User:IntegralPython/sandbox: Difference between revisions

Revision as of 02:22, 2 February 2022

Latest revision as of 02:27, 6 June 2022

(4 intermediate revisions by one other user not shown)

Line 6:

==Open set condition==

In [[fractal geometry]], the '''open set condition''' ('''OSC''') is a commonly imposed condition on self-similar fractals. In some sense, the condition imposes restrictions on the overlap in a fractal construction.<ref>{{cite journal |last1=Bandt |first1=Christoph |last2= Viet Hung |first2= Nguyen |last3 = Rao |first3 = Hui | title=On the Open Set Condition for Self-Similar Fractals | journal=Proceedings of the American Mathematical Society | volume=134 | year=2006 | pages=1369–74 | issue=5 |doi=10.1090/S0002-9939-05-08300-0 |jstor=4097989 | url=http://www.jstor.org/stable/4097989| url-access=limited}}</ref> Specifically, given an iterative[[iterated function system]] of [[contraction mapping| contractive mappings]] fψ_''i'', the open set condition requires that there exists a nonempty, open set S satisfying two conditions:

#<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V, </math>

# Each <math>\psi_i (V)</math> is pairwise disjoint.

Introduced in 1946 by P.A.P Moran,<ref>{{cite journal | last=Moran | first=P.A.P. | title=Additive Functions of Intervals and Hausdorff Measure | journal=Proceedings-Cambridge Philosophical Society | volume=42 | year=1946 | pages=15–23 | doi=10.1017/S0305004100022684}}</ref> the open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket. It is also used to simplify computation of the packing measure.<ref>{{cite journal| last1=Llorente|first1=Marta|last2=Mera|first2=M. Eugenia| last3=Moran| first3=Manuel| title= On the Packing Measure of the Sierpinski Gasket | journal= University of Madrid | url=https://eprints.ucm.es/id/eprint/58898/1/version%20final(previa%20prueba%20imprenta).pdf}}</ref>

The open set condition is used to compute the dimensions of certain self-similar fractals, notably the Sierpinski Gasket.

https://www.jstor.org/stable/4097989?read-now=1&refreqid=excelsior%3A19b81930eca74e0a264d556ab56211ae&seq=1#page_scan_tab_contents

An equivalent statement of the open set condition is to require that the s-dimensional [[Hausdorff measure]] of the set is greater than zero.<ref>

{{cite web |url=https://www.math.cuhk.edu.hk/conference/afrt2012/slides/Wen_Zhiying.pdf |title=Open set condition for self-similar structure |last= Wen |first=Zhi-ying |publisher=Tsinghua University |access-date= 1 February 2022 }} </ref>

===Computing Hausdorff measure===

Line 27 ⟶ 26:

:<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. </math>

Taking [[natural logarithm]]s of both sides of the above equation, we can solve for ''s'', that is: ''s'' = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC. In general a set ''E'' which is a fixed point of a mapping

: <math> A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) </math>

is self-similar if and only if the intersections

:<math> H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0, </math>

where ''s'' is the Hausdorff dimension of ''E'' and ''H^s'' denotes [[Hausdorff measure]]. This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally. Indeed, under the same conditions as above, the unique fixed point of ψ is self-similar.

==Hand-eye calibration problem==

Line 47 ⟶ 36:

====Separable solutions====

Given the equation {{math|AX{{=}}ZB}}, it is possible to decompose the equation into a purely rotational and translational part; methods utilizing this are referred to as separable methods. Where {{math|'''R'''_A}} represents a 3×3 rotation matrix and {{math|'''t'''_A}} a 3×1 translation vector, the equation can be broken into two parts:<ref>[{{cite journal | url=https://arxiv.org/pdf/1907.12425.pdf] | arxiv=1907.12425 | doi=10.1007/s00138-017-0841-7 | title=Solving the robot-world hand-eye(s) calibration problem with iterative methods | year=2017 | last1=Tabb | first1=Amy | last2=Ahmad Yousef | first2=Khalil M. | journal=Machine Vision and Applications | volume=28 | issue=5–6 | pages=569–590 | s2cid=20150713 }}</ref>

