如何在pygame中找到三角形的中点，然后递归地重复执行它以形成谢尔宾斯基三角形？

Question

如何找到最初绘制的三角形的中点？我需要创建一个谢尔宾斯基三角形，其中一个三角形内有多个三角形。到目前为止，我有第一个三角形的代码，如下所示：

import pygame

pygame.init()

colors = [pygame.Color(0, 0, 0, 255),       # Black

          pygame.Color(255, 0, 0, 255),     # Red

          pygame.Color(0, 255, 0, 255),     # Green

          pygame.Color(0, 0, 255, 255),     # Blue

          pygame.Color(255, 255, 255, 255)] # White

# Each of these constants is the index to the corresponding pygame Color object

# in the list, colors, defined above.

BLACK = 0

RED = 1

GREEN = 2

BLUE = 3

WHITE = -1

height = 640

width = 640

size = [width, height]

screen = pygame.display.set_mode(size)

screen.fill(WHITE)

def draw_triangle(p1, p2, p3, color, line_width, screen):

    p1 = [5, height - 5]

    p2 = [(width - 10) / 2, 5]

    p3 = [width - 5, height - 5]

    pygame.draw.polygon(screen, 0, [p1, p2, p3], 2)

    pygame.display.flip()

def find_midpoint(p1, p2):

def sierpinski(degree, p1, p2, p3, color, line_width, screen):

剩下的两个函数都是完成谢尔宾斯基三角形所需要的。首先，创建一个函数来查找中点，然后创建一个在这些三角形内创建多个三角形（称为谢尔宾斯基三角形）的函数。

Answer 1

另一个三角形内的“中点三角形”由一个三角形定义，该三角形的坐标是周围三角形各边的中点：因此，对于三角形的每条线/边，计算中点：def lineMidPoint( p1, p2 ):    """ Return the mid-point on the line p1 to p2 """    # Ref: https://en.wikipedia.org/wiki/Midpoint    x1, y1 = p1    x2, y2 = p2    x_mid = round( ( x1 + x2 ) / 2 )    y_mid = round( ( y1 + y2 ) / 2 )    return ( x_mid, y_mid )在您的情况下，这将被多次调用p1,p2并p3生成 3 个“角”三角形：# midpoints of each sizemid_p1 = lineMidPoint( p1, p2 )mid_p2 = lineMidPoint( p2, p3 )mid_p3 = lineMidPoint( p3, p1 ) # The 3 "corner" trianglesupper_triangle = [ mid_p1, p2, mid_p2 ]left_triangle  = [ p1, mid_p1, mid_p3 ]right_triangle = [ mid_p3, mid_p2, p3 ]# The inner triangle (for the sake of completeness)inner_triangle = [ mid_p1, mid_p2, mid_p3 ]然后，您需要将其包装在递归调用中，并进行某种深度救助。就像是：def drawTriangle( window, colour, points, bailout=5 ):    if ( bailout > 0 ):        # Calculate the 3 inner corner-triangles        p1, p2, p3 = points        mid_p1 = lineMidPoint( p1, p2 )        mid_p2 = lineMidPoint( p2, p3 )  # mid-point of each side        mid_p3 = lineMidPoint( p3, p1 )         # triangles between the original corners, and new mid-points        upper_triangle = [ mid_p1, p2, mid_p2 ]          left_triangle  = [ p1, mid_p1, mid_p3 ]        right_triangle = [ mid_p3, mid_p2, p3 ]        drawTriangle( window, colour, upper_triangle, bailout-1 )        drawTriangle( window, colour, left_triangle,  bailout-1 )        drawTriangle( window, colour, right_triangle, bailout-1 )    else:        pygame.draw.lines( window, colour, True, points )  # draw triangle我认为这画出了一个谢尔宾斯基三角形

