# Fractals/Computer graphic techniques/2D/zoom

Zoom is an example of plane transformation and is related with plane description

# Applications

Zoom can be used for making :
* static images
** normal, flat ( non-transformed)
** transformed : exponential images
*** feigenbaum-stretch by Claude Heiland-Allen
*** Exponential map from the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2018
*** Exponential mapping with Kalles Fraktaler by Claude Heiland-Allen
* video
Fractals/Computer_graphic_techniques/2D/exp

# Q&A

* how-to-adjust-panning-while-zooming
* How many iterations of the Mandelbrot set for an accurate picture at a certain zoom?
** automatic dwell limit ( MuEncy)

# Notation

If you want to zoom
Note that base of zoom factor is important when one wants to optimize zooming by using previously computed pixels.
Usually other notation is used :
" Some programs use a power of 2, others use scientific notation. " Dinkydau
See also
* parameter files
* graphic files with parameters

# Pixel size

Pixel size definition:
* size of a pixel in a world coordinate.
* pixel spacing: "px_spacing is the width of the image (in real coordinates) divided by the number of pixels in a row"
Properities:
* For linear scale and same resolution on both axes
* "Pixel spacing is no longer fixed when applying a conformal mapping with non-constant scaling."
| n = Zoom exponent | mag = Zoom level | Image size in pixels = iSize | Image size = fSize | PixelSize | Decimal digits | Binary digits ( precision) | radius=fSize/2 = 1/mag |
| -1 | 6.66666667e-01 | 1000 | 3 | 0.003 | | | 1.5 |
| 0 | 1 = 10^0 = 1.0e0 | 1000 | 2.0 | 0.002 | | | 1.0 |
| 1 | 10 = 10^1 =1.0e1 | 1000 | 0.2 | 0.0002 | | | 0.1 |
| 2 | 100 = 10^2 = 1.0e2 | 1000 | 0.02 | 0.00002 | 5 | 17 | 0.01 |
| 3 | 1000 = 10^3 = 1.0e3 | 1000 | 0.002 | 0.0000002 | 6 | 21 | 0.001 |
| 6 | 4.42685e+06 = 4.42685*10^6 | 1000 | | | 9 | 31 | 0.0000002259 |
| 275 | 2.1*10^{275} = 2.1E275 | 1000 | 1.42857142857143*10^{−275} | 1.42857142857143*10^{−278} | 278 | 945 | |
| 2167 | 1E2167 | | | | | | |
| n (integer) | 10^n = 1.0en | iSize (integer) | fSize (float) | fSize/iSize | n+3 | (int) 3.4*(n+3) | 1/10^n |

# Maximal zoom

Maximal zoom factor is :fractalforums : world-record-list-for-deepest-mandelbrot-set-zooms
* 10^15
* 10^17
* 10^999
* 10^1 500
* 10^2 286
* e^10 000
* 4.69 e^100 062
Maximal zoom is correlated with :
* precision of number types used for calulations ( single , double, Quadruple precision, multiple-precision (MP) )
** manipulating numbers below $10^{-308}$ in double format shall be avoided.
* algorithm

## Examples

* zoom factor for video
* zoom factor
6 digit exponent kfr file :

# Optimisation

* optimisation
* perturbation technique
* zoom optimization from Xaos by Jan Hubicka

# Precision

What precision do I need for zoom ?
* -log2(pixel_size) = precision in bits
* prec = 53 - log2(view_radius))
* it also depends on algorithm used to make an image
* "the distance between two adjacent points on the ray is calculated with at least 4 significant bits."
* "Floating point number precision loses accuracy at approximately 1e15, distorted images occur." Microfractal
* manipulating numbers below $10^{-308}$ in double format shall be avoided.
* for deeper zoom use
** perturbation techique
** reference complression algorithm
And result is :

# Zoom types

* manual/automatic
* discrete (steps) / continous

## autozoom

* fractalforums.org: various-autozoom-functions

## continous zoom

* Mandelbrot zoom with center at (-0.743643887037158704752191506114774, 0.131825904205311970493132056385139) and magnification 1 .. 3.18 × 1031 using double-double precision
* Self-similarity in the Mandelbrot set shown by zooming in. The display center pans from −1,0 to −1.31 while the view magnifies from 0.5 × 0.5 to 0.12 × 0.12 to approximate the Feigenbaum ratio

