text:
Fractals/Iterations in the complex plane/Mandelbrot set/mset distortion

This page is about: warped midget, distortion of the island ( part of the Mandelbrot set and parameter plane).

=name = There are many names of mini copies of Mandelbrot set within itself:
 * Small Mandelbrot sets
 * mini Mandelbrot set
 * baby Mandelbrot set
 * baby-brot
 * Bug
 * Island : "word 'island' was introduced by Benoit Mandelbrot in his description of the Mandelbrot set in The Fractal Geometry of Nature" Robert Munafo
 * island mu-molecules
 * embedded copy of the Mandelbrot Set
 * Mandelbrotie
 * Midget
 * minibrot
 * primitive M-copies
 * slightly deformed copies of the original shape
 * Near-copies of the Mandelbrot set within itself

Different names for distortion
 * eccentricity
 * compare with eccentricity of of a conic section is a non-negative real number that uniquely characterizes its shape
 * warping
 * deformation

Names for distorted midgets:
 * warped midget

=Examples=

See also:
 * commons:Category:Near-copies_of_the_Mandelbrot_set_within_itself

Sequence of locations from the elephant valley, with the period doubling ( by Claude Heiland-Allen )

4 	-1.565201668337550811e-01+ 1.032247108922831780e+00 i 	1.697e-02 8 	4.048996651751222142e-01 + 1.458203637665893004e-01 i 	2.743e-03 16 	2.925037532341934199e-01 + 1.492506899834379792e-02 i 	3.484e-04 32 	2.602618199285006706e-01 + 1.667791320926506355e-03 i 	4.113e-05 64 	2.524934589775105209e-01 + 1.971526796077277316e-04 i 	4.920e-06 128 	2.506132008410751344e-01 + 2.396932642510365971e-05 i 	5.997e-07 256 	2.501519680089798192e-01 + 2.954962325906881015e-06 i 	7.398e-08 512 	2.500378219137852631e-01 + 3.668242052764790239e-07 i 	9.185e-09 1024 	2.500094340031833728e-01 + 4.569478652064613658e-08 i 	1.144e-09 2048 	2.500023558032561377e-01 + 5.701985912706845832e-09 i 	1.428e-10 4096 	2.500005886128087162e-01 + 7.121326948562731412e-10 i 	1.783e-11 8192 	2.500001471109009610e-01 + 8.897814201389539302e-11 i 	2.228e-12
 * 1) period	 center 			                radius

List by marcm200 cardioid      angle period p      p<-2p->4p         eccentricity   cardioid center -- 5             177.1             1.35           0.35925922475800742273-0.64251373713854231795*i 6             176.5             1.47           0.4433256333996235532-0.37296241666284651872*i 7             176.1             1.55           0.43237619264199450564-0.22675990443534863039*i 8             175.8             1.52           0.40489966517512215871-0.14582036376658927268*i 10            175.4             1.65           0.35681724849231194474-0.069452865466830299157*i 12            175.1             1.72           0.32558950955066034982-0.038047880934755723414*i 15            174.8             1.79           0.29844800890399547644-0.018383367322073254635*i 17            174.7             1.80           0.28756611704687790043-0.012281055409848253349*i 21            174.6             1.84           0.27436979919551240936-0.0062585874913430950342*i 25            174.5             1.87           0.2652783219046058732 -0.0037120599898783183101*i 31            174.4             1.90           0.26095224231422198269-0.0018410978175303128503*i 43            174.34            1.91           0.2556052938434967281 -0.00066797029763820438275*i

Locations:
 * 0,25000102515011806826817597033225524583655 + 0,0000000016387052819136931666219461i Zoom = 6,871947673*(10^10) so radius = 0.14556040756914119* 10^{-10}
 * 0.2925294 + 0.0149698 i @ 0.0006675
 * c = 0.292503753234193 -0.014925068998344i period = 16,  size of image +0.0005
 * by marcm200
 * period 7 c =0.43237619264199450564-0.22675990443534863039i
 * period 14 c=0.4325688150887696537-0.22873440581356344059i
 * period 28 c=0.43266973566541755414-0.22933968667591089763i

=Self-similarity=

The Mandelbrot set is self-similar under magnification in the neighborhoods of the Misiurewicz points. It is also conjectured to be self-similar around generalized Feigenbaum points (e.g., −1.401155 or −0.1528 + 1.0397i), in the sense of converging to a limit set. The Mandelbrot set in general is not strictly self-similar but it is quasi-self-similar, as small slightly different versions of itself can be found at arbitrarily small scales. These little copies of the Mandelbrot set are all slightly different, mostly because of the thin threads connecting them to the main body of the set.

"this combination of familiarity (small copies of the Mandelbrot set) and novelty (subtly distorted, differently decorated) that make the Mandelbrot set much more interesting than the fractals generated by IFS."

=Q&A=

are they exact copies?
They are:
 * not exact copies
 * slightly deformed copies
 * approximately self-similar copies
 * quasiconformally equivalent to the whole Mandelbrot set
 * quasi-self-similar
 * Baby Mandelbrot Sets are Not Exact Self Similar. A baby Mandelbrot sets are homeomorphic to the original one

Why does the Mandelbrot set contain copies of itself?
Why does the Mandelbrot set contain (slightly deformed) copies of itself?
 * "renormalization explains why the baby Mandelbrot set appears: all of the quadratic polynomials in a baby Mandelbrot set renormalize and all renormalize in essentially the same way."
 * "the fractal is a reflection of reflections of reflections ..."
 * "iteration creates complexity. Mandelbrot set is based on checkin behaviour of infinite sequences. The behaviour of that sequences is very complex. In fact, it may be chaotic. We are in the presence of sensitive dependence on initial conditions (a.k.a. the butterfly effect.)"
 * "While the process is very simple, the iterative nature of the process leads to very different results depending on the point on the complex plane at which you start."

Why there are copies of Mandelbrot set in other fractals ?
The Mandelbrot set is universal so it can be find in:
 * an iterated complex mapping
 * the bifurcation locus for a parametrized family of holomorphic functions: rational families of functions, transcendental families
 * the bifurcation locus of any holomorphic family of rational maps
 * quadratic-like maps of Douady and Hubbard

How to describe minibrot ?
Measures
 * period of main pseudocardioid
 * center of main pseudocardioid
 * window ( radius and center) of parameter plane
 * angles of external rays that land on cusp of the pseudocardiod
 * size of main pseudocardioid
 * orientation
 * distortion

Size and orientation
Scaling for period-n windows of chaotic dynamical systems
 * Windows of periodicity scaling by Evgeny Demidov
 * J.A.Yorke, C.Grebogi, E.Ott, and L.Tedeschini-Lalli  "Scaling Behavior of Windows in Dissipative Dynamical Systems" Phys.Rev.Lett. 54, 1095 (1985)
 * B.R.Hunt, E.Ott Structure in the Parameter Dependence of Order and Chaos for the Quadratic Map  J.Phys.A 30 (1997), 7067.
 * Deriving the size estimate by Claude Heiland-Allen
 * approximate selfsimilarity by Claude Heiland-Allen
 * m-island-zoom.c from mandelbrot-graphics library by Claude Heiland-Allen

The formula for the size and orientation:

$$ r = \frac{1}{\beta \Lambda_p^2}$$ Input and related values:
 * nucleus $$c$$ of a minibrot's main pseudocardioid
 * period $$p$$ of a minibrot's main pseudocardioid
 * $$\beta $$
 * $$ \Lambda_n $$
 * $$\lambda_n $$

Output:
 * complex valued size and orientation estimation r ( complex number). To compute it just use arg and abs on the result

where

$$z_1 = 0 ; z_{n+1} = z_n^2 + c $$

$$\lambda_n = 2 z_n $$

$$ \Lambda_n =\lambda_2 \lambda_3 \cdots \lambda_n $$ $$\beta = 1 + \Lambda_2^{-1} + \Lambda_3^{-1} + \dots + \Lambda_p^{-1} $$

How to use it ?

"if the nucleus of the baby is c and the complex size is r, there is another miniature copy near the baby around $$c+rc$$ with size approximately $$r^2$$" 	Claude Heiland-Allen Example minibrots:
 * period 3 near −2
 * period 4 near i
 * period 5 near −1.5+0.5i

See also c function m_size from mandelbrot-numerics library by Claude Heiland-Allen

area
The area $$A$$ of the island of period p+1 is approximated by the formula by Robert Munafo:

$$A_{p+1} \approx 0.05 \frac{sin^2(\frac{\pi}{p})}{p^4}$$

How to measure the distorsion ?
Measure
 * express these in terms of the derivatives of the iteration at the preimage of 0
 * the ratios of radii of the satellites to the main cardioid
 * angular positions of secondary ( child) components
 * in degrees(or radians) because the parts of the minibrot are simply rotated ( conformal transformations )
 * the ratio of two distances to the cusp
 * "Warped midgets in the Mandelbrot set have been measured, using an algorithm that allows the positions of the head, and cardioid atoms (north and south) of any midget to be found, once one has placed the cursor on the computer terminal somewhere inside any midget. We describe two distortions of midgets: linear distortions and angular distortions. When the north and south angles are plotted in the north/south angle plane, families of points are formed. The angle and distance measures of warped midgets from the Sea Horse Valley of the Mandelbrot set and from other sea horse valleys of midgets, whether on the Spike or on tendrils above atoms, all fall closely together in one part of the north/south plane. Measures of warped midgets from tendrils above the major atoms on the surface of the Cardioid fall closely together in another part of the north/south plane. This different way of looking at the Mandelbrot set offers an interesting way of studying the distortions of midgets." A. G. Davis Philip, Michael Frame, Adam Robucci: Warped midgets in the Mandelbrot set. Computers & Graphics 18(2): 239-248 (1994)

eccentricity of the pseudocardioid
The eccentricity of the pseudocardioid was calculated in a geometrical manner and defined as the ratio of two distances from the cusp to the (to do )

$$ ecc = \frac{cusp_B}{cusp_A}$$

position of the child

 * 2 components :
 * parent ( pseudocardioid with period = p ) = p-componenent
 * child ( pseudocircle with period = 2*p) = 2p-component
 * two lines inside parent (pseudocardioid)
 * axis of the lower part of the parent (pseudocardioid) = line thru 2 points: nucleus and cusp of pseudocardioid = lower axis
 * axis of the upper part of the parent = line thru 2 points: nucleus and root point ( between child and parent components) = upper axis
 * angle between lines measured in turns
 * it is 0 for no distortion = all 3 points are on the same line ( like in case of whole M-set with main cardioid as a parent and period 2 component as a child componenent)
 * it is > 0 for distortion

