如何使用Python绘制分形图案

import numpy as npfrom PIL import Imageimport mathdef plot_line(from_coordinates, to_coordinates, thickness, colour, pixels):    # 找出像素阵列的边界    max_x_coordinate = len(pixels[0])    max_y_coordinate = len(pixels)    # 两点之间沿着x轴和y轴的距离    horizontal_distance = to_coordinates[1] - from_coordinates[1]    vertical_distance = to_coordinates[0] - from_coordinates[0]    # 两点之间的总距离    distance =  math.sqrt((to_coordinates[1] - from_coordinates[1])**2                 + (to_coordinates[0] - from_coordinates[0])**2)    # 每次给一个新的像素上色时，将向前走多远    horizontal_step = horizontal_distance/distance    vertical_step = vertical_distance/distance    # 此时，将进入循环以在像素数组中绘制线    # 循环的每一次迭代都会沿着线添加一个新的点    for i in range(round(distance)):        # 这两个坐标是直线中心的坐标        current_x_coordinate = round(from_coordinates[1] + (horizontal_step*i))        current_y_coordinate = round(from_coordinates[0] + (vertical_step*i))        # 一旦得到了点的坐标，        # 就在坐标周围画出尺寸为thickness的图案        for x in range (-thickness, thickness):            for y in range (-thickness, thickness):                x_value = current_x_coordinate + x                y_value = current_y_coordinate + y                if (x_value > 0 and x_value < max_x_coordinate and                     y_value > 0 and y_value < max_y_coordinate):                    pixels[y_value][x_value] = colour# 定义图像的大小pixels = np.zeros( (500,500,3), dtype=np.uint8 )# 画一条线plot_line([0,0], [499,499], 1, [255,200,0], pixels)# 把像素阵列变成一张真正的图片img = Image.fromarray(pixels)# 显示得到的图片，并保存它img.show()img.save('Line.png')

def draw_triangle(center, side_length, thickness, colour, pixels):    # 等边三角形的高度是，h = ½(√3a)    # 其中a是边长    triangle_height = round(side_length * math.sqrt(3)/2)    # 顶角    top = [center[0] - triangle_height/2, center[1]]    # 左下角    bottom_left = [center[0] + triangle_height/2, center[1] - side_length/2]    # 右下角    bottom_right = [center[0] + triangle_height/2, center[1] + side_length/2]    # 在每个角之间画一条线来完成三角形    plot_line(top, bottom_left, thickness, colour, pixels)    plot_line(top, bottom_right, thickness, colour, pixels)    plot_line(bottom_left, bottom_right, thickness, colour, pixels)

def rotate(coordinate, center_point, degrees):    # 从坐标中减去旋转的点    x = (coordinate[0] - center_point[0])    y = (coordinate[1] - center_point[1])    # Python的cos和sin函数采用弧度而不是度数    radians = math.radians(degrees)    # 计算旋转点    new_x = (x * math.cos(radians)) - (y * math.sin(radians))    new_y = (y * math.cos(radians)) + (x * math.sin(radians))    # 将在开始时减去的偏移量加回旋转点上    return [new_x + center_point[0], new_y + center_point[1]]

def draw_triangle(center, side_length, degrees_rotate, thickness, colour,                   pixels, shrink_side_by, iteration, max_depth):    # 等边三角形的高度是，h = ½(√3a)    # 其中'a'是边长    triangle_height = side_length * math.sqrt(3)/2    # 顶角    top = [center[0] - triangle_height/2, center[1]]    # 左下角    bottom_left = [center[0] + triangle_height/2, center[1] - side_length/2]    # 右下角    bottom_right = [center[0] + triangle_height/2, center[1] + side_length/2]    if (degrees_rotate != 0):        top = rotate(top, center, degrees_rotate)        bottom_left = rotate(bottom_left, center, degrees_rotate)        bottom_right = rotate(bottom_right, center, degrees_rotate)    # 三角形各边之间的坐标    lines = [[top, bottom_left],[top, bottom_right],[bottom_left, bottom_right]]    line_number = 0    # 在每个角之间画一条线来完成三角形    for line in lines:        line_number += 1        plot_line(line[0], line[1], thickness, colour, pixels)        # 如果还没有达到max_depth，就画一些新的三角形        if (iteration < max_depth and (iteration < 1 or line_number < 3)):            gradient = (line[1][0] - line[0][0]) / (line[1][1] - line[0][1])            new_side_length = side_length*shrink_side_by            # 正在绘制的三角形线的中心            center_of_line = [(line[0][0] + line[1][0]) / 2,                               (line[0][1] + line[1][1]) / 2]            new_center = []            new_rotation = degrees_rotate            # 需要旋转traingle的数量            if (line_number == 1):                new_rotation += 60            elif (line_number == 2):                new_rotation -= 60            else:                new_rotation += 180            # 在一个理想的世界里，这将是gradient=0，            # 但由于浮点除法的原因，无法            # 确保永远是这种情况            if (gradient < 0.0001 and gradient > -0.0001):                if (center_of_line[0] - center[0] > 0):                    new_center = [center_of_line[0] + triangle_height *                                  (shrink_side_by/2), center_of_line[1]]                else:                    new_center = [center_of_line[0] - triangle_height *                                   (shrink_side_by/2), center_of_line[1]]            else:                # 计算直线梯度的法线                difference_from_center = -1/gradient                # 计算这条线距中心的距离                # 到新三角形的中心                distance_from_center = triangle_height * (shrink_side_by/2)                # 计算 x 方向的长度，                # 从线的中心到新三角形的中心                x_length = math.sqrt((distance_from_center**2)/                                      (1 + difference_from_center**2))                # 计算出x方向需要走哪条路                if (center_of_line[1] < center[1] and x_length > 0):                    x_length *= -1                # 现在计算Y方向的长度                y_length = x_length * difference_from_center                # 用新的x和y值来偏移线的中心                new_center = [center_of_line[0] + y_length,                               center_of_line[1] + x_length]            draw_triangle(new_center, new_side_length, new_rotation,                           thickness, colour, pixels, shrink_side_by,                           iteration+1, max_depth)