserved in this mapping. " Mika Seppä^{[29]}
 log : "To illustrate the complexity of the boundary of the Mandelbrot set, Figure 8 renders the image of dM under the transformation log(z  c) for a certain c in the boudary of Mandelbrot set. Note the cusp on the main cardioid in the upper right; looking to the left in the figure corresponds to zooming in towards the point c. (Namely, c = 0.39054087...  0.58678790i... the point on the boundary of the main cardioid corresponding to the golden mean Siegel disk.). Note the cusp on the main cardioid in the upper right; looking to the left in the figure corresponds to zooming in towards the point c. "^{[30]}
 "Legendary side scrolling fractal zoom. 1 Month + (Interpolator+Video Editor) = Log(z). This means logarithmic projection for this location, that gives this interesting sidescrolling plane ^^)"^{[31]}
 " There are no program that can render this fractal on log(Z) plane. But you can make it in Ultra Fractal or in similar software with programmable distributive. Formula is:C = exp(D), for D  is your zoomable coordinates" SeryZone X
Formal mathematical description[edit  edit source]
Transformation is usually denoted as
Here transformation is complex: exponential function and translation
where:
 c is a parameter point from c plane ( flat image in Cartesian coordinate). It is a a parameter of quadratic map
 is a parameter in a new transformed plane ( logpolar image )
 is a constant parameter ( translation)
 e is Euler's number, is a mathematical constant approximately equal to 2.71828,
where:
 is natural logarithm
description for programmers[edit  edit source]
filter mercator (image in) in(xy*xy:[cos(pi/2/Y*y),1]) end
Maxima CAS src code:
(%i1) kill(all); (%i2) display2d:false; (%i3) ratprint : false; /* remove "rat :replaced " */ (%i4) c:cx +cy*%i$ (%i5) c0:c0x+c0y*%i$ (%i6) realpart(c0 + exp(c)); (%o6) %e^cx*cos(cy)+c0x (%i7) imagpart(c0 + exp(c)); (%o7) %e^cx*sin(cy)+c0y (%i8) cabs(c0 + exp(c)); (%o8) sqrt((%e^cx*sin(cy)+c0y)^2+(%e^cx*cos(cy)+c0x)^2) (%i9) carg(c0 + exp(c)); (%o9) atan2(%e^cx*sin(cy)+c0y,%e^cx*cos(cy)+c0x)
Notes
 the mapping is periodic because there are trigonometric functions inside
(%i1) kill(all); (%i2) display2d:false; (%i3) ratprint : false; /* remove "rat :replaced " */ (%i4) ce:ceex+cey*%i; (%i5) c0:c0x+c0y*%i; (%i6) realpart(log(ce  c0)); (%o6) log((ceyc0y)^2+(ceexc0x)^2)/2 (%i7) imagpart(log(ce  c0)); (%o7) atan2(ceyc0y,ceexc0x)
exponential grid scan[edit  edit source]
The grid scan with exponential coordiante mapping is different then the standard scan for flat images. Here is example by by Robert Munafo^{[32]}
/*
Plot a single pixel, row i and column j. Use as many rows as you need for the image to show the whole Mandelbrot set.
ctr_r and ctr_i are the real and imaginary coordinates of the center of the view we want to plot
px_radius is half the width of the image (in real coordinates)
px_spacing is the width of the image (in real coordinates) divided by the number of pixels in a row
*/
void pixel_53(int i, int j, int itmax)
{
double cr, ci, o_r, o_i, angle, radius;
/* compute angle and radius */
angle = ((double) j) * px_spacing / px_radius * 3.14159265359;
radius = ((double) i) * px_spacing / px_radius * 3.14159265359;
/* compute offsets */
o_r = cos(angle) * px_radius * exp(radius);
o_i = sin(angle) * px_radius * exp(radius);
ci = ctr_i + o_i;
cr = ctr_r + o_r;
evaluate_and_plot(cr, ci, itmax, i, j);
}
Videos[edit  edit source]
 Mercator  Scrolling Mandelbrot Fractal by Mathtown
 Mappings by the exponential function by William Nesse
 Mapping properties of the exponential function by MatheMagician
 The function f(z)=exp(z) by Michael Robinson
 Complex: visualizing exponential map and local inverse by James Cook
 Mandelbrot Mercator Testing by Fractal MathPro
Images[edit  edit source]

without transformation

real (= 1D) version

1D: 3 windows with nonlinear, i.e. logarithmic xaxis scale:

complex (= 2D) version

normal projection

polar projection

M(3,1) Misiurewicz at 1E10 magnification (bottom) use the exponential map


Mercator Effect in GIMP with MathMap

Map of the region close to the MyrbergFeigenbaum point. The buds of the main Mandelbrot set converge geometrically towards this place (at a rate set by Feigenbaum's constant), producing the regularly spaced buds at the top and bottom. Since I did not have an exact coordinate for it, the map is centered a little bit away.

This image corresponds to a 10^30 zoom around a point (I just used 30 digits of precision, so there are artefacts at the left edge

This image corresponds to a 10^30 zoom around a point (I just used 30 digits of precision, so there are artefacts at the left edge).

