???????? 花紋的生成可以使用貼圖的方式，同樣也可以使用方程，本文列出了幾種常用曲線的方程式，以取代貼圖方式完成特定花紋的生成。

???????? 注意極坐標的使用.................

???????? 前面部分基礎資料，參考：Python：Matplotlib 畫曲線和柱狀圖（Code）

???????? Pyplot教程：https://matplotlib.org/gallery/index.html#pyplots-examples?

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??????? 顧名思義，蝴蝶曲線（Butterfly curve ）就是曲線形狀如同蝴蝶。蝴蝶曲線如圖所示，以方程描述，是一條六次平面曲線。如果大家覺得這個太過簡單，別著急，還有第二種。如圖所示，以方程描述，這是一個極坐標方程。通過改變這個方程中的變量θ，可以得到不同形狀與方向的蝴蝶曲線。如果再施以復雜的組合和變換，我們看到的就完全稱得上是一幅藝術品了。

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Python代碼：

import numpy as np import matplotlib.pyplot as plt import os,sys,caffeimport matplotlib as mpl from mpl_toolkits.mplot3d import Axes3D #draw lorenz attractor # %matplotlib inline from math import sin, cos, pi import mathdef mainex():#drawSpringCrurve();#畫柱坐標系螺旋曲線#HelicalCurve();#采用柱坐標系#尖螺旋曲線#Votex3D();#phoenixCurve();#ButterflyCurve();#ButterflyNormalCurve();#dicareCurve2d();#WindmillCurve3d();#HelixBallCurve();#球面螺旋線#AppleCurve();#HelixInCircleCurve();#使用scatter，排序有問題seperialHelix();def drawSpringCrurve():#碟形彈簧#圓柱坐標 #方程：#import matplotlib as mpl#from mpl_toolkits.mplot3d import Axes3D#import numpy as np#import matplotlib.pyplot as pltmpl.rcParams['legend.fontsize'] = 10;fig = plt.figure();ax = fig.gca(projection='3d');# Prepare arrays x, y, z#theta = np.linspace(-4 * np.pi, 4 * np.pi, 100)#z = np.linspace(-2, 2, 100)#r = z**2 + 1t = np.arange(0,100,1);r = t*0 +20;theta = t*3600 ;z = np.arange(0,100,1);for i in range(100):z[i] =(sin(3.5*theta[i]-90))+24*t[i];x = r * np.sin(theta);y = r * np.cos(theta);ax.plot(x, y, z, label='SpringCrurve');ax.legend();plt.show();def HelicalCurve():#螺旋曲線#采用柱坐標系t = np.arange(0,100,1);r =t ;theta=10+t*(20*360);z =t*3;x = r * np.sin(theta);y = r * np.cos(theta);mpl.rcParams['legend.fontsize'] = 10;fig = plt.figure();ax = fig.gca(projection='3d');ax.plot(x, y, z, label='HelicalCurve');ax.legend();plt.show();def ButterflyCurve():#蝶形曲線，使用球坐標系#或許公式是錯誤的，應該有更加復雜的公式t = np.arange(0,4,0.01);r = 8 * t;theta = 3.6 * t * 2*1 ;phi = -3.6 * t * 4*1;x = t*1;y = t*1;#z = t*1;z =0for i in range(len(t)):x[i] = r[i] * np.sin(theta[i])*np.cos(phi[i]);y[i] = r[i] * np.sin(theta[i])*np.sin(phi[i]);#z[i] = r[i] * np.cos(theta[i]);mpl.rcParams['legend.fontsize'] = 10;fig = plt.figure();ax = fig.gca(projection='3d');ax.plot(x, y, z, label='ButterflyCurve');#ax.scatter(x, y, z, label='ButterflyCurve');ax.legend();plt.show();def ButterflyNormalCurve():#蝶形曲線，使用球坐標系#或許公式是錯誤的，應該有更加復雜的公式#螺旋曲線#采用柱坐標系#t = np.arange(0,100,1);theta=np.arange(0,6,0.1);#(0,72,0.1);r =theta*0;z =theta*0;x =theta*0;y =theta*0;for i in range(len(theta)):r[i] = np.power(math.e,sin(theta[i]))- 2*cos(4*theta[i]) + np.power( sin(1/24 * (2*theta[i] -pi ) ) , 5 );#x[i] = r[i] * np.sin(theta[i]);#y[i] = r[i] * np.cos(theta[i]);x = r * np.sin(theta);y = r * np.cos(theta);mpl.rcParams['legend.fontsize'] = 10;fig = plt.figure();ax = fig.gca(projection='3d');ax.plot(x, y, z, label='ButterflyNormalCurve');ax.legend();plt.show();def phoenixCurve():#蝶形曲線，使用球坐標系t = np.arange(0,100,1);r = 8 * t;theta = 360 * t * 4 ;phi = -360 * t * 8;x = t*1;y = t*1;z = t*1;for i in range(len(t)):x[i] = r[i] * np.sin(theta[i])*np.cos(phi[i]);y[i] = r[i] * np.sin(theta[i])*np.sin(phi[i]);z[i] = r[i] * np.cos(theta[i]);mpl.rcParams['legend.fontsize'] = 10;fig = plt.figure();ax = fig.gca(projection='3d');ax.plot(x, y, z, label='phoenixCurve');ax.legend();plt.show();def dicareCurve2d():r = np.arange(0, 2, 0.01)theta = 2 * np.pi * rax = plt.subplot(111, projection='polar')ax.plot(theta, r)ax.set_rmax(2)ax.set_rticks([0.5, 