In mathematical language, the slope m of the line through 2 points
 * $$ z_1 = x_1 + y_1*i$$
 * $$ z_2 = x_2 + y_2*i$$

is

 * $$m=\frac{y_2-y_1}{x_2-x_1}.$$

Angle between two lines of slope $$m_1$$ and $$m_2$$ is

$$ \tan^{-1}\dfrac{m_1 - m_2}{1+m_1 m_2} $$

Here is c function

How distorted can a minibrot be?
What is maxima distortion of minibrot ?

How to find minibrot ?

 * locate = find

How to find distorted minibrot ?

 * using mandelbrot-graphics library and program : m-warped-midgets

What is the relation between distortion and other measures of the island ?

 * the eccentricity grows with increasing period, but not linearly, not even monotone.
 * the biggest island of the wake is the most distorted. The period of the biggest minibrot in the m/n wake is  n+1

=See also=
 * How to find the angles of external rays that land on the root point of any Mandelbrot set's component which is not accessible from main cardioid ( M0) by a finite number of boundary crossing ?

=References=


text:
Javagony/For loop

Introduction
As noted in the preface, Javagony does not allow any loops. In this chapter you will learn the alternative for a for loop in Javagony. We will be creating a recursive function that will provide the functionality of the following for loop:

You could write i++, however I have written i+=1 on purpose to demonstrate that you can write also i+=2 or any other number you want. We will be using recursion as well as knowledge from the previous chapter to emulate the functionality of this loop.

Implementation
Notice that we have reused code from the previous chapter, Comparing 2 integers to check which one is greater.

This function gets three values: firstNum, lastNum, and step. For this section, assume that step is a positive number. We will discuss negative values of step in the next section of this chapter. This code in regular java can be written as following:

First the function prints the value firstNum. This is in order to show that it iterates over all the numbers that a regular for loop would. You can replace it with the code you want running. Then we increase firstNum by step, and use the knowledge we gained in the previous chapter to check if firstNum is lower than lastNum.

Notice that I wrote firstNum+1 instead of just firstNum. This is because I want firstNum to be less that lastNum, but this method checks if firstNum is less or equal to lastNum. If firstNum+1<=lastNum then we can conclude that firstNum<lastNum. In this code we are only working with integer numbers. Omitting the +1 will result an Javagony version of the following regular Java code:

If firstNum (which at this point contains the value of the next number) is less than lastNum, then we proceed to the next iteration.

If step<0
Let’s now change the values given to the function in the main method to 0,-10,-1 and observe how the code will work.

If we run the code now, we will see that it prints the following output: 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10

But this is not what we want. We don’t want the code to iterate over the last value. Let’s check how the code runs to see how to fix it.

The code stars with the variable firstValue as 0 and prints it. Then it increases the variable by -1, which is the same as subtracting 1 from it and we get -1.

Now, we get to the comparing. The value of lastNum (in this case -10) is subtracted from the maximum value that is possible to store in an int. This is the same as adding 10. One would expect this to raise an error, but instead an integer overflow will occur, and we will get a negative number.

Then, using the method forFunc which does not allow for integer overflow we add the value of firstNum+1 (in our case the result is 0) to the large negative number, and if an integer overflow does not occur, we proceed to the next iteration.

What will happen after the code prints -9? Well, it will change the value of firstNum to -10, and then check if -9 plus this large negative number will result in an integer overflow. However, it does not and goes one to iterate on i=-10.

How to fix it? Well, we can fix it by omitting the +1, but then we will have the same trouble with step>0. We can have two functions, one that omits the +1, and one that doesn’t.

The first one will be a replacement of the following loop if step=1:

and if step=-1:

The second one will be a replacement of the following loop if step=1:

and if step=-1:

Now, you have got all the options covered, you just need to figure out which one is right for you in the next production app you write, since from now on you will write codes only in Javagony. Not really.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Javagony/For loop


text:
No Man's Sky/Wanted levels

A wanted level is how high a threat the sentinels think you are. It starts at zero, where they won't attack you, except on extreme sentinel planets, but it will be set to one by doing certain things:
 * Attacking a sentinel
 * Letting a sentinel scan you on an extreme sentinel planet
 * Continuing to mine resources while a sentinel is watching you
 * Attacking resource pods or reinforced doors in supply depots and secure facilities
 * Destroying a freighter cargo pod
 * Destroying any non-pirate starship
 * Attacking the space station (won't use the same wave system as mentioned below)
 * Get caught with smuggled contraband during an inventory scan, and refusing to give them up

It will be set to three by collecting a gravitino ball (unless you farmed it yourself).

Attacking the space station, by accident or on purpose, will summon sentinels from the station, and they don't stop. This is a good way to farm their drops, as you can board the space station to escape them.

As your wanted level sets to one, nearby sentinels will immediately become hostile. Destroying all of them will bring reinforcements. Either run, or hide, if you want to escape.

Wave types
What type of waves you get depend on whether you are in your starship or not. Some actions can only be triggered in a starship, most only on ground, some in both.

You can stop sentinels attacking you in space by entering the space anomaly, your freighter, or the space station. However, your freighter and the anomaly cannot be summoned (or dismissed) during combat.

Waves - on the ground
First you need to destroy all nearby sentinels, then reinforcements will appear. If you are scanned an an extreme sentinel planet, or there are no nearby sentinels when you angered them, wave one begins immediately. Otherwise, wave one begins after a 10 second countdown, saying "reinforcements arriving" (once all nearby hostile sentinels are killed). Running will cause them to lose you, and they will search for you. If you can stay hidden for a certain amount of time (which depends on the wave, being 13 seconds at wave one, and up to sixty on wave five), your wanted level resets to 0, and any sentinels that were chasing you will turn peaceful again.

The first wave is quite simple: a few healing sentinels and combat sentinels. Take the healers out first. They are the small black ones. Take out the combat ones (they are larger and orange) afterwards.

Killing all the wave one sentinels will increase your wanted level, and give you up to 10 seconds to escape or prepare. This next one is a lot like wave one, but with more sentinels. Sentinel summoners (black triangular ones) will occasionally summon two more combat sentinels, so they should be taken out first.

Wave three is tougher, with 10 seconds to escape or prepare. It is like wave 2, but you also need to deal with a dog-like sentinel. Try to get high ground, maybe on a building, and shoot down the flying ones first. Then take out the dog-sentinel.

Wave four is really tough, with 6 seconds to escape or prepare, and having loads of flying sentinels and a large mech. Take out the flying ones first, especially the healers and summoners. The mech shoots lasers at you, and even has a flame thrower at half health. Aim for the black spots, since they are weak points.

Wave five is the hardest wave, with only 3 seconds to prepare. You are unlikely to be able to escape in that time. A dog, a mech, and maybe 15 flying sentinels are fighting you. If you thought that was hard, wait until you see what else you've got to face: a massive, fifty-foot armoured walker. After taking out some of the smaller targets (mainly the healers, summoners, the dogs, and the mech), aim for the back or top of the head of the walker. Anywhere else is impervious to attack.

Completing wave 5 will temporarily disable sentinels on the planet, set you wanted level back to 0, and show you were the sentinel nest, now unguarded, is. But before you rush off, consider collecting your rewards first. Combat sentinels, dog sentinels, and sentinel meches all drop a round pod of valuable resources. Nanites, ammo, salvaged glass, and more, can be found. Walkers also drop their pieces, and the large piece (their body) can be destroyed for more loot.

Consider also going to the marked sentinel pillar, since they have resource pods of rare sentinel drops, and a terminal can be interacted with to collect information about the history of the sentinels, from their point of view. You can also use the pillar to extract a new multitool as well.

Waves - in space
It is less easy to fight sentinels in space, and the recovered loot is limited, but at least you need to face less of them.

Wave one: you need to destroy one single sentinel interceptor.

Wave two: you need to fight two sentinel interceptors.

Wave three: you need to destroy three stronger sentinel interceptors.

Wave four: you need to destroy a larger amount of sentinel interceptors.

Wave five: you fight maybe 10 sentinel interceptors, not all at once, and a freighter with loads of turrets on it. Take out the turrets and a few interceptors. Attacking the freighter damages it, and if you deal enough damage, it will explode, causing all sentinel interceptors to flee.

In between each wave, you have a short time where you can pulse jump away, or better (sentinels can still drag you out of a pulse jump when more arrive): enter a space station/freighter/the anomaly. If that cannot be done, your only hope it to fight, or activate an emergency warp. After activating the emergency warp, remember to repair the emergency warp drive starship upgrade.

