翻訳と辞書 Words near each other ・ Mandi (Lok Sabha constituency) ・ Mandi (Mirpur) ・ Mandi (Vidhan Sabha constituency) ・ Mandi Ahmed Abad railway station ・ Mandi Bahauddin ・ Mandi Bahauddin District ・ Mandi Bahauddin railway station ・ Mandelbaum Effect ・ Mandelbaum Gate ・ Mandelbaum House ・ Mandelbox ・ Mandelbrodt ・ Mandelbrot ・ Mandelbrot Competition ・ Mandelbrot set ・ Mandelbulb ・ Mandelholz Dam ・ Mandelia Faunal Reserve ・ Mandelic acid ・ Mandelieu-la-Napoule ・ Mandelin reagent ・ Mandelkubb ・ Mandell ・ Mandell Berman ・ Mandell Creighton ・ Mandell L. and Madeleine H. Berman Foundation ・ Mandelli's mouse-eared bat ・ Mandello ・ Mandello del Lario ・ Mandello Vitta Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Mandelbulb ： ウィキペディア英語版 Mandelbulb The Mandelbulb is a three-dimensional fractal, constructed by Daniel White and Paul Nylander using spherical coordinates in 2009. A canonical 3-dimensional Mandelbrot set does not exist, since there is no 3-dimensional analogue of the 2-dimensional space of complex numbers. It is possible to construct Mandelbrot sets in 4 dimensions using quaternions. However, this set does not exhibit detail at all scales like the 2D Mandelbrot set does. White and Nylander's formula for the "''n''th power" of the vector $=\; \backslash langle\; x,\; y,\; z\backslash rangle$ in is :$^n\; :=\; r^n\backslash langle\backslash sin(n\backslash theta)\backslash cos(n\backslash phi),\backslash sin(n\backslash theta)\backslash sin(n\backslash phi),\backslash cos(n\backslash theta)\backslash rangle$ where $r=\backslash sqrt$, $\backslash phi=\backslash arctan(y/x)=\backslash arg\; (x+yi)$, and $\backslash theta=\backslash arctan(\backslash sqrt/z)=\backslash arccos(z/r)$. The Mandelbulb is then defined as the set of those $$ in for which the orbit of $\backslash langle\; 0,\; 0,\; 0\backslash rangle$ under the iteration $\backslash mapsto\; ^n+$ is bounded.〔 see "formula" section〕 For ''n'' > 3, the result is a 3-dimensional bulb-like structure with fractal surface detail and a number of "lobes" depending on ''n''. Many of their graphic renderings use ''n'' = 8. However, the equations can be simplified into rational polynomials when n is odd. For example, in the case ''n'' = 3, the third power can be simplified into the more elegant form: :$\backslash langle\; x,\; y,\; z\backslash rangle^3\; =\; \backslash left\backslash langle\backslash \; \backslash frac\; ,\backslash frac,z(z^2-3x^2-3y^2)\backslash right\backslash rangle$. ==Quadratic formula== Other formulae come from identities which parametrise the sum of squares to give a power of the sum of squares such as: :$$ (x^2-y^2-z^2)^2+(2 x z)^2+(2xy)^2 = (x^2+y^2+z^2)^2 which we can think of as a way to square a triplet of numbers so that the modulus is squared. So this gives, for example: :$$ x\rightarrow x^2-y^2-z^2+x_0 :$$ y\rightarrow 2 x z+y_0 :$$ z\rightarrow 2 x y +z_0 or various other permutations. This 'quadratic' formula can be applied several times to get many power-2 formulae. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mandelbulb」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.