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.................

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。

Python代碼：

import numpy as np

import matplotlib.pyplot as plt

import os,sys,caffe

import matplotlib as mpl

from mpl_toolkits.mplot3d import Axes3D #draw lorenz attractor

# %matplotlib inline

from math import sin, cos, pi

import math

def 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 plt

mpl.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 + 1

t = 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 =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.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 * r

ax = plt.subplot(111, projection='polar')

ax.plot(theta, r)

ax.set_rmax(2)

ax.set_rticks([0.5, 1, 1.5, 2]) # Less radial ticks

ax.set_rlabel_position(-22.5) # Move radial labels away from plotted line

ax.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*20

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='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 = 36

for 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 = 1.2

for 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:]]) / 2.0

sl += np.index_exp[:]

return x

# prepare some coordinates, and attach rgb values to each

r, g, b = np.indices((17, 17, 17)) / 16.0

rc = 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 components

colors = np.zeros(sphere.shape + (3,))

colors[..., 0] = rc

colors[..., 1] = gc

colors[..., 2] = bc

# and plot everything

fig = plt.figure();

ax = fig.gca(projection='3d');

ax.voxels(r, g, b, sphere,

facecolors=colors,

edgecolors=np.clip(2*colors - 0.5, 0, 1), # brighter

linewidth=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 argparse

mainex();

畫圖結果：

總結

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