Salvaged glass
This item acts a bit like geodes. When broken in your inventory, they can drop powerful upgrades, sentinel pillar planetary charts, pungeum and other sentinel drops.

Sentinel upgrades are very powerful. They target up to four stats, mostly combat related, and the maximum increase is quite high. They come in multitool upgrades and exosuit upgrades. Despite what the word "forbidden" on the tooltip may suggest, having or equipping one will not anger sentinels.

title: No Man's Sky/Wanted levels


text:
Òla, Ionatà!

'''Òla, Ionatà! Audian by the Direct Method'''

1
Uno òmme es a una ciutà. L'òmme éscrive un jurnau. L'òmme ve una ciutà.

Es l'òmme a una ciutà? Òc, ell es a una ciutà.

Es l'òmme enn'òmme? No, ell n'es enn'òmme. Ell es a una ciutà.

Es l'òmme ennu tren? No, ell n'es ennu tren. Ell es ennu tren. L'òmme es a una ciutà.

Éscrive l'òmme un jurnau? Òc, ell éscrive un jurnau.

Éscrive le jurnau un'òmme? No, le jurnau n'écrive un'òmme. Un jurnau n'éscrive. Un'òmme éscrive. L'òmme éscrive le jurnau.

Ve la ciutà au òmme? No, la ciutà no ve au òmme. Una ciutà no ve. Un'òmme ve. Ve l'òmme la ciutà? Òc, l'òmme ve la ciutà.

Ond es l'òmme? Ell es a una ciutà. Qué éscrive l'òmme? Éscrive un jurnau. Qué ve l'òmme? Ve una ciutà. Es a una ciutà e éscrive un jurnau e ve una ciutà.

Es l'òmme ennu jurnau? No, ell éscrive un jurnau. Ve l'òmme un'òmme? No, ell no ve un'òmme. Ell ve una ciutà.

|1

title: Òla, Ionatà!


text:
Salut Jonathan!

Kapitros

 * Preface
 * Chapter: 1
 * Chapter: 2
 * Chapter: 3

title: Salut Jonathan!


text: Category:Book:Salut Jonathan!

title: Category:Book:Salut Jonathan!


text:
Salut Jonathan!/Prefaco

'''Salut Jonathan! Neo by the Direct Method'''

Salut Jonathan! (Hello, Jonathan!) is the story of a young Englishman who goes to visit his friend in another European country...or so it would seem.

It is written entirely in the language Neo, a language created by the Arturo Alfandari that was originally published in 1961 and updated in 1965/66. The book Salute, Jonathan! was originally written in Interlingue by Mithridates and published on Wikibooks in 2019.

Salut Jonathan! begins with very simple language. The first words in the book are Un vir starar in un civ (a man stands in a city), and it continues from there. The words the reader knows are repeated and new words are added slowly and in context, so that no dictionary is required to read it. Neo was created to be an international language, with a very simple and straightforward grammar. Thanks to its simplicity and regularity, you will know how to read and use it by the end of Salut, Jonathan!.

The method in the book is based on the same used in books such as English by the Nature Method, Le français par la méthode nature, and L'italiano secondo il metodo natura, as well as Hans Ørberg's Lingua Latina per se Illustrata.

Curious about the true nature of the story in the book? You will begin to get clues as to the true story around Chapter 4 as the language becomes more and more complex and an actual story begins to form. Salut, Jonathan! is in fact a full translation of a classic book that you are likely very familiar with. Read on to find out!

title: Salut Jonathan!/Prefaco


text:
ORCID Advocacy Toolkit/What is Advocacy?

What is advocacy?

What better place to start than the wikipedia definition?

Steps to effective advocacy:

 * 1) Define your goals
 * 2) Understand your audience
 * 3) Develop a vision and strategy
 * 4) Identify resources and gaps
 * 5) Plan and develop the campaign
 * 6) Anchor change in the culture
 * 7) Evaluate progress regularly

Source: Open Research Toolkit Module 11 https://mfr.osf.io/render?url=https://osf.io/5d78q/?direct%26mode=render%26action=download%26mode=render

Advocacy approaches
Build into a range of Library workshops i.e. not a stand alone session about ORCiD; users need a reason to attend the session beyond ORCiD.

e.g. Sessions on Raising Research Visibility, Open Research, Research Data Management


text:
Talk:Public International Law/Approaches/New Haven School

English
Hi everyone I need help for home school this is important for me to go beg to school is soon as possible please don't let me down I really need help to you guysp


text:
User:Maxi9090

i like video games. Maxi9090 (discuss • contribs) 19:57, 15 November 2022 (UTC)

title: User:Maxi9090


text:
Cookbook:Shirataki

Shirataki, also called konjac noodles or miracle noodles, are a variety of Japanese noodles made from the konjac yam.

Characteristics
Shirataki are made primarily from glucomannan extracted from konjac yams. This gives them a translucent, gelatinous texture with almost no flavor. Some varieties may contain added soy/tofu, which can give them a more opaque and white appearance. The noodles can come in a variety of shapes and are lighter in texture than wheat or rice noodles. They may have a slight fishy odor, which should fade after a thorough rinse.

Production
Konjac corms are dried and ground to extract glucomannan flour. This flour is them mixed with water and limewater (calcium hydroxide solution) to make a gelatinous mixture. This mixture is then shaped into noodles and cooked.

Procurement and storage
Shirataki noodles tend to be available either dried or packed in liquid. Both have relatively short shelf lives and should be cooked and eaten quickly.

Use
Shirataki have been eaten in Japan for centuries. Compared to other noodle varieties, they don't have starch on the surface, and this prevents sticking and alteration of sauce texture. Their mild flavor makes them great at absorbing the other flavors in a dish.

Noodles packed in liquid should be rinsed well and added directly to the final dish for a very brief cooking period. They do not need to be boiled separately like many other types of noodles do. Some like to pan-fry them briefly to evaporate any excess water and avoid diluting the final product.

title: Cookbook:Shirataki


text:
Template:ChessMove

/|

title: Template:ChessMove


text:
User:Exec Codes/sandbox

test

title: User:Exec Codes/sandbox


text:
User:Christophercbrown

I’m a product manager working in the open research services team within the digital resources directorate at Jisc.

I manage the UK ORCID Consortium and lead on developing a national persistent identifier research strategy. I also support a number of projects, activities and community groups in areas such as research notebooks, research data, research information, research analytics, open research, standards development, and work that supports researchers throughout the whole research lifecycle.

I actively support the work of international organisations, such as the Research Data Alliance (where I co-chair the National PID Strategies Working Group), engage with international groups and projects, and keep a watching brief on developments with persistent identifiers and research notebooks to ensure Jisc is positioned correctly with national and international developments.

title: User:Christophercbrown


text:
ORCID Advocacy Toolkit/contributors

People who have contributed to this project: Thanks to everyone else not mentioned here!

title: ORCID Advocacy Toolkit/contributors


text:
JavaScript/Dates/Exercises

1. You can show the offset between your timezone and UTC by using getTimezoneOffset. Show this offset.

2. Show the offset again, but don't use getTimezoneOffset. Instead, work with Date.now and Date.UTC.

title: JavaScript/Dates/Exercises


text:
User:BuiltByBromine

Just a kid from the United States of America.

title: User:BuiltByBromine


text:
High School Mathematics Extensions/Peano's Axioms

The Peano Axioms, also called Peano's Axioms, Dedekind-Peano Axioms, or the Peano postulates, are a series of axioms attempting to formalize the natural numbers ($$\mathbb{N}$$) using simple, intuitive axioms.

Introduction
During the time of Peano, mathematical notation as a whole—especially mathematical logic—was a new field, and common notions of second-order logic did not exist. Peano's own notation for logic was unpopular, but he did popularize something similar to the common way of denoting elements of a set ($$A \in B$$), using $$\epsilon$$. While we will not be using Peano's old notation, we will be expressing the axioms in a common notation which you should be familiar with from the set theory chapters of this book.

The Peano axioms describe the properties of the natural set, commonly denoted N or in blackboard bold, $$\mathbb{N}$$. The first axiom states that 0 is a natural number.

 * 1. $$0 \in \mathbb{N}$$. ("zero is a natural number")

The next four axioms describe the most important properties of the natural set, the equivalence relation.

 * 2. $$\forall x \in \mathbb{N}: x = x$$. ("all natural numbers equal themselves")
 * 3. $$\forall x,y,z \in \mathbb{N}: x=y, y=z \implies x=z$$. ("equality is transitive")
 * 4. $$\forall x,y \in \mathbb{N}: x=y \implies y=x$$. ("equality is symmetric")
 * 5. $$\forall x \in \mathbb{N}: x=y \implies y \in \mathbb{N}$$. ("if x is natural, and x equals y, y is natural")

After this, Peano introduces the idea of a successor function, commonly denoted $$\mathbf{S}$$, $$\widehat{S}$$, or $$\mathrm{succ}$$.

 * 6. $$\forall x \in \mathbb{N}: \mathrm{succ}(x) \in \mathbb{N}$$. ("the successor of any natural number is a natural number")
 * 7. $$\forall x, y \in \mathbb{N}: \mathrm{succ}(x) = \mathrm{succ}(y) \iff x = y$$. ("the successor function is an injection")
 * 8. $$\mathrm{succ}(x) = 0 \implies \bot$$. ("there is no x such that the successor of x is 0")

The obvious intuition is that we can compose the entire natural set by successively applying the successor function, but none of the axioms actually present this. Thus, there is motivation to also add the axiom of induction, which is commonly misattributed as part of Peano's axioms, but it was not described in Peano's original construction.

