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学习Python中用numpy与matplotlib遇到的一些数学函数与函数的绘图

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學習Python中的一些數(shù)學函數(shù)與函數(shù)的繪圖

主要用到numpy 與 matplotlib
如果有什么不正確，歡迎指教。

圖片不知道怎樣批量上傳，一個一個怎么感覺很小，請見諒

自行復制拷貝，到vs，jupyter notebook, spyder都可以

函數(shù) $y = x ? s i n x$

函數(shù) $y = 3x^4 - 4x^3 + 1$

函數(shù) $y = x^4$

函數(shù) $\sqrt[3]{x}$

函數(shù) $y=2x^3-9x^2+12x-3$

函數(shù) $y=2x^3-6x^2-18x+7$

三次拋物線 $y = x^3$

半立方拋物線 $y^2 = ax^3$

概率曲線 $y=e^{-x^2}$

箕舌線 $=\frac{8a^3}{x^2+4a^2}$

蔓葉線 $y^2(2a-x)=x^3$ 或 $x=2asin2θ,y=2a2tan2θsin4θx=2asin^2\theta, y=2a^2tan^2\theta sin^4\theta$

修改：之前x的取值是 $?3.6≤x≤3.6-3.6\leq x \leq3.6$ , 實際上x的值是 $≥0\geq 0$

$ρ=2a?tan2θ\rho=2a-tan^2\theta$

笛卡兒葉形線畫圖極坐標 $\frac{3asin\theta cos\theta}{sin^3\theta + cos^3\theta}$

笛卡兒葉形線直角坐標 $x^3+y^3-3axy=0$ 或 $x=3at1+t3,y=3at21+t3x=\frac{3at}{1+t^3}, y=\frac{3at^2}{1+t^3}$

%matplotlib inline %config InlineBackend.figure_format = "svg" import numpy as np import matplotlib.pyplot as plt from scipy import signal plt.rcParams['figure.figsize'] = (8, 4.5) plt.rcParams['figure.dpi'] = 150 plt.rcParams['font.sans-serif'] = ['Simhei'] #替代字體 plt.rcParams['axes.unicode_minus'] = False #解決坐標軸負數(shù)的鉛顯示問題#極坐標 # plt.subplot(111, polar = True) # plt.ylim([0, 6])a = 1 # t = np.arange(-0.1, 4 * np.pi, np.pi / 180) t = np.linspace(-0.5, 200, 5000) x = (3 * a * t) / (1 + t ** 3) y = (3 * a * t ** 2) / (1 + t ** 3) plt.plot(x, y, label = r'$ x=\frac{3at}{1+t^3}, y=\frac{3at^2}{1+t^3} $')a = 1.5 x1 = (3 * a * t) / (1 + t ** 3) y1 = (3 * a * t ** 2) / (1 + t ** 3) plt.plot(x1, y1, label = r'$ x=\frac{3at}{1+t^3}, y=\frac{3at^2}{1+t^3} $')# plt.grid() plt.legend() plt.show()

星形線（內(nèi)擺線的一種） $x23+y23=a23x^\frac{2}{3}+y^\frac{2}{3}=a^\frac{2}{3}$ 或 ${x=acos3θy=asin3θ\begin{cases} x=acos^3\theta \\ y=asin^3\theta \end{cases}$

%matplotlib inline %config InlineBackend.figure_format = "svg" import numpy as np import matplotlib.pyplot as plt from scipy import signal plt.rcParams['figure.figsize'] = (8, 4.5) plt.rcParams['figure.dpi'] = 150 plt.rcParams['font.sans-serif'] = ['Simhei'] #替代字體 plt.rcParams['axes.unicode_minus'] = False #解決坐標軸負數(shù)的鉛顯示問題a = 1 theta = np.arange(0, 2 * np.pi, np.pi / 180) x = a * np.cos(theta) ** 3 y = a * np.sin(theta) ** 3 plt.plot(x, y, label = r'$ x=acos^3\theta, y=asin^3\theta \quad|a=1$')a = 2 x1 = a * x y1 = a * y plt.plot(x1, y1, label = r'$ x=acos^3\theta, y=asin^3\theta \quad|a=2$')a = 3 x2 = a * x y2 = a * y plt.plot(x2, y2, label = r'$ x=acos^3\theta, y=asin^3\theta \quad|a=3$')ax = plt.gca() ax.spines['right'].set_color('none') ax.spines['top'].set_color('none') ax.xaxis.set_ticks_position('bottom') ax.spines['bottom'].set_position(('data', 0)) ax.yaxis.set_ticks_position('left') ax.spines['left'].set_position(('data', 0))plt.axis('equal') plt.legend() plt.show()

