Matrix67 博客里漸變的分形動畫效果是怎麼實現的？

01-11

7 個分形圖形的動畫演示
http://www.matrix67.com/blogimage_2014/fractal_gifs/koch.gif
類似這種效果是比較容易的

就是想辦法做了一下插值而已

放上我當時仿的代碼吧

order = 5; step = 8; next = If[#2 == 0, {#1}, N@{{#1[[1]], 2/3 #1[[1]] + 1/3 #1[[2]]}, {2/3 #1[[1]] + 1/3 #1[[2]], 2/3 #1[[1]] + 1/3 #1[[2]] + Exp[#2 I [Pi]/3]/3 (#1[[2]] - #1[[1]])}, {1/3 #1[[1]] + 2/3 #1[[2]] + Exp[-#2 I [Pi]/3]/3 (#1[[1]] - #1[[2]]), 1/3 #1[[1]] + 2/3 #1[[2]]}, {1/3 #1[[1]] + 2/3 #1[[2]], #1[[2]]}}] ; nextstep = next[#, 1] ; node = NestList[ Flatten[nextstep /@ #, 1] , {{0, 1}, {1, Exp[-I [Pi]/3]}, {Exp[-I [Pi]/3], 0}}, order]; result = Table[ Flatten[next[#, FractionalPart[k]] /@ node[[IntegerPart[k]]], 1], {k, 1, order + 1 - 1/step, 1/step}]; preplot = Map[{Re[#], Im[#]} , result, {3}];


data = Table[

   Graphics[{Blue, Line /@ (preplot[[i]])},

    PlotRange -&> {{0, 1}, {-Sqrt[3]/2, Sqrt[3]/6}},

    ImageSize -&> 600], {i, Length@preplot}];

data = Flatten[{data, Reverse[data]}, 1];

ListAnimate[data, AnimationRate -&> 12]

圖片見http://fmn.rrimg.com/fmn066/20141208/1345/original_HJ6M_78d300007343125f.gif

然後另外一個做了點漸變的處理，代碼如下

(*triangle*) order = 5; step = 100; next = Table[ N@{#[[1]] + #[[2]] Exp[2 [Pi] I k/6], #[[2]]/3, (-1)^k}, {k, 6}] ; node = NestList[Flatten[next /@ #, 1] , {{0, 1, 1}}, order]; result = Table[ Flatten[{{#[[1]], #[[2]], #[[3]] + 2*FractionalPart[k]} /@ node[[IntegerPart[k]]], node[[IntegerPart[k]]]}, 1], {k, Table[IntegerPart[g] + Sin[FractionalPart[g] [Pi]/2]^2, {g, 1, order - 1/step, 1/step}]}]; preplot = Map[Function[{x}, {Re[x], Im[x]}] /@ Table[#[[1]] + 3/2 #[[2]] Exp[ 2 [Pi] I/3*g + [Pi] I/6*(#[[3]] - 1)*Sign[#[[3]] - 1]], {g, 3}] , result, {2}];


data = Table[

   Graphics[{FaceForm[Opacity[0]],

     EdgeForm[{GrayLevel[0.33]}], Polygon[#]  /@ (preplot[[k + 1]]),

     EdgeForm[{GrayLevel[0.33]}], Polygon[#]  /@ (preplot[[k + 2]]),

     EdgeForm[{GrayLevel[0.33]}], Polygon[#]  /@ (preplot[[k + 3]])

     }, PlotRange -&> {{-1, 1}, {-1, 1}}*3/2]

   , {k, 1, Length[result], 5}];

data = Flatten[{data, Reverse[data]}, 1];

ListAnimate[data, AnimationRate -&> 30]