We can simply say that the natural set is constructible by successively applying the successor.
 * 9. The natural set, $$\mathbb{N}$$ is constructible by successively applying the successor function.


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Book:Electrical engineering

This is a redirect page for a book category.
 * 1) Redirect Shelf:Electrical engineering

title: Book:Electrical engineering


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Fractals/Computer graphic techniques/2D/coordinate

Coordinate of a plane =Types=

general types
 * linear/nonlinear ( scale of the axis )
 * dimensions : 1D, 2D, 3D, ...
 * orientation for 3D:
 * Handedness ( Right Hand Coordinate System (RHS) or Left Hand Coordinate System (LHS))
 * origin location
 * top-left origin coordinate system ( the origin is the upper-left corner of the screen and x and y go positive to the right and down)
 * bottom-left origin coordinate system ( The origin is the lower-left corner of the window or client area)
 * middle = center origin coordinate system ( The 0,0 coordinate is in the middle. x increases from left to right. y increases from bottom to top.)

There are different 2D coordinate systems :
 * cartesian coordinates ( rectangular, orthogonal)
 * elliptic coordinate ( Curvilinear )
 * parabolic
 * polar
 * Log-polar coordinates
 * Bipolar coordinates
 * normalized homogeneous coordinates

types in computer graphic
 * world coordinates ( with floating point values )
 * device coordinate
 * screen coordinates ( with integer values ) used for pixels of the screen or bitmap or array

Relation between these two types creates new items :
 * pixel size ( pixel spacing ) related with zoom
 * mapping ( converting) between screen and integer coordinates
 * clipping
 * rasterisation

These are not:
 * geospatial coordinates
 * geographic coordinate system (GCS)
 * spatial reference system (SRS) or coordinate reference system (CRS)

cartesian
Cartesian coordinates are used in Euclidean geometry.

Cartesian coordinate system
 * 1D, 2D, 3D, ...
 * linear scale on each axis ( preserve the shape )
 * coordinate
 * orthogonal ( rectangular)
 * polar

Elements
 * origin (0,0) and it's location
 * units
 * orientation
 * width, height

Homogenous Coordinates
Homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in 1827 are a system of coordinates used in projective geometry.

They have the advantage that:
 * the coordinates of points, including points at infinity, can be represented using finite coordinates
 * formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts
 * Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix
 * The use of normalized homogeneous coordinates avoids overflow and underflow errors in the iteration of rational functions

In homogeneous coordinates a point on a 2D plane is a 3-tuple ( a finite ordered list (sequence) of 3 numbers) : $$x,y,w$$

Elements
 * extended Euclidean plane or the real projective plane

conversions
convertion from homogeneous coordinates $$ (x,y,w) $$ to Cartesian coordinates $$ (x,y) $$, we simply divide each Cartesian coordinate $$ x$$ and $$y$$ by $$w$$

$$ (\frac{x}{w},\frac{y}{w})$$

To convert the Cartesian coordinates $$ (x,y) $$ to the homogeneous coordinates, add a third coordinate of 1 at the end :

$$ (x,y,1) $$

The matrix operations in homogeneous coordinates

 * interactive guide to homogeneous coordinates by Words and Buttons Online
 * interactive guide to homogeneous coordinates by Words and Buttons Online
 * mathworks : matrix-representation-of-geometric-transformations
 * Matrix Operations for Image Processing by Paul Haeberli Nov 1993

Log-polar coordinates
In mathematics, log-polar coordinates (or logarithmic polar coordinates) is a coordinate system in two dimensions, where a point is identified by two numbers:
 * one for the logarithm of the distance to a certain point
 * one for an angle

Log-polar coordinates are closely connected to polar coordinates, which are usually used to describe domains in the plane with some sort of rotational symmetry. In areas like harmonic and complex analysis, the log-polar coordinates are more canonical than polar coordinates. See also Exponential mapping

Definition
Log-polar coordinates in the plane consist of a pair of real numbers (ρ,θ), where ρ is the logarithm of the distance between a given point and the origin and θ is the angle between a line of reference (the x-axis) and the line through the origin and the point. The angular coordinate is the same as for polar coordinates, while the radial coordinate is transformed according to the rule

 * $$ r = e^\rho$$.

where $$ r $$ is the distance to the origin.

Log polar mapping: the approximate mapping from Cartesian space ( x,y) to polar space or (ρ,θ) space

$$ x,y \to \rho, \theta \to log(\rho), \theta$$

$$ \rho = \sqrt{(x - x_c)^2 + (y - y_c)^2 }$$ $$ \theta = \frac{y - y_c}{x - x_c}$$ where
 * where $$\rho$$ is the radial distance from the center of the image say $$(x_c, y_c) $$
 * $$\theta$$ denotes the angle.

image analysis
Already at the end of the 1970s, applications for the discrete spiral coordinate system were given in image analysis ( image registration ). To represent an image in this coordinate system rather than in Cartesian coordinates, gives computational advantages when rotating or zooming in an image. Also, the photo receptors in the retina in the human eye are distributed in a way that has big similarities with the spiral coordinate system. It can also be found in the Mandelbrot fractal (see picture to the right).

Log-polar coordinates can also be used to construct fast methods for the Radon transform and its inverse.

Discrete geometry
In order to solve a PDE numerically in a domain, a discrete coordinate system must be introduced in this domain. If the domain has rotational symmetry and you want a grid consisting of rectangles, polar coordinates are a poor choice, since in the center of the circle it gives rise to triangles rather than rectangles.

However, this can be remedied by introducing log-polar coordinates in the following way:
 * Divide the plane into a grid of squares with side length 2$$\pi$$/n, where n is a positive integer
 * Use the complex exponential function to create a log-polar grid in the plane. The left half-plane is then mapped onto the unit disc, with the number of radii equal to n. It can be even more advantageous to instead map the diagonals in these squares, which gives a discrete coordinate system in the unit disc consisting of spirals, see the figure to the right.

screen or image or device coordinate

 * The coordinate values in an image are always positive
 * The origin = point 90,0) is located at the upper left corner of the screen
 * The coordinates are integer values
 * the coordinate system on the screen is left-handed, i.e. the x coordinate increases from left to right and the y coordinate increases from top to bottom.

Raster 2D graphic

world coordinate
or simply :

=coordinate transformations= See also:
 * plane transformations

There are often many different possible coordinate systems for describing geometrical figures. The relationship between different systems is described by coordinate transformations, which give formulas for the coordinates in one system in terms of the coordinates in another system. For example, in the plane, if Cartesian coordinates (x, y) and polar coordinates (r, θ) have the same origin, and the polar axis is the positive x axis, then the coordinate transformation from polar to Cartesian coordinates is given by x = r cosθ and y = r sinθ.

With every bijection from the space to itself two coordinate transformations can be associated:
 * Such that the new coordinates of the image of each point are the same as the old coordinates of the original point (the formulas for the mapping are the inverse of those for the coordinate transformation)
 * Such that the old coordinates of the image of each point are the same as the new coordinates of the original point (the formulas for the mapping are the same as those for the coordinate transformation)

For example, in 1D, if the mapping is a translation of 3 to the right, the first moves the origin from 0 to 3, so that the coordinate of each point becomes 3 less, while the second moves the origin from 0 to −3, so that the coordinate of each point becomes 3 more. This is a list of some of the most commonly used coordinate transformations.

2-dimensional
Let (x, y) be the standard Cartesian coordinates, and (r, θ) the standard polar coordinates.

From polar coordinates

 * $$\begin{align}

x &= r\cos\theta \\ y &= r\sin\theta \\[5pt] \frac{\partial(x, y)}{\partial(r, \theta)} &= \begin{bmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{bmatrix} \\[5pt] \text{Jacobian} = \det{\frac{\partial(x, y)}{\partial(r, \theta)}} &= r \end{align}$$

From log-polar coordinates
The formulas for transformation from Cartesian coordinates to log-polar coordinates are given by

 * $$\begin{cases} \rho = \ln\left(\sqrt{ x^2 + y^2}\right), \\ \theta = \operatorname{atan2}(y,\, x). \end{cases}$$

and the formulas for transformation from log-polar to Cartesian coordinates are

 * $$\begin{cases}x = e^{\rho}\cos\theta, \\ y = e^{\rho}\sin\theta.\end{cases}$$

By using complex numbers (x, y) = x + iy, the latter transformation can be written as
 * $$ x + iy = e^{\rho+i\theta} $$

i.e. the complex exponential function.

From this follows that basic equations in harmonic and complex analysis will have the same simple form as in Cartesian coordinates. This is not the case for polar coordinates.

 * $$\begin{align}

x &= e^\rho\cos\theta, \\ y &= e^\rho\sin\theta. \end{align}$$

By using complex numbers $$(x, y) = x + iy'$$, the transformation can be written as
 * $$ x + iy = e^{\rho + i\theta}$$

That is, it is given by the complex exponential function.