擺線 ${x=a(θ?sinθ)y=a(1?cosθ)\begin{cases} x=a(\theta-sin\theta) \\ y=a(1-cos\theta) \end{cases}$

%matplotlib inline %config InlineBackend.figure_format = "svg" import numpy as np import matplotlib.pyplot as plt from scipy import signal plt.rcParams['figure.figsize'] = (8, 4.5) plt.rcParams['figure.dpi'] = 150 plt.rcParams['font.sans-serif'] = ['Simhei'] #替代字體 plt.rcParams['axes.unicode_minus'] = False #解決坐標軸負數(shù)的鉛顯示問題#極坐標 # plt.subplot(111, polar = True) # plt.ylim([0, 6])a = 1 theta = np.arange(0, 4 * np.pi, np.pi / 180) x = a * (theta - np.sin(theta)) y = a * (1 - np.cos(theta)) plt.plot(x, y, label = r'$ x=a(\theta-sin\theta),\quad y=a(1-cos\theta) \quad|a=1$')a = 2 x1 = a * x y1 = a * y plt.plot(x1, y1, label = r'$ x=a(\theta-sin\theta),\quad y=a(1-cos\theta) \quad|a=2$')a = 3 x2 = a * x y2 = a * y plt.plot(x2, y2, label = r'$ x=a(\theta-sin\theta),\quad y=a(1-cos\theta) \quad|a=3$')ax = plt.gca() ax.spines['right'].set_color('none') ax.spines['top'].set_color('none') ax.xaxis.set_ticks_position('bottom') ax.spines['bottom'].set_position(('data', 0)) ax.yaxis.set_ticks_position('left') ax.spines['left'].set_position(('data', 0))plt.axis('equal') plt.legend() plt.show()

心形線（外擺線的一種） $KaTeX parse error: Can't use function '$' in math mode at position 27: …a\sqrt{x^2+y^2}$? 或 $ \rho=a(1-c…$

%matplotlib inline %config InlineBackend.figure_format = "svg" import numpy as np import matplotlib.pyplot as plt from scipy import signal plt.rcParams['figure.figsize'] = (8, 4.5) plt.rcParams['figure.dpi'] = 150 plt.rcParams['font.sans-serif'] = ['Simhei'] #替代字體 plt.rcParams['axes.unicode_minus'] = False #解決坐標軸負數(shù)的鉛顯示問題#極坐標 plt.subplot(111, polar = True) # plt.ylim([0, 6])a = 1 theta = np.arange(0, 2 * np.pi, np.pi / 180) y = a * (1 - np.cos(theta)) plt.plot(theta, y, label = r'$ \rho=a(1-cos\theta) $')y1 = a * (1 - np.sin(theta)) plt.plot(theta, y1, label = r'$ \rho=a(1-sin\theta) $')# a = 2 # x1 = a * x # y1 = a * y # plt.plot(x1, y1, label = r'$ x=a(\theta-sin\theta),\quad y=a(1-cos\theta) \quad|a=2$')# a = 3 # x2 = a * x # y2 = a * y # plt.plot(x2, y2, label = r'$ x=a(\theta-sin\theta),\quad y=a(1-cos\theta) \quad|a=3$')# ax = plt.gca() # ax.spines['right'].set_color('none') # ax.spines['top'].set_color('none') # ax.xaxis.set_ticks_position('bottom') # ax.spines['bottom'].set_position(('data', 0)) # ax.yaxis.set_ticks_position('left') # ax.spines['left'].set_position(('data', 0))# plt.axis('equal') plt.legend() plt.show()