Answer 2

我不确定争论的目的degree是什么，也许是为了限制递归深度？这是一个基于您的问题的示例，使用递归 sierpinski 函数：import pygamedef draw_triangle(p1, p2, p3, color, line_width, screen):&nbsp; &nbsp; pygame.draw.polygon(screen, color, [p1, p2, p3], line_width)def midpoint(p1, p2):&nbsp; &nbsp; """ Return the mid-point on the line p1 to p2 """&nbsp; &nbsp; x1, y1 = p1&nbsp; &nbsp; x2, y2 = p2&nbsp; &nbsp; x_mid = (x1 + x2) // 2&nbsp; &nbsp; y_mid = (y1 + y2) // 2&nbsp; &nbsp; return (x_mid, y_mid)def sierpinski(degree, p1, p2, p3, color, line_width, screen):&nbsp; &nbsp; # p1 → bottom left, p2 → bottom right, p3 → top&nbsp; &nbsp; # recursive function so check for exit condition first&nbsp; &nbsp; if abs(p1[0] - p2[0]) <= 2 and abs(p2[0] - p3[0]) <= 2 and abs(p1[0] - p3[0]) <= 2:&nbsp; &nbsp; &nbsp; &nbsp; return&nbsp; &nbsp; draw_triangle(p1, p2, p3, color, line_width, screen)&nbsp; &nbsp; a = midpoint(p1, p2)&nbsp; &nbsp; b = midpoint(p1, p3)&nbsp; &nbsp; c = midpoint(p2, p3)&nbsp; &nbsp; # skip the centre triangle&nbsp; &nbsp; sierpinski(degree, p1, a, b, color, line_width, screen)&nbsp; &nbsp; sierpinski(degree, p2, a, c, color, line_width, screen)&nbsp; &nbsp; sierpinski(degree, p3, b, c, color, line_width, screen)height = 640width = 640pygame.init()screen = pygame.display.set_mode((width, height), pygame.RESIZABLE)pygame.display.set_caption("Sierpiński")clock = pygame.time.Clock()update_screen = Truerunning = Truewhile running:&nbsp; &nbsp; for event in pygame.event.get():&nbsp; &nbsp; &nbsp; &nbsp; if event.type == pygame.QUIT:&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; running = False&nbsp; &nbsp; &nbsp; &nbsp; elif event.type == pygame.VIDEORESIZE:&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; width, height = event.dict["size"]&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; screen = pygame.display.set_mode((width, height), pygame.RESIZABLE)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; update_screen = True&nbsp; &nbsp; if update_screen:&nbsp; &nbsp; &nbsp; &nbsp; # only draw the screen when required&nbsp; &nbsp; &nbsp; &nbsp; screen.fill(pygame.color.Color("white"))&nbsp; &nbsp; &nbsp; &nbsp; # determine initial points based on window size&nbsp; &nbsp; &nbsp; &nbsp; p1 = [5, height - 5]&nbsp; &nbsp; &nbsp; &nbsp; p2 = [(width - 10) // 2, 5]&nbsp; &nbsp; &nbsp; &nbsp; p3 = [width - 5, height - 5]&nbsp; &nbsp; &nbsp; &nbsp; sierpinski(None, p1, p2, p3, pygame.color.Color("black"), 1, screen)&nbsp; &nbsp; &nbsp; &nbsp; pygame.display.update()&nbsp; &nbsp; &nbsp; &nbsp; update_screen = False&nbsp; &nbsp; # limit framerate&nbsp; &nbsp; clock.tick(30)pygame.quit()为了简洁起见，我删除了颜色处理，而是使用pygame.color.Color接受其构造函数的字符串参数。我还使用整数除法//来代替round(…).根据递归函数的深度或复杂性，您可以重新绘制每一帧，但我想展示一个限制示例，以防函数复杂性增加。最后，我最近一直在调整屏幕大小，这似乎与一次绘制有关，所以我也将其包括在内。编辑：我修改了该sierpinski函数以支持degree指定递归 dep 的参数def sierpinski(degree, p1, p2, p3, color, line_width, screen):&nbsp; &nbsp; # p1 → bottom left, p2 → bottom right, p3 → top&nbsp; &nbsp; # recursive function so check for exit condition first&nbsp; &nbsp; if degree is None:&nbsp; &nbsp; &nbsp; &nbsp; if abs(p1[0] - p2[0]) <= 2 and abs(p2[0] - p3[0]) <= 2 and abs(p1[0] - p3[0]) <= 2:&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; return&nbsp; &nbsp; else:&nbsp; &nbsp; &nbsp; &nbsp; if degree == 0:&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; return&nbsp; &nbsp; &nbsp; &nbsp; else:&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; degree -= 1&nbsp; &nbsp; …然后我添加了一些事件处理，以便可以使用鼠标滚轮来增加和减少度数，这显示在标题栏上：elif event.type == pygame.MOUSEBUTTONUP:&nbsp; &nbsp; if event.button == 4:&nbsp; # wheel up&nbsp; &nbsp; &nbsp; &nbsp; if degree is None:&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; degree = 8&nbsp; &nbsp; &nbsp; &nbsp; else:&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; degree += 1&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; if degree > maximum_degree:&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; degree = maximum_degree&nbsp; &nbsp; &nbsp; &nbsp; update_screen = True&nbsp; &nbsp; elif event.button == 5: # wheel down&nbsp; &nbsp; &nbsp; &nbsp; if degree is None:&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; degree = 3&nbsp; &nbsp; &nbsp; &nbsp; else:&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; degree -= 1&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; if degree < minimum_degree:&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; degree = minimum_degree&nbsp; &nbsp; &nbsp; &nbsp; update_screen = True…