From bipolar coordinates

 * $$\begin{align}

x &= a \frac{\sinh \tau}{\cosh \tau - \cos \sigma} \\ y &= a \frac{\sin \sigma}{\cosh \tau - \cos \sigma} \end{align}$$

From 2-center bipolar coordinates

 * $$\begin{align}

x &= \frac{1}{4c}\left(r_1^2 - r_2^2\right) \\ y &= \pm \frac{1}{4c}\sqrt{16c^2 r_1^2 - (r_1^2 - r_2^2 + 4c^2)^2} \end{align}$$

From Cesàro equation

 * $$\begin{align}

x &= \int \cos \left[\int \kappa(s) \,ds\right] ds \\ y &= \int \sin \left[\int \kappa(s) \,ds\right] ds \end{align}$$

From Cartesian coordinates

 * $$\begin{align}

r &= \sqrt{x^2 + y^2} \\ \theta' &= \arctan\left|\frac{y}{x}\right| \end{align}$$ Note: solving for $$\theta'$$ returns the resultant angle in the first quadrant ($0 < \theta < \frac{\pi}{2}$ ). To find $$\theta$$, one must refer to the original Cartesian coordinate, determine the quadrant in which $$\theta$$ lies (for example, (3,−3) [Cartesian] lies in QIV), then use the following to solve for $$\theta$$:

 * For $$\theta'$$ in QI:
 * $$\theta = \theta'$$
 * For $$\theta'$$ in QII:
 * $$\theta= \pi - \theta'$$
 * For $$\theta'$$ in QIII:
 * $$\theta = \pi + \theta' $$
 * For $$\theta'$$ in QIV:
 * $$\theta = 2\pi - \theta' $$

The value for $$\theta$$ must be solved for in this manner because for all values of $$\theta$$, $$\tan\theta$$ is only defined for $-\frac{\pi}{2}<\theta<+\frac{\pi}{2}$, and is periodic (with period $$\pi$$). This means that the inverse function will only give values in the domain of the function, but restricted to a single period. Hence, the range of the inverse function is only half a full circle.

Note that one can also use
 * $$\begin{align}

r &= \sqrt{x^2 + y^2} \\ \theta' &= 2 \arctan \frac{y}{x + r} \end{align}$$

From 2-center bipolar coordinates

 * $$\begin{align}

r &= \sqrt{\frac{r_1^2 + r_2^2 - 2c^2}{2}} \\ \theta &= \arctan \left[\sqrt{\frac{8c^2(r_1^2 + r_2^2 - 2c^2)}{r_1^2 - r_2^2} - 1}\right] \end{align}$$

Where 2c is the distance between the poles.

To log-polar coordinates from Cartesian coordinates

 * $$\begin{align}

\rho &= \log\sqrt{x^2 + y^2}, \\ \theta &= \arctan \frac{y}{x}. \end{align}$$

In Cartesian coordinates

 * $$\begin{align}

\kappa &= \frac{x'y - y'x}{({x'}^2 + {y'}^2)^\frac{3}{2}} \\ s &= \int_a^t \sqrt{{x'}^2 + {y'}^2}\, dt \end{align}$$

In polar coordinates

 * $$\begin{align}

\kappa &= \frac{r^2 + 2{r'}^2 - rr''}{(r^2 + {r'}^2)^\frac{3}{2}} \\ s &= \int_a^\varphi \sqrt{r^2 + {r'}^2}\, d\varphi \end{align}$$

3-dimensional
Let (x, y, z) be the standard Cartesian coordinates, and (ρ, θ, φ) the spherical coordinates, with θ the angle measured away from the +Z axis (as, see conventions in spherical coordinates). As φ has a range of 360° the same considerations as in polar (2 dimensional) coordinates apply whenever an arctangent of it is taken. θ has a range of 180°, running from 0° to 180°, and does not pose any problem when calculated from an arccosine, but beware for an arctangent.

If, in the alternative definition, θ is chosen to run from −90° to +90°, in opposite direction of the earlier definition, it can be found uniquely from an arcsine, but beware of an arccotangent. In this case in all formulas below all arguments in θ should have sine and cosine exchanged, and as derivative also a plus and minus exchanged.

All divisions by zero result in special cases of being directions along one of the main axes and are in practice most easily solved by observation.

From spherical coordinates

 * $$\begin{align}

x &= \rho \, \sin\theta \, \cos\varphi \\ y &= \rho \, \sin\theta \, \sin\varphi \\ z &= \rho \, \cos\theta \\ \frac{\partial(x, y, z)}{\partial(\rho, \theta, \varphi)} &= \begin{pmatrix} \sin\theta\cos\varphi & \rho\cos\theta\cos\varphi & -\rho\sin\theta\sin\varphi \\ \sin\theta\sin\varphi & \rho\cos\theta\sin\varphi &  \rho\sin\theta\cos\varphi \\ \cos\theta        &  -\rho\sin\theta        &  0 \end{pmatrix} \end{align}$$

So for the volume element:

dx\;dy\;dz = \det{\frac{\partial(x, y, z)}{\partial(\rho, \theta, \varphi)}} d\rho\;d\theta\;d\varphi = \rho^2 \sin\theta \; d\rho \; d\theta \; d\varphi $$

From cylindrical coordinates

 * $$\begin{align}

x &= r \, \cos\theta \\ y &= r \, \sin\theta \\ z &= z \, \\ \frac{\partial(x, y, z)}{\partial(r, \theta, z)} &= \begin{pmatrix} \cos\theta & -r\sin\theta & 0 \\ \sin\theta & r\cos\theta & 0 \\ 0 &           0 & 1       \end{pmatrix} \end{align}$$

So for the volume element:

dV = dx\;dy\;dz = \det{\frac{\partial(x, y, z)}{\partial(r, \theta, z)}} dr\;d\theta\;dz = r \; dr \; d\theta \; dz $$

From Cartesian coordinates

 * $$\begin{align}

\rho  &= \sqrt{x^2 + y^2 + z^2} \\ \theta &= \arctan \left( \frac{\sqrt{x^2 + y^2}}{z} \right)=\arccos \left( {\frac{z}{\sqrt{x^2 + y^2 + z^2}}} \right) \\ \varphi  &= \arctan \left( {\frac{y}{x}} \right) = \arccos \left( \frac{x}{\sqrt{x^2 + y^2}}\right) = \arcsin \left( \frac{y}{\sqrt{x^2 + y^2}}\right) \\ \frac{\partial\left(\rho, \theta, \varphi\right)}{\partial\left(x, y, z\right)} &= \begin{pmatrix} \frac{x}{\rho}                   & \frac{y}{\rho}                    &  \frac{z}{\rho} \\ \frac{xz}{\rho^2\sqrt{x^2 + y^2}} & \frac{yz}{\rho^2\sqrt{x^2 + y^2}} & -\frac{\sqrt{x^2 + y^2}}{\rho^2} \\ \frac{-y}{x^2 + y^2}             & \frac{x}{x^2 + y^2}               &  0 \\ \end{pmatrix} \end{align}$$

See also the article on atan2 for how to elegantly handle some edge cases.

So for the element:
 * $$d\rho\ d\theta\ d\varphi=\det\frac{\partial(\rho,\theta,\varphi)}{\partial(x,y,z)}dx\ dy\ dz=\frac{1}{\sqrt{x^2+y^2}\sqrt{x^2+y^2+z^2}}dx\ dy\ dz$$

From cylindrical coordinates

 * $$\begin{align}

\rho  &= \sqrt{r^2 + h^2} \\ \theta &= \arctan\frac{r}{h} \\ \varphi  &= \varphi \\ \frac{\partial(\rho, \theta, \varphi)}{\partial(r, h, \varphi)} &= \begin{pmatrix} \frac{r}{\sqrt{r^2 + h^2}} & \frac{h}{\sqrt{r^2 + h^2}} & 0 \\ \frac{h}{r^2 + h^2}       & \frac{-r}{r^2 + h^2}       & 0  \\ 0                         & 0                          & 1 \\            \end{pmatrix} \\ \det \frac{\partial(\rho, \theta, \varphi)}{\partial(r, h, \varphi)} &= \frac{1}{\sqrt{r^2+h^2}} \end{align}$$

From Cartesian coordinates

 * $$\begin{align}

r &= \sqrt{x^2 + y^2} \\ \theta &= \arctan{\left(\frac{y}{x}\right)} \\ z &= z \quad \end{align}$$

\frac{\partial(r, \theta, h)}{\partial(x, y, z)} = \begin{pmatrix} \frac{x}{\sqrt{x^2 + y^2}} & \frac{y}{\sqrt{x^2 + y^2}} & 0 \\ \frac{-y}{x^2 + y^2}      & \frac{x}{x^2+y^2}          & 0 \\ 0                         & 0                          & 1    \end{pmatrix} $$

From spherical coordinates

 * $$\begin{align}

r &= \rho \sin \varphi \\ h &= \rho \cos \varphi \\ \theta &= \theta \\ \frac{\partial(r, h, \theta)}{\partial(\rho, \varphi, \theta)} &= \begin{pmatrix} \sin\varphi & \rho\cos\varphi & 0 \\ \cos\varphi & -\rho\sin\varphi & 0 \\ 0       &  0            & 1 \\            \end{pmatrix} \\ \det\frac{\partial(r, h, \theta)}{\partial(\rho, \varphi, \theta)} &= -\rho \end{align}$$

Arc-length, curvature and torsion from Cartesian coordinates

 * $$\begin{align}

s &= \int_0^t \sqrt{{x'}^2 + {y'}^2 + {z'}^2}\, dt \\[3pt] \kappa &= \frac{\sqrt{(zy'-yz')^2 + (xz' - zx')^2 + (yx' - xy')^2}}{({x'}^2 + {y'}^2 + {z'}^2)^\frac{3}{2}} \\[3pt] \tau &= \frac{x(y'z - yz') + y(xz' - x'z) + z'(x'y - xy')}{{(x'y - xy')}^2 + {(xz'- x'z)}^2 + {(y'z - y''z')}^2} \end{align}$$

