python 画希尔伯特曲线

给你一个问题：
让你在一个N×NN\times N的点阵，让你画一条连续曲线，使得这条曲线经过这个点阵中的每个点，并且每个点只经过一次，N满足条件：N=2k，k⊂ZN=2^k， k\subset \mathbb{Z}。

德国的数学家 David Hilbert就构造了一条满足上面要求的曲线。如下图

1阶希尔伯特曲线

2阶希尔伯特曲线

3阶希尔伯特曲线

4阶希尔伯特曲线

看出来了没有，其实希尔伯特曲线的构造方法是一个递归过程，要构造n阶的希尔伯特曲线，先构造4个n-1阶的希尔伯特曲线，这4个n-1阶的希尔伯特曲线通过特定的顺序连接起来。其实连接顺序就只有4种，跟1阶希尔伯特相似，不过开口方向不同而已。

有了这个认识，编程就不难了，python的代码如下：

# encoding=utf-8from tkinter import *def show_order(startx, starty, shape, order):global segx, segyif order == 1:if shape == 1:canvas.create_line(10 + startx * segx, 10 + starty * segy,10 + startx * segx, 10 + (starty + 1) * segy,tags="pic")canvas.create_line(10 + startx * segx, 10 + (starty + 1) * segy,10 + (startx + 1) * segx, 10 + (starty + 1) * segy,tags="pic")canvas.create_line(10 + (startx + 1) * segx, 10 + (starty + 1) * segy,10 + (startx + 1) * segx, 10 + starty * segy,tags="pic")elif shape == 2:canvas.create_line(10 + startx * segx, 10 + starty * segy,10 + (startx + 1) * segx, 10 + starty * segy,tags="pic")canvas.create_line(10 + (startx + 1) * segx, 10 + starty * segy,10 + (startx + 1) * segx, 10 + (starty + 1) * segy,tags="pic")canvas.create_line(10 + (startx + 1) * segx, 10 + (starty + 1) * segy,10 + startx * segx, 10 + (starty + 1) * segy,tags="pic")elif shape == 3:canvas.create_line(10 + startx * segx, 10 + (starty + 1) * segy,10 + startx * segx, 10 + starty * segy,tags="pic")canvas.create_line(10 + startx * segx, 10 + starty * segy,10 + (startx + 1) * segx, 10 + starty * segy,tags="pic")canvas.create_line(10 + (startx + 1) * segx, 10 + starty * segy,10 + (startx + 1) * segx, 10 + (starty + 1) * segy,tags="pic")elif shape == 4:canvas.create_line(10 + (startx + 1) * segx, 10 + starty * segy,10 + startx * segx, 10 + starty * segy,tags="pic")canvas.create_line(10 + startx * segx, 10 + starty * segy,10 + startx * segx, 10 + (starty + 1) * segy,tags="pic")canvas.create_line(10 + startx * segx, 10 + (starty + 1) * segy,10 + (startx + 1) * segx, 10 + (starty + 1) * segy,tags="pic")else:if shape == 1:show_order(startx, starty, 2, order-1)canvas.create_line(10 + startx * segx, 10 + (starty + EXP[order-1] - 1) * segy,10 + startx * segx, 10 + (starty + EXP[order-1]) * segy,tags="pic")show_order(startx, starty + EXP[order-1], 1, order-1)canvas.create_line(10 + (startx + EXP[order-1] - 1) * segx, 10 + (starty + EXP[order-1]) * segy,10 + (startx + EXP[order-1]) * segx, 10 + (starty + EXP[order-1]) * segy,tags="pic")show_order(startx + EXP[order-1], starty + EXP[order-1], 1, order-1)canvas.create_line(10 + (startx + EXP[order] - 1) * segx, 10 + (starty + EXP[order-1]) * segy,10 + (startx + EXP[order] - 1) * segx, 10 + (starty + EXP[order-1] - 1) * segy,tags="pic")show_order(startx + EXP[order-1], starty, 4, order-1)elif shape == 2:show_order(startx, starty, 1, order-1)canvas.create_line(10 + (startx + EXP[order-1] - 1) * segx, 10 + starty * segy,10 + (startx + EXP[order-1]) * segx, 10 + starty * segy,tags="pic")show_order(startx + EXP[order-1], starty, 2, order-1)canvas.create_line(10 + (startx + EXP[order-1])*segx, 10 + (starty + EXP[order-1] - 1) * segy,10 + (startx + EXP[order-1])*segx, 10 + (starty + EXP[order-1]) * segy,tags="pic")show_order(startx + EXP[order-1], starty + EXP[order-1], 2, order-1)canvas.create_line(10 + (startx + EXP[order-1] - 1) * segx, 10 + (starty + EXP[order] - 1)*segy,10 + (startx + EXP[order-1]) * segx, 10 + (starty + EXP[order] - 1) * segy,tags="pic")show_order(startx, starty + EXP[order-1], 3, order-1)elif shape == 3:show_order(startx, starty + EXP[order-1], 2, order-1)canvas.create_line(10 + startx * segx, 10 + (starty + EXP[order-1]) * segy,10 + startx * segx, 10 + (starty + EXP[order-1] - 1) * segy,tags="pic")show_order(startx, starty, 3, order-1)canvas.create_line(10 + (startx + EXP[order-1] - 1) * segx, 10 + (starty + EXP[order-1] - 1) * segy,10 + (startx + EXP[order-1]) * segx, 10 + (starty + EXP[order-1] - 1) * segy,tags="pic")show_order(startx + EXP[order-1], starty, 3, order-1)canvas.create_line(10 + (startx + EXP[order] - 1) * segx, 10 + (starty + EXP[order-1] - 1) * segy,10 + (startx + EXP[order] - 1) * segx, 10 + (starty + EXP[order-1]) * segy,tags="pic")show_order(startx + EXP[order-1], starty + EXP[order-1], 4, order-1)elif shape == 4:show_order(startx + EXP[order-1], starty, 1, order-1)canvas.create_line(10 + (startx + EXP[order-1])*segx, 10 + starty*segy,10 + (startx + EXP[order-1] - 1) * segx, 10 + starty * segy,tags="pic")show_order(startx, starty, 4, order-1)canvas.create_line(10 + (startx + EXP[order-1] - 1) * segx, 10 + (starty + EXP[order-1] - 1) * segy,10 + (startx + EXP[order-1] - 1) * segx, 10 + (starty + EXP[order-1]) * segy,tags="pic")show_order(startx, starty+EXP[order-1], 4, order-1)canvas.create_line(10 + (startx + EXP[order-1] - 1) * segx, 10 + (starty + EXP[order] - 1) * segy,10 + (startx + EXP[order-1]) * segx, 10 + (starty + EXP[order] - 1) * segy,tags="pic")show_order(startx + EXP[order-1], starty + EXP[order-1], 3, order-1)def display():if ord1.get() == "":returncanvas.delete("pic")global segx, segysegx = (WIDTH - 20) / (EXP[int(ord1.get())] - 1)segy = (HEIGHT - 20) / (EXP[int(ord1.get())] - 1)show_order(0, 0, 1, int(ord1.get()))
segx = None
segy = NoneEXP = 10 * [1]
for i in range(1, 10):EXP[i] = EXP[i-1] * 2WIDTH = 700
HEIGHT = 700window = Tk()
window.title("希尔伯特曲线")
canvas = Canvas(window, width=WIDTH, height=HEIGHT, bg="white")
canvas.pack()frame = Frame(window)
frame.pack(anchor=W)
lbl1 = Label(frame, text="Enter the order: ")
lbl1.pack(side=LEFT)
ord1 = StringVar()
entry1 = Entry(frame, textvariable=ord1, width=10)
entry1.pack(side=LEFT)
but1 = Button(frame, text="Display", command=display)
but1.pack()
window.mainloop()

程序运行截图

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