圖片見http://fmn.rrimg.com/fmn066/20141208/1940/original_Qe7T_14360000c734118c.gif

下面這張是另外一個風格的，

order = 7; step = order*8; p = 1; q = 0.9; next = If[#2 == 0, {#1}, N@{{#1[[1]], #1[[1]] + p/2 ( #1[[2]] - #1[[1]]) Exp[q I #2 #1[[3]] [Pi]/3], -#1[[ 3]]}, {#1[[1]] + p/2 ( #1[[2]] - #1[[1]]) Exp[q I #2 #1[[3]] [Pi]/3], p/2 (#1[[1]] - #1[[2]]) Exp[-q I #2 #1[[3]] [Pi]/3] + #1[[ 2]], #1[[3]]}, {p/2 (#1[[1]] - #1[[2]]) Exp[-q I #2 #1[[3]] [Pi]/3] + #1[[ 2]], #1[[2]], -#1[[3]]}}] ; nextstep = next[#, 1] ; node = NestList[Flatten[nextstep /@ #, 1] , {{0, 1, 1}}, order]; result = Table[ Flatten[next[#, FractionalPart[k]] /@ node[[IntegerPart[k]]], 1], {k, Table[ IntegerPart[g] + Sin[FractionalPart[g] [Pi]/2]^2, {g, 1, order - 1/step, 1/step}]}]; preplot = Map[{Re[#], Im[#]} , result[[;; , ;; , {1, 2}]], {3}]; data = Table[ Graphics[{Blue, Line /@ (preplot[[i]])}, PlotRange -&> {{0, 1}, {0, Sqrt[3]/2}}, ImageSize -&> 600], {i, Length@preplot}]; data = Flatten[{data, Reverse[data]}, 1];

ListAnimate[data, AnimationRate -&> 24]

圖片見http://fmn.rrimg.com/fmn066/20150129/2115/original_xCZs_648800016f571191.gif

圖比較大，請耐心載入

另外你可以看一下這本書 The Algorithmic Beauty of Plants

網上有電子版，那裡面描述了如何用分形結構（其實應該不叫分形了吧...叫L-system）去畫樹木...畫的非常像。但是我嘗試了一下，沒什麼好辦法將其連續化

隨便截了個

不是非常了解，僅看過Mandelbrot 的大自然的分形幾何學 (豆瓣)55，嘗試著回答一下。

你需要看一下分形圖形的生成機制，分形圖形最重要的一點就是自相似性（self-similarity），就是次一層級的圖形和原來的圖形相似，可以採用迭代器的方式進行產生。

題主給的例子是在一般的分形教材都會接觸到的，叫科赫曲線（Koch curve），就是每次將線段三等分，中間的那三分之一外延做正三角形。

比如迭代5次之後是這個樣子，代碼網上有很多，我貼一個作者 Dagmar Scherzinger 用 MATLAB 寫的吧

function koch(level)

% KOCH(LEVEL) Draw the Koch curve fractal. LEVEL denotes the number
% of iterations to perform.
% Example:
% koch(5);
xl = zeros(10,1);
xr = xl;
yl = xl;
yr = yl;
xr(level) = 1;

r = sqrt(1/3^2 - 1/6^2);
clf;
set(gca,"FontSize",14);
set(gcf,"Color",[1,1,1]);
hold on;
subkoch(xl,xr,yl,yr,level,r);
title("Koch Curve");
text(0.5,-0.05,(["Number of iterations: " num2str(level)]), ...
"HorizontalAlign","center","FontSize",12);
hold off;

axis equal; axis tight; axis off;
%------------------------------------------------------------
function subkoch(xl,xr,yl,yr,level,r)
%Zeichne eine Linie auf der niedrigsten Level der Rekursion
if (level&<2)
plot([xl(1) xr(1)],[-yl(1) -yr(1)],"b-")
return
end
%Verzweige in niedrigere Stufen

level=level-1;
%Linker Zweig
xl(level)=xl(level+1);
yl(level)=yl(level+1);
xr(level)=1/3*xr(level+1)+2/3*xl(level+1);
yr(level)=1/3*yr(level+1)+2/3*yl(level+1);
subkoch(xl,xr,yl,yr,level,r);
%Mittlerer linker Zweig
xl(level)=xr(level);
yl(level)=yr(level);

xr(level)=.5*xr(level+1)+.5*xl(level+1)-r*(yl(level+1)-yr(level+1));
yr(level)=.5*yr(level+1)+.5*yl(level+1)+r*(xl(level+1)-xr(level+1));
subkoch(xl,xr,yl,yr,level,r);
%Mittlerer rechter Zweig
xl(level)=xr(level);
yl(level)=yr(level);
xr(level)=2/3*xr(level+1)+1/3*xl(level+1);
yr(level)=2/3*yr(level+1)+1/3*yl(level+1);
subkoch(xl,xr,yl,yr,level,r);
%Rechter Zweig

xl(level)=xr(level);
yl(level)=yr(level);
xr(level)=xr(level+1);
yr(level)=yr(level+1);
subkoch(xl,xr,yl,yr,level,r);
level=level+1;
return;

再來幾個分形圖