:{{math|'''R'''_A'''R'''_X{{=}}'''R'''_Z'''R'''_B}}

:{{math|'''R'''_A'''t'''_X+'''t'''_A{{=}}'''R'''_Z'''t'''_B+'''t'''_Z}}

Line 86 ⟶ 75:

{{collapse top | Ugandan knuckles}}

Ugandan Knuckles is an [[internet meme]] from January 2018 depicting a deformed version of [[Knuckles the Echidna]]. Players would go in hords to the virtual reality video game ''[[VRChat]]'' to troll other players. The people would say quotes such as "Do you know the way?", which originate from the 2010 Ugandan action film ''[[Who Killed Captain Alex?]]'', as well as "spitting" on other users whom they felt did not know "de way".<ref name="dailydot">{{Cite web|url=https://www.dailydot.com/unclick/ugandan-knuckles-vrchat-meme/|title=How Ugandan Knuckles turned VRChat into a total trollfest|last=Hathaway|first=Jay|date=11 January 2018|website=The Daily Dot|archive-url=|archive-date=|dead-url=|access-date=13 January 2018}}</ref><ref>{{Cite web|url=https://heavy.com/games/2018/01/controversial-ugandan-knuckles-meme/|title=Controversial ‘Ugandan'Ugandan Knuckles’Knuckles' Meme Has Infested VRChat|last=MacGregor|first=Collin|date=9 January 2018|website=Heavy.com|archive-url=|archive-date=|dead-url=|access-date=13 January 2018}}</ref> The meme was a significant trend followed by several news organisations, including ''[[USA Today]]''.<ref>https://www.usatoday.com/story/tech/news/2018/02/09/ugandan-knuckles-do-you-know-de-wey-meme-explained/307575002/ Retrieved October 9 2018</ref>

===History===

Line 92 ⟶ 81:

===Controversy===

Because of its use of a fake Ugandan accent as well as the quotations from ''Who Killed Captain Alex?'', the meme was widely criticized for being racially insensitive;<ref name="dailydot"/><ref name=Polygon2/> ''[[Polygon (website)|Polygon]]'' described it as problematic.<ref name=Polygon2>{{cite web|url=https://www.polygon.com/2018/1/8/16863932/ugandan-knuckles-meme-vrchat|title=‘Ugandan'Ugandan Knuckles’Knuckles' is overtaking VRChat|last=Alexander|first=Julia|publisher=[[Vox Media, Inc.]]|work=Polygon|date=October 9, 2018|accessdate=January 9, 2018}}</ref> On January 27 2018, the company [[Razer Inc.|Razer]] was brought under fire for posting a Ugandan Knuckles meme that was widely criticised as a racist misstep.<ref>https://gizmodo.com/does-razer-know-it-posted-a-racist-meme-1822485212 Retrieved October 9 2018</ref>

The original creator of the 3D avatar, [[DeviantArt]] user "tidiestflyer", showed regret over the character, saying that he hoped it would not be used to annoy players of ''VRChat'' and that he enjoys the game and does not want to see anyone's rights get taken away because of the avatar.<ref>{{Cite web|url=http://www.gamerevolution.com/news/362289-creator-vrchats-ugandan-knuckles-meme-regrets-decision|title=Creator of VRChat’sVRChat's ‘Ugandan'Ugandan Knuckles’Knuckles' Meme Regrets His Decision|last=Tamburro|first=Paul|date=8 January 2018|website=GameRevolution|archive-url=|archive-date=|dead-url=|access-date=9 October 2018}}</ref> In response to the trolling in the game, the developers of ''VRChat'' published an open letter on ''[[Medium (website)|Medium]]'', stating that they were developing "new systems to allow the community to better self moderate" and asking users to use the built-in muting features.<ref>{{Cite web |url=https://www.polygon.com/2018/1/10/16875716/vrchat-safety-concerns-open-letter-players-behavior |title=VRChat team speaks up on player harassment in open letter |last=Alexander |first=Julia |date=January 10, 2018 |website=Polygon |access-date=October 9, 2018}}</ref>

{{collapse bottom}}