10^60 zoom. The banded structure occurs because the zoom passes through the embedded Julia sets around smaller Mandelbrot sets; each new Mandelbrot zoomed past adds their own Julia set to the sequence. The center point is c=1.256827152259138864846434197797294538253477389787308085590211144291 +0.37933802890364143684096784819544060002129071484943239316486643285025i

Zooms of the Mandelbrot set around a point projected so that the distance to the point forms a logarithmic scale. Details on the left edge are ~10^15 times smaller than on the right edge. The rightmost black blob in the pictures is the main "continent" of the set. Satelite sets of different sizes can be seen linked by fractal chains of varying complexity. The colors are set using the distance estimator method.
How to read location from exponential image ?[edit  edit source]
 fractalforums.org : canyoufindthelocationofthisfractal ?
 "An inverse mapping (using the complex logarithm function) is used to compute the location (in the strip) of needed data for any given location in the normal orthogonal coordinate space." Robert Munafo^{[33]}
Tips:
 the image is not symmetric ( up and down) so imaginary part of c0 is not zero
Implementations[edit  edit source]
Fractal programs
 UltraFractal
 Polar_Projection in fractalzoomer
 emndl: exponential strip visualisation of the Mandelbrot set, program by Claude HeilandAllen
 pfract  console c program by Mika Seppä
 emndl: mathr's exponential strip mandelbrot set renderer
kf and zoomasm[edit  edit source]
 The rendering of the final video can be accelerated by computing exponentially spaced rings around the zoom center, before reprojecting to a sequence of flat images.
 kf2.15 supports rendering EXR keyframes in exponential map form
 zoomasm can assemble above keyframes into a zoom video. zoomasm works from EXR, including raw iteration data, and colouring algorithms can be written in OpenGL shader source code fragments
 kfextras by Claude HeilandAllen  has the exponential map (aka log polar or mercator projection) convertor
OpenCV[edit  edit source]
 opencv 3.4 warpPolar() function
 logpolar transform using Camera by Aresh T. Saharkhiz
 stackoverflow question: reprojectingpolartocartesiangrid
 descr
 stackoverflow question: howtotunelogpolarmappinginopencv
 stackoverflow question : convertinganimagefromcartesiantopolarlimbdarkening
numpy[edit  edit source]
ImageMagic[edit  edit source]
 pano2fisheye  Transforms a spherical panorama to a fisheye view = ? Reverse polar coordinates?
 distorts depolar
 snibgo's ImageMagick pages: Polar distortions
processing[edit  edit source]
scikit[edit  edit source]
Mathworks[edit  edit source]
References[edit  edit source]
 ↑ Logarithmic_scale in wikipedia
 ↑ matlab ref: loglog plot
 ↑ Exponential_map_(Riemannian_geometry)
 ↑ Exponential/`Raise to Power' Operator by R. Fisher, S. Perkins, A. Walker and E. Wolfart ( 2003)
 ↑ exponentialdataandlogarithmicscales by Stephen Redmond
 ↑ 24 Views of the Complex Exponential Function by Pacific Tech
 ↑ semilog plot in wikipedia
 ↑ Log–log plot in wikipedia
 ↑ exponentialdataandlogarithmicscales by Stephen Redmond
 ↑ LogScale Reading Posted July 17, 2015 by Scott
 ↑ wolfram Exp visualizations
 ↑ scientificlib : Biholomorphism
 ↑ Exponential mapping with Kalles Fraktaler by Claude HeilandAllen
 ↑ exponential map Robert P. Munafo, 2010 Dec 5. From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872022.
 ↑ Transverse_Mercator_projection in wikipedia
 ↑ kfextras for manipulating output from Kalles Fraktaler 2
 ↑ Mercator: Extreme An interactive visualization of the extreme distortions of the Mercator projection by Drew Griscom Roos
 ↑ The Mercator Projection by John J. G. Savard
 ↑ flickr : Mercator Mandelbrot Maps by Anders Sandberg
 ↑ log(z)MandelbrotZooms by Thorsten Forstemann
 ↑ youtube video : Mandelbrot deep zoom to 2^142 or 5.5*10^42. Log(z) by SeryZone X
 ↑ Mandelbrot Zoom 333  log(z)(Sery's Edition) by SeryZone Arts
 ↑ Mandelbrot Mercator Testing by Fractal MathPro
 ↑ A Map of the Universe by J. Richard Gott III, Mario Jurić, David Schlegel, Fiona Hoyle, Michael Vogeley, Max Tegmark, Neta Bahcall, Jon Brinkmann
 ↑ xkcd : Depth
 ↑ Mercator Mandelbrot by Anders Sandberg
 ↑ flickr album : Mercator Mandelbrot Maps by Anders Sandberg
 ↑ Mandelbrot set images and videos by David Madore
 ↑ FRACTALS IN POLAR COORDINATES by Mika Seppä
 ↑ FRONTIERS IN COMPLEX DYNAMICS by CURTIS T. MCMULLEN
 ↑ youtube video : Mandelbrot deep zoom to 2^142 or 5.5*10^42. Log(z) by SeryZone X
 ↑ Exponential Map from the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 2010 Dec 5.
 ↑ exponential map From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 19872022.