=implementations=

wgpu
wgpu uses the coordinate systems of D3D and Metal:
 * Render: center is 0, and radius is 1, so right upper is (1,1) and left lower point is ( -1,-1)
 * Texture: 0 is left upper point, right upper is (1,0), left lower is (0,1)

CSS box model for HTML

 * user coordinate system
 * local coordinate system
 * transformation matrix

HTML canvas’ coordinate system
 * s a two-dimensional grid
 * x increases from left to right: from x=0 (far left) to x=+960 (far right)
 * y decreases from bottom to top: goes from y=+720 (bottom) to y=0 (top).
 * The origin = 0,0 coordinate is in the top left ( The upper-left corner of the canvas )

OpenGL
5 different coordinate systems in OpenGL:
 * Local or Object
 * World
 * View or Eye
 * Clip
 * Screen

OpenGl :
 * "OpenGL is right handed in object space and world space, But in window space (aka screen space) we are suddenly left handed"
 * the transformation pipeline: local coordinates -> world coordinates -> view coordinates -> clip coordinates -> screen coordinates "Each coordinate system serves a purpose for one or more OpenGL features"
 * using
 * homogeneous coordinates
 * Normalized Device Coordinates ( NDC) only coordinates (x,y,z) : -1 ≤ x,y,z ≤ +1. Any coordinates that are outside this range will be discarded or clipped = won't be visible on screen
 * matrix calculus

OpenCV

 * The origin of the OpenCV image coordinate system is at the top-left corner of the image

SVG
SVG coordinate systems
 * The default coordinate system in SVG is much the same as in HTML
 * canvas the space or area where all SVG elements are drawn
 * viewport defines a drawing region characterized by size (width, height), and an origin, measured in abstract user units. Term SVG viewport is distinct from the "viewport" term used in CSS
 * viewBox

p5.js library
2D coordinate system
 * canvas
 * origin ( 0,0) it's location

=See also=
 * 2D plane transformations

=References=


text:
Cookbook:Linguine

Linguine, literally "little tongues", is a type of Italian pasta consisting of long, ovular noodles. It is made by extruding pasta dough.

title: Cookbook:Linguine


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AI Art Generation Handbook/Communities

There are many communities that are active in AI art generations, here are some of them.

Facebook
Stable Diffusion - The original Stable Diffusion group (Based on creation date) in Facebook that share the artwork done in Stable Diffusion

Stable Diffusion Universe- The splinter group of Stable Diffusion Group

AI Art: Prompt Craft (MidJourney, Stable Diffusion, Dall E, NightCafe) - A page for sharing AI generated images and the prompts used to create them.

Dalle2 Artwork • No Censorship - A FB group that share DALL-E works that are not censored (NSFW)

AI Revolution - MidJourney AI, DALL-E 2, Stable Diffusion - A place to share workflow for image generation and processing

Youtube
Here are the Youtube of some AI art which at some point share their technical workflow on AI art

https://www.youtube.com/c/1littlecoder - 1littlecoder

https://www.youtube.com/c/Aitrepreneur - Aitrepreneur

https://www.youtube.com/@NerdyRodent - NerdyRodent

https://www.youtube.com/@OlivioSarikas - Olivio Sarikas

https://www.youtube.com/@marat_ai - marat_ai

https://www.youtube.com/c/MattVideoProductions202 - MattVidPro AI

https://www.youtube.com/@MonzonMedia - Monzon Media

https://www.youtube.com/@sebastiankamph - Sebastian Kamph

https://www.youtube.com/@tingtingin - TingTingin

https://www.youtube.com/@EndangeredAI/videos - EndangeredAI

Reddit
r/AIart

r/StableDiffusion

r/DefendingAIart

r/AIGeneratedArt

r/bigsleep

r/dalle

r/dalle2

r/DNDAI - Reddit's community where showcase of AI Art generation to showcase the Dungeon and Dragon (DnD) character, background, game assets using AI

r/DeepDream

r/MediaSynthesis

r/sdforall An alternate sub-reddit when Stable Diffusion

Miscellaneous
Dataset Search Engine

LAION Aesthetic - A subset 12M (0.21% of aesthetical pleasing images)

ROM 1504 - A full search engine for all of the trained images in Stable Diffusion

HaveIBeenTrained - Reverse image search from 5B images to see if the images has been used in training and the site creator claimed it can be opted out
 * List of Artists Supported by DALL.E 2
 * Generate Unique Dall-E Prompts with Ease
 * Dall-E 2 Prompt Book

title: AI Art Generation Handbook/Communities


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Fractals/Computer graphic techniques/2D/transformation

Plane transformations

=Names=
 * transformations
 * Geometric Image Transformations
 * Geometrical transformations
 * coordinate transformations
 * projections
 * Cartographic Map projection ( in case of 3D to 2D graphic)
 * mappings

=Theory=

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

$$f:\mathbb{C}\to\mathbb{C}$$

It is implemented as point transformations.

It is usually denoted as

$$ w = f(z)$$

where
 * $$z = x + y i $$ is a point of complex plane ( input)
 * $$w = u + v i $$ is a point of image complex plane ( output = result)

=Example code =

For instance, Mercator:

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

=classifications= 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

=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"

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.

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 $$s = s_x+ s_y*i$$

$$x' = x * s_x $$ $$y' = y * s_y$$

becomes:

$$\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} s_x & 0 & 0 \\ 0 & s_y & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}$$

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 map

 * $$z' \to z + t $$

where
 * $$z = x + y*i $$
 * $$z' = x' + y'*i $$
 * $$t = t_x + t_y*i $$

Using this system, translation can be expressed with matrix multiplication. The functional form

$$x' = x + t_x $$ $$ y' = y + t_y$$

becomes:

$$\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & t_x \\ 0 & 1 & t_y \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}.$$

rotation
For rotation by an angle θ counterclockwise (positive direction) about the origin the functional form is

$$x' = x \cos \theta - y \sin \theta$$ $$y' = x \sin \theta + y \cos \theta$$.

Written in matrix form, this becomes:

$$\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin\theta & 0\\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}$$

Similarly, for a rotation clockwise (negative direction) about the origin, the functional form is

$$x' = x \cos \theta + y \sin \theta$$ $$y' = -x \sin \theta + y \cos \theta$$

the matrix form is:

$$\begin{bmatrix} x' \\ y' \\ 1 \end{bmatrix} = \begin{bmatrix} \cos \theta &  \sin\theta & 0 \\ -\sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ 1 \end{bmatrix}$$

These formulae assume that the x axis points right and the y axis points up.

the counter-clockwise rotation matrix in normalised coordinate:

$$\begin{bmatrix} \cos \theta & -\sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

C code without matrix

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:
 * Jason Davies: maps
 * barrel-distortion
 * swirl
 * log-polar map
 * Parabolic Cylindrical Coordinates
 * Parabolic Coordinates
 * Numerical approximation of conformal mappings
 * description
 * inflection
 * Altmetric: 25Citations: 1More detail Article | OPEN Exponential Sensitivity and its Cost in Quantum Physics by András Gilyén, Tamás Kiss & Igor Jex
 * Blaschke product
 * 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 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 $$f(z)$$ is conformal at infinity if the function $$f(1/z)$$ 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 $$f$$ as being conformal at infinity if it leaves angles unchanged at the "North Pole" of the 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

cylindrical projection
Cylindrical projection ( or cylindric p. ) maps from sphere ( without poles) to cylinder

Joukowsky transformation (map)
Description :
 * John D Cook

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


text:
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 : Fractals/Computer_graphic_techniques/2D/exp
 * 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

=Q&A= =Notation= If you want to zoom baseSize // constant value = extend in real and imaginary axis when zoomLevel is zero. zoomLevel // initial value = 0.0 zoomFactor = 2^(-zoomLevel) // initial value = 1.0 = no zoom sizeX = zoomFactor * baseSize sizeY = zoomFactor * baseSize minImaginary = centerY - sizeY/2 maxImaginary = centerY + sizeY/2
 * 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)

Note that base of zoom factor is important when one wants to optimize zooming by using previously computed pixels.

Usually other notation is used :

baseSize // constant value = extend in real and imaginary axis when zoomLevel is zero. zoomLevel // initial value = 0.0 zoomFactor = 10^zoomLevel // initial value = 1.0 = no zoom sizeX = baseSize / zoomFactor sizeY = baseSize / zoomFactor minImaginary = centerY - sizeY/2 maxImaginary = centerY + sizeY/2

" Some programs use a power of 2, others use scientific notation. " Dinkydau

magnification = 2^(6901.591) = 3.8539742140068607747447368372813E2077 = 3.8539742140068607747447368372813 *10^(2077)

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."

iSize = iXmax - iXmin ; // width of image in pixels ( integer or screen ) coordinate ; pixels are numbered from iXmin to iXmax; image is rectangular fSize = CxMax-CxMin // width of image in world ( floating point ) coordinate PixelWidth = fWidth / iWidth PixelSize = PixelWidth //