$x=sinθ,y=cosθ+x23x=sin\theta,\quad y=cos\theta+\sqrt[3]{x^2}$

%matplotlib inline %config InlineBackend.figure_format = "svg" import numpy as np import matplotlib.pyplot as plt from scipy import signal plt.rcParams['figure.figsize'] = (8, 4.5) plt.rcParams['figure.dpi'] = 150 plt.rcParams['font.sans-serif'] = ['Simhei'] #替代字體 plt.rcParams['axes.unicode_minus'] = False #解決坐標軸負數(shù)的鉛顯示問題a = 1 theta = np.arange(0, 2 * np.pi, np.pi / 180) x = np.sin(theta) y = np.cos(theta) + np.power((x ** 2), 1 / 3) #原來用 y = np.cos(theta) + np.power(x, 2/3)只畫出一半，而且報power的錯 plt.plot(x, y, label = r'$x=sin\theta,\quady=cos\theta+\sqrt[3]{x^2}$')# ax = plt.gca() # ax.spines['right'].set_color('none') # ax.spines['top'].set_color('none') # ax.xaxis.set_ticks_position('bottom') # ax.spines['bottom'].set_position(('data', 0)) # ax.yaxis.set_ticks_position('left') # ax.spines['left'].set_position(('data', 0))plt.axis('equal') #等比例會好看點 plt.legend() plt.show()

阿基米德螺線 $ρ=aθ\rho=a\theta$

對數(shù)螺線 $ρ=eaθ\rho=e^{a\theta}$

雙曲螺旋線 $ρθ=a\rho\theta=a$

伯努利雙紐線 $x^2+y^2)^2=2a^2xy$ 或 $x^2+y^2)^2=a^2(x^2-y^2)$ 或 $ρ2=a2sin2θ\rho^2=a^2sin2\theta$ 或 $ρ2=a2cos2θ\rho^2=a^2cos2\theta$

三葉玫瑰線 $ρ=acos3θ\rho=acos3\theta$ 或 $ρ=asin3θ\rho=asin3\theta$

%matplotlib inline %config InlineBackend.figure_format = "svg" import numpy as np import matplotlib.pyplot as plt from scipy import signal # plt.rcParams['figure.figsize'] = (8, 4.5) # plt.rcParams['figure.dpi'] = 150 plt.rcParams['font.sans-serif'] = ['Simhei'] #替代字體 plt.rcParams['axes.unicode_minus'] = False #解決坐標軸負數(shù)的鉛顯示問題#極坐標 plt.subplot(111, polar = True) # plt.ylim([0, 30])a = 1 theta = np.linspace(0, 2 * np.pi, 200) rho = a * np.cos(3 * theta) plt.plot(theta, rho, label = r'$ \rho=acos3\theta \quad |a=1 $', linestyle = 'solid')# a = 2 # rho1 = a * rho # plt.plot(theta, rho1, label = r'$ \rho=acos3\theta \quad |a=2 $', linestyle = 'solid')a = 1 rho2 = a * np.sin(3 * theta) plt.plot(theta, rho2, label = r'$ \rho=asin3\theta \quad |a=1 $', linestyle = 'solid')plt.legend() plt.show()

四葉玫瑰線 $ρ=acos2θ\rho=acos2\theta$ 或 $ρ=asin2θ\rho=asin2\theta$

%matplotlib inline %config InlineBackend.figure_format = "svg" import numpy as np import matplotlib.pyplot as plt from scipy import signal # plt.rcParams['figure.figsize'] = (8, 4.5) # plt.rcParams['figure.dpi'] = 150 plt.rcParams['font.sans-serif'] = ['Simhei'] #替代字體 plt.rcParams['axes.unicode_minus'] = False #解決坐標軸負數(shù)的鉛顯示問題#極坐標 plt.subplot(111, polar = True) # plt.ylim([0, 30])a = 1 theta = np.linspace(0, 2 * np.pi, 200) rho = a * np.cos(4 * theta) plt.plot(theta, rho, label = r'$ \rho=acos2\theta \quad |a=1 $', linestyle = 'solid')a = 2 rho1 = a * rho plt.plot(theta, rho1, label = r'$ \rho=acos2\theta \quad |a=2 $', linestyle = 'solid')# a = 1 # rho2 = a * np.sin(4 * theta) # plt.plot(theta, rho2, label = r'$ \rho=asin2\theta \quad |a=1 $', linestyle = 'solid')plt.legend() plt.show()