## discrete zoom

discrete zoom
* notation
** sequence of: plane description ( text ) or images with rectangle ( graphic documentation)
** continous:
** composition of zoom and translation or Poincaré half-plane metric
* A zoom sequence of the Mandelbrot set
* Iterative zoom into fractal pattern generated by four coupled quadratic maps
Mandelbrot set: initial image of a zoom sequence with a frame indicating the position of the next image. Image of a sequence as appendix to the frameless main image sequence. For image parameters see corresponding main image.
Corresponding frameless main image sequence: Initial image / zoom step 1 2 3 4 5 6 7 8 9 10 11 12 13 14.
To the next image of the zoom sequence with frames.
The whole image sequence with frames:
| Start to 1 | Step 1 to 2 | Step 2 to 3 | Step 3 to 4 | Step 4 to 5 |
| Step 5 to 6 | Step 6 to 7 | Step 7 to 8 | Step 8 to 9 | Step 9 to 10 |
| Step 10 to 11 | Step 11 to 12 | Step 12 to 13 | Step 13 to 14 | |

# Zoom examples
## deep zoom

Baby Monster by DinkydauSet: zoom into parameter plane of complex quadratic polynomial :
Cross Hair by Claude Heiland-Allen . Below is the content of kfr file ( parameter for Kalles Fraktaler )
List :
* Mandelbrot Examples by David Eck ( corners )
* perturbator_deep_zoom_stress_test by Claude Heiland-Allen
* Millionaires: an occasional series of Mandelbrot set deep zooms with central period just over a million by Claude Heiland-Allen

## shallow

shallow locations with high iteration counts by Claude Heiland-Allen
* Re: 3.56992006738525396399695724115347205e-01
* Im: 6.91411005282446050826514373514151521e-02
* Zoom: 1e19 so it needs arbitrary precision numbers ( like MPFR) with 72 binary bits when using simple method ( without perturbation)
* Period: 1000000
* Iterations: 1100100100
* it is a dense set
* The angled internal address of the mini in the middle is (morally, if not factually) something like 1 1/9 10 2/9 100 4/9 1000 5/9 10000 7/9 100000 8/9 1000000, so the external angles could be computed relatively easily I suppose.

# References

* ↑ geeksforgeeks : how-to-zoom-in-on-a-point-using-scale-and-translate
* ↑ stackoverflow : how-do-i-zoom-into-the-mandelbrot-set
* ↑ Fractal Forum > Fractal Art > Images Showcase (Rate My Fractal) > Baby Monster
* ↑ Exponential Map by Robert P. Munafo, 2010 Dec 5. From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2020
* ↑ Stretching cusps by Claude Heiland-Allen
* ↑ zoom level definition
* ↑ Fractal Journey by ORSON WANG
* ↑ Zoom level discussion - Maximum zoom level supported by Fractal eXtreme
* ↑ fractalforums : history-of-deep-zoom-records
* ↑ fractint deep_zoom
* ↑ zooming with exteded precision in Delphi
* ↑ video Mandelbrot Zoom 999 on youtube by phaumann
* ↑ DeepZooming with FractInt
* ↑ Mandelbrot set detail Glimpse of variety by DinkydauSet
* ↑ e10000 by by Claude Heiland-Allen
* ↑ fractalforums: kalles-fraktaler-gallery/6-digit-exponent
* ↑ w:wiki/Talk:Floating_point/Archive_4#No_mention_of_Double-double No_mention_of_Double-double - talk
* ↑ stackoverflow : mandelbrot-set-reaches-to-its-limit-way-too-soon
* ↑ Fractalshades-doc : math
* ↑ fractalforums : problem with zooming transform one view rect-to another
* ↑ fractalforums "basic-location-and-zoom-question
* ↑ optimizing_zoom_animations by Claude Heiland-Allen
* ↑ fractalforums about-precision-of-doubles
* ↑ fractalforums.org : single-precision-is-not-enough-for-bilinear-approximation
* ↑ fractalforums.org : metallica
* ↑ Fractalshades-doc : math
* ↑ poincare half-plane metric for zoom animation by Claude Heiland-Allen
* ↑ Baby Monster by DinkydauSet
* ↑ Cross Hair by Claude Heiland-Allen
* ↑ Kalles Fraktaler 2 by Karl Runmo
* ↑ fractalforums.org : challenging-locations-for-current-acceleration-methods