=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 : Re: -1.[encoded data, 133443 characters] Im: 7.[encoded data, 133449 characters] Zoom: 4.69449307780E100062 Iterations: 3149655 IterDiv: 1.000000 SmoothMethod: 1 ColorMethod: 0 ColorOffset: 0 Rotate: 0.000000 Ratio: 360.000000 Colors: 18,222,214,246,246,246,41,13,43,9,7,9,232,45,28,246,246,246,29,35,214,9,8,9,224,31,238,247,246,251,228,213,241,9,9,9,243,210,45,246,246,246,25,216,31,9,9,9,238,34,246,247,246,252,212,43,240,9,9,9,225,222,36,246,246,246,218,25,244,9,3,9,15,8,9,246,244,244,218,14,13,9,8,3,247,214,17,246,246,246,37,22,239,9,6,9,44,30,25,246,246,246,221,228,236,8,9,9,9,210,213,244,246,247,18,222,241,3,8,11,12,1,254,246,246,250,218,246,224,9,9,9,231,15,17,250,246,246,240,234,208,10,9,9,242,34,210,252,246,246,246,236,14,10,9,3,238,45,14,249,247,246,225,246,247,9,10,10,217,227,229,246,246,248,22,11,230,7,7,9,34,209,10,246,246,245,37,236,26,9,9,5,219,213,16,246,246,244,3,45,14,7,9,6,214,214,37,246,246,246,224,13,214,9,9,9,246,224,24,246,246,246,3,31,227,0,9,9,2,37,37,243,246,246,12,24,233,4,9,9,24,229,9,246,247,244,34,225,18,9,9,5,224,5,27,249,246,246,241,41,45,9,9,9,32,246,245,246,246,250,25,37,230,9,9,9,226,34,238,246,246,251,23,246,238,9,9,9,238,2,47,250,245,246,237,38,15,9,9,9,9,220,233,246,247,246,235,235,31,9,9,5,20,208,15,246,246,246,28,219,31,9,9,4,218,36,8,246,246,246,45,33,242,9,7,9,222,23,35,246,246,246,22,232,243,9,9,9,210,39,45,246,246,246,24,41,20,9,9,9,241,246,227,246,248,246,19,48,215,9,9,9,219,29,241,246,246,246,213,224,45,9,9,9,45,208,232,246,247,250,221,240,241,9,11,9,37,246,3,246,246,243,229,13,9,9,9,6,7,240,44,243,253,246,9,255,9,9,9,2,236,37,15,249,246,246,229,232,237,9,9,9,18,230,213,245,246,246,22,5,45,9,9,9,242,254,218,246,254,246,14,246,23,2,9,9,12,31,214,246,246,249,232,21,248,10,9,11,250,231,241,246,246,246,28,30,27,5,5,9,19,15,228,246,246,246,30,41,219,9,9,9,229,249,215,251,249,249,246,219,246,11,9,9,244,41,26,246,246,246,24,235,236,4,10,9,11,252,7,246,247,246,246,43,237,9,9,9,8,214,0,245,246,245,27,218,30,9,9,9,214,41,236,246,246,251,13,18,240,2,9,9,9,234,17,243,246,246,7,210,239,9,9,10,240,44,235,247,246,247,212,33,218,9,7,9,237,24,31,250,246,246,238,246,246,9,9,9,210,47,27,246,246,246,19,22,45,8,3,9,210,9,238,247,246,248,242,224,221,9,10,9,18,252,28,246,249,246,37,210,39,9,8,9,230,13,30,246,244,245,216,4,5,9,7,6,221,210,45,246,246,246,9,218,224,1,9,9,9,41,36,244,246,246,19,224,45,9,9,9,219,28,239,247,246,246,232,216,13,9,9,9,213,38,233,246,246,246,212,39,212,9,9,7,42,48,9,246,248,245,35,248,29,9,11,9,236,251,228,246,248,250,40,210,244,5,9,10,9,43,229,246,246,246,213,45,28,8,7,7,10,12,30,246,246,246,217,246,210,9,9,9,37,45,23,245,246,246,6,6,45,9,6,9,225,44,37,247,246,246,228,24,27,9,8,9,244,43,45,248,246,246,216,13,220,9,7,9,230,45,218,246,246,246,4,23,210,5,5,9,37,24,24,246,246,246,228,26,33,10,9,9,255,210,228,247,246,246,41,45,19,9,9,8,246,223,43,246,249,246,25,243,46,3,9,7,9,210,15,246,246,244,249,28,3,9,9,9,31,219,246,246,246,246,37,34,30,9,9,9,234,41,219,246,246,250,41,245,252,9,9,9,210,37,35,247,246,246,245,18,31,9,8,9,9,210,210,243,248,246,9,246,11,6,9,4,209,20,29,246,246,246,228,224,232,9,9,10,210,37,241,246,246,246,234,242,31,9,9,9,30,23,230,246,246,246,46,229,25,9,9,4,245,212,14,249,246,246,221,29,238,9,9,9,25,216,233,246,247,246,227,237,45,9,9,9,41,16,231,246,246,246, Smooth: 0 MultiColor: 0 BlendMC: 0 MultiColors: 17	0	1,7	0	0,2	0	0,-100	0	2,5	0	0,-100	0	2, Power: 2 FractalType: 0 Slopes: 1 SlopePower: 1500 SlopeRatio: 50 SlopeAngle: 45 imag: 1 real: 1 SeedR: 0 SeedI: 0 FactorAR: 1 FactorAI: 0

=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 :

Magnification = 6.666667e-01 ; radius = 1.500000e+00 ; PixelSpacing = 3.000000e-03 ; PixelSpacingBits = 8 use float Magnification = 1.333333e+00 ; radius = 7.500000e-01 ; PixelSpacing = 1.500000e-03 ; PixelSpacingBits = 9 use float Magnification = 2.666667e+00 ; radius = 3.750000e-01 ; PixelSpacing = 7.500000e-04 ; PixelSpacingBits = 10 use float Magnification = 5.333333e+00 ; radius = 1.875000e-01 ; PixelSpacing = 3.750000e-04 ; PixelSpacingBits = 11 use float Magnification = 1.066667e+01 ; radius = 9.375000e-02 ; PixelSpacing = 1.875000e-04 ; PixelSpacingBits = 12 use float ... Magnification = 7.158279e+08 ; radius = 1.396984e-09 ; PixelSpacing = 2.793968e-12 ; PixelSpacingBits = 38 use float Magnification = 1.431656e+09 ; radius = 6.984919e-10 ; PixelSpacing = 1.396984e-12 ; PixelSpacingBits = 39 use float Magnification = 2.863312e+09 ; radius = 3.492460e-10 ; PixelSpacing = 6.984919e-13 ; PixelSpacingBits = 40 use float Magnification = 5.726623e+09 ; radius = 1.746230e-10 ; PixelSpacing = 3.492460e-13 ; PixelSpacingBits = 41 use double Magnification = 1.145325e+10 ; radius = 8.731149e-11 ; PixelSpacing = 1.746230e-13 ; PixelSpacingBits = 42 use double Magnification = 2.290649e+10 ; radius = 4.365575e-11 ; PixelSpacing = 8.731149e-14 ; PixelSpacingBits = 43 use double Magnification = 4.581298e+10 ; radius = 2.182787e-11 ; PixelSpacing = 4.365575e-14 ; PixelSpacingBits = 44 use double Magnification = 9.162597e+10 ; radius = 1.091394e-11 ; PixelSpacing = 2.182787e-14 ; PixelSpacingBits = 45 use double Magnification = 1.832519e+11 ; radius = 5.456968e-12 ; PixelSpacing = 1.091394e-14 ; PixelSpacingBits = 46 use double Magnification = 3.665039e+11 ; radius = 2.728484e-12 ; PixelSpacing = 5.456968e-15 ; PixelSpacingBits = 47 use double Magnification = 7.330078e+11 ; radius = 1.364242e-12 ; PixelSpacing = 2.728484e-15 ; PixelSpacingBits = 48 use double Magnification = 1.466016e+12 ; radius = 6.821210e-13 ; PixelSpacing = 1.364242e-15 ; PixelSpacingBits = 49 use double Magnification = 2.932031e+12 ; radius = 3.410605e-13 ; PixelSpacing = 6.821210e-16 ; PixelSpacingBits = 50 use double Magnification = 5.864062e+12 ; radius = 1.705303e-13 ; PixelSpacing = 3.410605e-16 ; PixelSpacingBits = 51 use long double Magnification = 1.172812e+13 ; radius = 8.526513e-14 ; PixelSpacing = 1.705303e-16 ; PixelSpacingBits = 52 use long double Magnification = 2.345625e+13 ; radius = 4.263256e-14 ; PixelSpacing = 8.526513e-17 ; PixelSpacingBits = 53 use long double Magnification = 4.691250e+13 ; radius = 2.131628e-14 ; PixelSpacing = 4.263256e-17 ; PixelSpacingBits = 54 use long double Magnification = 9.382499e+13 ; radius = 1.065814e-14 ; PixelSpacing = 2.131628e-17 ; PixelSpacingBits = 55 use long double Magnification = 1.876500e+14 ; radius = 5.329071e-15 ; PixelSpacing = 1.065814e-17 ; PixelSpacingBits = 56 use long double Magnification = 3.753000e+14 ; radius = 2.664535e-15 ; PixelSpacing = 5.329071e-18 ; PixelSpacingBits = 57 use long double Magnification = 7.505999e+14 ; radius = 1.332268e-15 ; PixelSpacing = 2.664535e-18 ; PixelSpacingBits = 58 use long double Magnification = 1.501200e+15 ; radius = 6.661338e-16 ; PixelSpacing = 1.332268e-18 ; PixelSpacingBits = 59 use long double Magnification = 3.002400e+15 ; radius = 3.330669e-16 ; PixelSpacing = 6.661338e-19 ; PixelSpacingBits = 60 use long double Magnification = 6.004800e+15 ; radius = 1.665335e-16 ; PixelSpacing = 3.330669e-19 ; PixelSpacingBits = 61 use arbitrary precision number, like MPFR Magnification = 1.200960e+16 ; radius = 8.326673e-17 ; PixelSpacing = 1.665335e-19 ; PixelSpacingBits = 62 use arbitrary precision number, like MPFR Magnification = 2.401920e+16 ; radius = 4.163336e-17 ; PixelSpacing = 8.326673e-20 ; PixelSpacingBits = 63 use arbitrary precision number, like MPFR Magnification = 4.803840e+16 ; radius = 2.081668e-17 ; PixelSpacing = 4.163336e-20 ; PixelSpacingBits = 64 use arbitrary precision number, like MPFR Magnification = 9.607679e+16 ; radius = 1.040834e-17 ; PixelSpacing = 2.081668e-20 ; PixelSpacingBits = 65 use arbitrary precision number, like MPFR Magnification = 1.921536e+17 ; radius = 5.204170e-18 ; PixelSpacing = 1.040834e-20 ; PixelSpacingBits = 66 use arbitrary precision number, like MPFR Magnification = 3.843072e+17 ; radius = 2.602085e-18 ; PixelSpacing = 5.204170e-21 ; PixelSpacingBits = 67 use arbitrary precision number, like MPFR Magnification = 7.686143e+17 ; radius = 1.301043e-18 ; PixelSpacing = 2.602085e-21 ; PixelSpacingBits = 68 use arbitrary precision number, like MPFR Magnification = 1.537229e+18 ; radius = 6.505213e-19 ; PixelSpacing = 1.301043e-21 ; PixelSpacingBits = 69 use arbitrary precision number, like MPFR Magnification = 3.074457e+18 ; radius = 3.252607e-19 ; PixelSpacing = 6.505213e-22 ; PixelSpacingBits = 70 use arbitrary precision number, like MPFR Magnification = 6.148915e+18 ; radius = 1.626303e-19 ; PixelSpacing = 3.252607e-22 ; PixelSpacingBits = 71 use arbitrary precision number, like MPFR Magnification = 1.229783e+19 ; radius = 8.131516e-20 ; PixelSpacing = 1.626303e-22 ; PixelSpacingBits = 72 use arbitrary precision number, like MPFR Magnification = 2.459566e+19 ; radius = 4.065758e-20 ; PixelSpacing = 8.131516e-23 ; PixelSpacingBits = 73 use arbitrary precision number, like MPFR