%matplotlib inline %config InlineBackend.figure_format = "svg" import numpy as np import matplotlib.pyplot as plt from scipy import signal # plt.rcParams['figure.figsize'] = (8, 4.5) # plt.rcParams['figure.dpi'] = 150 plt.rcParams['font.sans-serif'] = ['Simhei'] #替代字體 plt.rcParams['axes.unicode_minus'] = False #解決坐標軸負數(shù)的鉛顯示問題#極坐標 plt.subplot(111, polar = True) # plt.ylim([0, 30])theta = np.linspace(0, 2 * np.pi, 200)a = 1 rho = a * np.sin(4 * theta) plt.plot(theta, rho, label = r'$ \rho=asin2\theta \quad |a=1 $', linestyle = 'solid')a = 2 rho1 = a * np.sin(4 * theta) plt.plot(theta, rho1, label = r'$ \rho=asin2\theta \quad |a=2 $', linestyle = '--')plt.legend() plt.show()

多葉玫瑰線 $ρ=acosnθ\rho=acosn\theta$ 或 $ρ=asinnθ,∣n=1,2,3...\rho=asinn\theta,\quad|n=1,2,3...$

%matplotlib inline %config InlineBackend.figure_format = "svg" import numpy as np import matplotlib.pyplot as plt from scipy import signal # plt.rcParams['figure.figsize'] = (8, 4.5) # plt.rcParams['figure.dpi'] = 150 plt.rcParams['font.sans-serif'] = ['Simhei'] #替代字體 plt.rcParams['axes.unicode_minus'] = False #解決坐標軸負數(shù)的鉛顯示問題#極坐標 plt.subplot(111, polar = True) # plt.ylim([0, 30])theta = np.linspace(0, 2 * np.pi, 200)a = 1 n = 20 rho = a * np.sin(10 * theta) plt.plot(theta, rho, label = r'$ \rho=asinn\theta \quad |a=1,n=10 $', linestyle = 'solid')plt.legend() plt.show()

函數(shù) 四葉線 $x=aρsinθ,y=aρcosθ,ρ=2sinsin2θx=a\rho sin\theta,y=a\rho cos\theta, \rho = \sqrt{2}sin{sin2\theta}$

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函數(shù) 四葉線 $x=aρsinθ,y=aρcosθ,ρ=2sin∣sin2θ∣x=a\rho sin\theta,y=a\rho cos\theta, \rho = \sqrt{2}sin{\vert sin2\theta \vert}$

%matplotlib inline %config InlineBackend.figure_format = "svg" import numpy as np import matplotlib.pyplot as plt from scipy import signal # plt.rcParams['figure.figsize'] = (8, 4.5) # plt.rcParams['figure.dpi'] = 150 plt.rcParams['font.sans-serif'] = ['Simhei'] #替代字體 plt.rcParams['axes.unicode_minus'] = False #解決坐標軸負數(shù)的鉛顯示問題#極坐標 # plt.subplot(111, polar = True) # plt.ylim([0, 30])a = 1 theta = np.linspace(-np.pi, np.pi, 1000) rho = np.sqrt(2) * np.sqrt(abs(np.sin(10*theta)) + 0.5) x = a * rho * np.sin(theta) y = a * rho * np.cos(theta) plt.plot(x, y, linestyle = 'solid')# label = r'$x=a\rho sin\theta,y=a\rho cos\theta, \rho = \sqrt{2}sin{sin2\theta}$',plt.axis('equal') # plt.legend() plt.show()

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python

学习Python中用numpy与matplotlib遇到的一些数学函数与函数的绘图

學習Python中的一些數(shù)學函數(shù)與函數(shù)的繪圖

函數(shù) y=x?sinxy = x - sinx y=x?sinx

函數(shù) y=3x4?4x3+1y = 3x^4 - 4x^3 + 1 y=3x4?4x3+1

函數(shù) y=x4y = x^4 y=x4

函數(shù) y=x3y = \sqrt[3]{x} y=3x?