1, 1.5, 2]) # Less radial ticksax.set_rlabel_position(-22.5) # Move radial labels away from plotted lineax.grid(True)ax.set_title("dicareCurve2d", va='bottom')plt.show();def WindmillCurve3d():#風車曲線t = np.arange(0,2,0.01);r =t*0+1 ;#r=1 ang =36*t;#ang =360*t;s =2*pi*r*t;x = t*1;y = t*1;for i in range(len(t)):x[i] = s[i]*cos(ang[i]) +s[i]*sin(ang[i]) ;y[i] = s[i]*sin(ang[i]) -s[i]*cos(ang[i]) ;z =t*0;mpl.rcParams['legend.fontsize'] = 10;fig = plt.figure();ax = fig.gca(projection='3d');ax.plot(x, y, z, label='WindmillCurve3d');ax.legend();plt.show();def HelixBallCurve():#螺旋曲線，使用球坐標系t = np.arange(0,2,0.005);r =t*0+4 ;theta =t*1.8 phi =t*3.6*20x = t*1;y = t*1;z = t*1;for i in range(len(t)):x[i] = r[i] * np.sin(theta[i])*np.cos(phi[i]);y[i] = r[i] * np.sin(theta[i])*np.sin(phi[i]);z[i] = r[i] * np.cos(theta[i]);mpl.rcParams['legend.fontsize'] = 10;fig = plt.figure();ax = fig.gca(projection='3d');ax.plot(x, y, z, label='HelixBallCurve');ax.legend();plt.show();def seperialHelix():#螺旋曲線，使用球坐標系t = np.arange(0,2,0.1);n = np.arange(0,2,0.1);r =t*0+4 ;theta =n*1.8 ;phi =n*3.6*20;x = t*0;y = t*0;z = t*0;for i in range(len(t)):x[i] = r[i] * np.sin(theta[i])*np.cos(phi[i]);y[i] = r[i] * np.sin(theta[i])*np.sin(phi[i]);z[i] = r[i] * np.cos(theta[i]);mpl.rcParams['legend.fontsize'] = 10;fig = plt.figure();ax = fig.gca(projection='3d');ax.plot(x, y, z, label='ButterflyCurve');ax.legend();plt.show();def AppleCurve():#螺旋曲線t = np.arange(0,2,0.01);l=2.5 b=2.5 x = t*1;y = t*1;z =0;#z=t*0;n = 36for i in range(len(t)):x[i]=3*b*cos(t[i]*n)+l*cos(3*t[i]*n) y[i]=3*b*sin(t[i]*n)+l*sin(3*t[i]*n)#x = r * np.sin(theta);#y = r * np.cos(theta);mpl.rcParams['legend.fontsize'] = 10;fig = plt.figure();ax = fig.gca(projection='3d');ax.plot(x, y, z, label='AppleCurve');ax.legend();plt.show();def HelixInCircleCurve():#園內螺旋曲線#采用柱坐標系t = np.arange(-1,1,0.01);theta=t*36 ;#360 deta 0.005鳥巢網 #36 deta 0.005 圓內曲線x = t*1;y = t*1;z = t*1;r = t*1;n = 1.2for i in range(len(t)):r[i]=10+10*sin(n*theta[i]);z[i]=2*sin(n*theta[i]);x[i] = r[i] * np.sin(theta[i]);y[i] = r[i] * np.cos(theta[i]);mpl.rcParams['legend.fontsize'] = 3;fig = plt.figure();ax = fig.gca(projection='3d');ax.plot(x, y, z, label='HelixInCircleCurve');#ax.scatter(x, y, z, label='HelixInCircleCurve');ax.legend();plt.show();def Votex3D():def midpoints(x):sl = ()for i in range(x.ndim):x = (x[sl + np.index_exp[:-1]] + x[sl + np.index_exp[1:]]) / 2.0sl += np.index_exp[:]return x# prepare some coordinates, and attach rgb values to eachr, g, b = np.indices((17, 17, 17)) / 16.0rc = midpoints(r)gc = midpoints(g)bc = midpoints(b)# define a sphere about [0.5, 0.5, 0.5]sphere = (rc - 0.5)**2 + (gc - 0.5)**2 + (bc - 0.5)**2 < 0.5**2# combine the color componentscolors = np.zeros(sphere.shape + (3,))colors[..., 0] = rccolors[..., 1] = gccolors[..., 2] = bc# and plot everythingfig = plt.figure();ax = fig.gca(projection='3d');ax.voxels(r, g, b, sphere,facecolors=colors,edgecolors=np.clip(2*colors - 0.5, 0, 1), # brighterlinewidth=0.5);ax.set(xlabel='r', ylabel='g', zlabel='b');plt.show();def drawFiveFlower():theta=np.arange(0,2*np.pi,0.02) #plt.subplot(121,polar=True) #plt.plot(theta,2*np.ones_like(theta),lw=2) #plt.plot(theta,theta/6,'--',lw=2) #plt.subplot(122,polar=True) plt.subplot(111,polar=True) plt.plot(theta,np.cos(5*theta),'--',lw=2) plt.plot(theta,2*np.cos(4*theta),lw=2) plt.rgrids(np.arange(0.5,2,0.5),angle=45) plt.thetagrids([0,45,90]);plt.show(); if __name__ == '__main__':import argparsemainex();

畫圖結果：

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