=Zoom types=
 * manual/automatic
 * discrete (steps) / continous

autozoom

 * fractalforums.org: various-autozoom-functions

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

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:

=Zoom examples=

deep zoom
Baby Monster by DinkydauSet: zoom into parameter plane of complex quadratic polynomial :

program : mandel machine x one week ( 4000x3000 ) algorithm : escape time precision = 7000 magnification = 2^(6901.591) = 3.8539742140068607747447368372813E2077 = 3.8539742140068607747447368372813 *10^(2077) centerRe =     0.4026223242141852148037824724584025835755135011342881027042525627039569502881670946745557412930327115393258794913548834454537746442641230778661250740808696667685167031459830818673342528174897920124209545137508922550549968221550363823064128549715028301461622593069839496780093952808237573205197164572673594064416772605893796305663235635527880132594672201779519565231079208219169990178251347386228988056919160862667394171741551928464324300754658056232719271701044207283382298845257395452425798232763791743779314353854383722879333193167946859425243707261854429090471722384741345982162350257163211439072535377347476863063971161548476045589432821456788895124531684440296396587503362225268140306547818930627963692042601813606866625158984034539104729252270036522838631376204929633117350514392534098621950298874921561848480721715466716034612641357006597673782157926191463771077735121499521415799630479650549024192802530771425233126786755033877940979179549256821540572419329726522803422394291516882903607750877675838298417678464016201757067174161515701977269714418733430534088597594663546370996913330820306039197466323342547246232849027813752870111841998012412400077840383443533115591420661427063968799614034835064533287768378088050419077884398682152379426978057892246066882890819414748458464211672718229513050617638747371353729091501912075437789263019440768014217748069621815249161576890441895414789868155359854876653884160567578413850597585526839599148529091786964892859271522516203113708568784835766360055829328924133199024188688844937378174198793904588311188393798698747503071556711010350663949412136302614381521786902283300674108990630100130448100638953468850976478379826240732566854132975350164428623726436941877225968956042433264823693750422381553726384657388053048021285450658976425595261910844568695172761539871359485880883688256972852102360571149799163601150837375544386512748326612079156006358192675222805435405708616040869538348696626733367203267216061599286266159729225868003852298996109238906739358339973449675179531666048724844664052031271387710787740075700568712301627930122446 centerIm = 0.3414661267010330639258781735300664742792277354243426548082347808924808079842063432641337745084491842274928006043526205335660078979452366866756969329129441667312564036874357179068281915055443525509599130331320610821012236201307641349065669509178006415633193858304674801753225345910395424302568889690765898697389344506716385444532416760848755811454604769082865518885687677043988923482755514450647501464529277548626228211775326421789838725360483456061933115401603120908257520719931828418717929225819541974242262149693721546436344398096633938554552017842417617179797586236724871081452420464280333125369667508863083941830427529275353658367633700463591140837752713130326658698539827580385561537043307893113743040288109423825683661313727523597674848484087312515923976610370831729746783850452751041682640731573001319728478688445020726240850018852640901126812894024112093280413500010748891409049642255971933031658182432137372842717466247939807168712399315317154441773203170047657667231871264210848409477591021867775954396341273647285869183323193925375356957738345603739082474205799801007250041854109282041356811030974599202640896266028843535401330225609771341865327032085135831902603640939226040984911910075236054433300263457714999659548633581618107640012502951867204419221851370812448359108387310012121047998792211175821507898822472067545566323088233165752919891462234618228444539224969139541397976528218516885487556195557179718372941445119064646137527396209369577296787387634243210140885928439211898265659865402840837723799792668419799443445421585714176943014398043568698898179253868782669987422851258992794036747294932260334459254264688360704250224826019403156200812024463700689508527620855762445523530566725311570637375162222843632726633580544365800704348649164263654725083759306974477081672768736376743801103735815276810809861222374937053298147754358060737309317498277835724639189151138998429279826787118135717747963875281826078928680133524179137446312206008103340479076493412664529374765456320670666843471865606806283972354009907403302001818229385746724505497627613149001435837120422584 iterationMax = 35 000 000

Cross Hair by Claude Heiland-Allen. Below is the content of kfr file ( parameter for Kalles Fraktaler )

Re: -1.94156695046089411519723545042405785041614289882129276726563682132425517088701591901639765783026799492137462058533435865811646793455507945 Im: -0.00017301617109765913618843129805085094212526763902256359889382674446838992551267030035919209576862092841923454016429306276671035659004195 Zoom: 3.15216049570E116 Iterations: 203416 IterDiv: 1.000000 SmoothMethod: 0 ColorMethod: 0 ColorOffset: 0 Rotate: 0.000000 Ratio: 360 Colors: 0,0,0,41,35,190,132,225,108,214,174,82,144,73,241,241,187,233,235,179,166,219,60,135,12,62,153,36,94,13,28,6,183,71,222,179,18,77,200,67,187,139,166,31,3,90,125,9,56,37,31,93,212,203,252,150,245,69,59,19,13,137,10,28,219,174,50,32,154,80,238,64,120,54,253,18,73,50,246,158,125,73,220,173,79,20,242,68,64,102,208,107,196,48,183,50,59,161,34,246,34,145,157,225,139,31,218,176,202,153,2,185,114,157,73,44,128,126,197,153,213,233,128,178,234,201,204,83,191,103,214,191,20,214,126,45,220,142,102,131,239,87,73,97,255,105,143,97,205,209,30,157,156,22,114,114,230,29,240,132,79,74,119,2,215,232,57,44,83,203,201,18,30,51,116,158,12,244,213,212,159,212,164,89,126,53,207,50,34,244,204,207,211,144,45,72,211,143,117,230,217,29,42,229,192,247,43,120,129,135,68,14,95,80,0,212,97,141,190,123,5,21,7,59,51,130,31,24,112,146,218,100,84,206,177,133,62,105,21,248,70,106,4,150,115,14,217,22,47,103,104,212,247,74,74,208,87,104,118,250,22,187,17,173,174,36,136,121,254,82,219,37,67,229,60,244,69,211,216,40,206,11,245,197,96,89,61,151,39,138,89,118,45,208,194,201,205,104,212,73,106,121,37,8,97,64,20,177,59,106,165,17,40,193,140,214,169,11,135,151,140,47,241,21,29,154,149,193,155,225,192,126,233,168,154,167,134,194,181,84,191,154,231,217,35,209,85,144,56,40,209,217,108,161,102,94,78,225,48,156,254,217,113,159,226,165,226,12,155,180,71,101,56,42,70,137,169,130,121,122,118,120,194,99, Smooth: 1 MultiColor: 0 BlendMC: 0 MultiColors: Power: 2 FractalType: 0

Re = -1.479,946,223,325,078,880,202,580,653,442,563,833,590,828,874,828,533,272,328,919,467,504,501,428,041,551,458,102,123,157,715,213,651,035,545,943,542,078,167,348,953,885,787,341,902,612,509,986,72 Im = 0.000,901,397,329,020,353,980,197,791,866,197,173,566,251,566,818,045,102,411,067,630,386,488,692,287,189,049,145,621,584,436,026,934,218,763,527,577,290,631,809,454,796,618,110,767,424,580,322,79 Magnification = 2^471 = 6.0971651373359223269171820894398 E141
 * 1) example by Dinkydau : http://www.fractalforums.com/announcements-and-news/superfractalthing-arbitrary-precision-mandelbrot-set-rendering-in-java/

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=