Python 绘制分形图

1. 曼德勃罗集

import numpy as np

import pylab as pl

import time

from matplotlib import cm



def iter_point(c):

    z = c

    for i in xrange(1, 100): # 最多迭代100次

        if abs(z)>2: break # 半径大于2则认为逃逸

        z = z*z+c

    return i # 返回迭代次数

    

def draw_mandelbrot(cx, cy, d):

    """

    绘制点(cx, cy)附近正负d的范围的Mandelbrot

    """

    x0, x1, y0, y1 = cx-d, cx+d, cy-d, cy+d 

    y, x = np.ogrid[y0:y1:200j, x0:x1:200j]

    c = x + y*1j

    start = time.clock()

    mandelbrot = np.frompyfunc(iter_point,1,1)(c).astype(np.float)

    print "time=",time.clock() - start

    pl.imshow(mandelbrot, cmap=cm.jet, extent=[x0,x1,y0,y1])

    pl.gca().set_axis_off()

    

x,y = 0.27322626, 0.595153338



pl.subplot(231)

draw_mandelbrot(-0.5,0,1.5)

for i in range(2,7):    

    pl.subplot(230+i)

    draw_mandelbrot(x, y, 0.2**(i-1))

pl.subplots_adjust(0.02, 0, 0.98, 1, 0.02, 0)

pl.show()

2. 分形树叶

import numpy as np

import matplotlib.pyplot as pl

import time



# 蕨类植物叶子的迭代函数和其概率值

eq1 = np.array([[0,0,0],[0,0.16,0]])

p1 = 0.01



eq2 = np.array([[0.2,-0.26,0],[0.23,0.22,1.6]])

p2 = 0.07



eq3 = np.array([[-0.15, 0.28, 0],[0.26,0.24,0.44]])

p3 = 0.07



eq4 = np.array([[0.85, 0.04, 0],[-0.04, 0.85, 1.6]])

p4 = 0.85



def ifs(p, eq, init, n):

    """

    进行函数迭代

    p: 每个函数的选择概率列表

    eq: 迭代函数列表

    init: 迭代初始点

    n: 迭代次数

    

    返回值： 每次迭代所得的X坐标数组， Y坐标数组， 计算所用的函数下标    

    """



    # 迭代向量的初始化

    pos = np.ones(3, dtype=np.float)

    pos[:2] = init

    

    # 通过函数概率，计算函数的选择序列

    p = np.add.accumulate(p)    

    rands = np.random.rand(n)

    select = np.ones(n, dtype=np.int)*(n-1)

    for i, x in enumerate(p[::-1]):

        select[rands<x] = len(p)-i-1

    

    # 结果的初始化

    result = np.zeros((n,2), dtype=np.float)

    c = np.zeros(n, dtype=np.float)

    

    for i in range(n):

        eqidx = select[i] # 所选的函数下标

        tmp = np.dot(eq[eqidx], pos) # 进行迭代

        pos[:2] = tmp # 更新迭代向量



        # 保存结果

        result[i] = tmp

        c[i] = eqidx

        

    return result[:,0], result[:, 1], c



start = time.clock()

x, y, c = ifs([p1,p2,p3,p4],[eq1,eq2,eq3,eq4], [0,0], 100000)

time.clock() - start

pl.figure(figsize=(6,6))

pl.subplot(121)

pl.scatter(x, y, s=1, c="g", marker="s", linewidths=0)

pl.axis("equal")

pl.axis("off")

pl.subplot(122)

pl.scatter(x, y, s=1,c = c, marker="s", linewidths=0)

pl.axis("equal")

pl.axis("off")

pl.subplots_adjust(left=0,right=1,bottom=0,top=1,wspace=0,hspace=0)

pl.gcf().patch.set_facecolor("#D3D3D3")

pl.show()

3. 其它分形图(科赫曲线、分形龙、谢尔宾斯基三角等)

from math import sin, cos, pi

import matplotlib.pyplot as pl

from matplotlib import collections



class L_System(object):

    def __init__(self, rule):

        info = rule['S']

        for i in range(rule['iter']):

            ninfo = []

            for c in info:

                if c in rule:

                    ninfo.append(rule[c])

                else:

                    ninfo.append(c)

            info = "".join(ninfo)

        self.rule = rule

        self.info = info



    def get_lines(self):

        d = self.rule['direct']

        a = self.rule['angle']

        p = (0.0, 0.0)

        l = 1.0

        lines = []

        stack = []

        for c in self.info:

            if c in "Ff":

                r = d * pi / 180

                t = p[0] + l*cos(r), p[1] + l*sin(r)

                lines.append(((p[0], p[1]), (t[0], t[1])))

                p = t

            elif c == "+":

                d += a

            elif c == "-":

                d -= a

            elif c == "[":

                stack.append((p,d))

            elif c == "]":

                p, d = stack[-1]

                del stack[-1]

        return lines



rules = [

    {

        "F":"F+F--F+F", "S":"F",

        "direct":180,

        "angle":60,

        "iter":5,

        "title":"Koch"

    },

    {

        "X":"X+YF+", "Y":"-FX-Y", "S":"FX",

        "direct":0,

        "angle":90,

        "iter":13,

        "title":"Dragon"

    },

    {

        "f":"F-f-F", "F":"f+F+f", "S":"f",

        "direct":0,

        "angle":60,

        "iter":7,

        "title":"Triangle"

    },

    {

        "X":"F-[[X]+X]+F[+FX]-X", "F":"FF", "S":"X",

        "direct":-45,

        "angle":25,

        "iter":6,

        "title":"Plant"

    },

    {

        "S":"X", "X":"-YF+XFX+", "Y":"+XF-YFY-FX+",

        "direct":0,

        "angle":90,

        "iter":6,

        "title":"Hilbert"

    },

    {

        "S":"L--F--L--F", "L":"+R-F-R+", "R":"-L+F+",

        "direct":0,

        "angle":45,

        "iter":10,

        "title":"Sierpinski"

    },

    

]



def draw(ax, rule, iter=None):

    if iter!=None:

        rule["iter"] = iter

    lines = L_System(rule).get_lines()

    linecollections = collections.LineCollection(lines)

    ax.add_collection(linecollections, autolim=True)

    ax.axis("equal")

    ax.set_axis_off()

    ax.set_xlim(ax.dataLim.xmin, ax.dataLim.xmax)

    ax.invert_yaxis()

    

fig = pl.figure(figsize=(7,4.5))

fig.patch.set_facecolor("papayawhip")



for i in xrange(6):

    ax = fig.add_subplot(231+i)

    draw(ax, rules[i])



fig.subplots_adjust(left=0,right=1,bottom=0,top=1,wspace=0,hspace=0)

pl.show()