函數(shù) y=2x3?9x2+12x?3y=2x^3-9x^2+12x-3 y=2x3?9x2+12x?3

函數(shù) y=2x3?6x2?18x+7y=2x^3-6x^2-18x+7 y=2x3?6x2?18x+7

三次拋物線 y=x3y = x^3 y=x3

半立方拋物線 y2=ax3y^2 = ax^3 y2=ax3

概率曲線 y=e?x2y=e^{-x^2} y=e?x2

箕舌線 y=8a3x2+4a2y =\frac{8a^3}{x^2+4a^2} y=x2+4a28a3?

蔓葉線 y2(2a?x)=x3y^2(2a-x)=x^3 y2(2a?x)=x3 或 x=2asin2θ,y=2a2tan2θsin4θx=2asin^2\theta, y=2a^2tan^2\theta sin^4\theta x=2asin2θ,y=2a2tan2θsin4θ

ρ=2a?tan2θ\rho=2a-tan^2\theta ρ=2a?tan2θ

笛卡兒葉形線畫圖 極坐標 r=3asinθcosθsin3θ+cos3θr = \frac{3asin\theta cos\theta}{sin^3\theta + cos^3\theta} r=sin3θ+cos3θ3asinθcosθ?

笛卡兒葉形線 直角坐標 x3+y3?3axy=0x^3+y^3-3axy=0 x3+y3?3axy=0 或 x=3at1+t3,y=3at21+t3x=\frac{3at}{1+t^3}, y=\frac{3at^2}{1+t^3} x=1+t33at?,y=1+t33at2?

星形線（內(nèi)擺線的一種） x23+y23=a23x^\frac{2}{3}+y^\frac{2}{3}=a^\frac{2}{3} x32?+y32?=a32? 或 {x=acos3θy=asin3θ\begin{cases} x=acos^3\theta \\ y=asin^3\theta \end{cases} {x=acos3θy=asin3θ?

擺線 {x=a(θ?sinθ)y=a(1?cosθ)\begin{cases} x=a(\theta-sin\theta) \\ y=a(1-cos\theta) \end{cases} {x=a(θ?sinθ)y=a(1?cosθ)?

心形線（外擺線的一種） KaTeX parse error: Can't use function '$' in math mode at position 27: …a\sqrt{x^2+y^2}$? 或 $ \rho=a(1-c…

x=sinθ,y=cosθ+x23x=sin\theta,\quad y=cos\theta+\sqrt[3]{x^2}x=sinθ,y=cosθ+3x2?

阿基米德螺線 ρ=aθ\rho=a\thetaρ=aθ

對數(shù)螺線 ρ=eaθ\rho=e^{a\theta} ρ=eaθ

雙曲螺旋線 ρθ=a\rho\theta=a ρθ=a

伯努利雙紐線 (x2+y2)2=2a2xy(x^2+y^2)^2=2a^2xy(x2+y2)2=2a2xy 或 (x2+y2)2=a2(x2?y2)(x^2+y^2)^2=a^2(x^2-y^2)(x2+y2)2=a2(x2?y2) 或 ρ2=a2sin2θ\rho^2=a^2sin2\theta ρ2=a2sin2θ 或 ρ2=a2cos2θ\rho^2=a^2cos2\thetaρ2=a2cos2θ

三葉玫瑰線 ρ=acos3θ\rho=acos3\thetaρ=acos3θ 或 ρ=asin3θ\rho=asin3\theta ρ=asin3θ

四葉玫瑰線 ρ=acos2θ\rho=acos2\theta ρ=acos2θ 或 ρ=asin2θ\rho=asin2\theta ρ=asin2θ

多葉玫瑰線 ρ=acosnθ\rho=acosn\theta ρ=acosnθ 或 ρ=asinnθ,∣n=1,2,3...\rho=asinn\theta,\quad|n=1,2,3...ρ=asinnθ,∣n=1,2,3...

函數(shù) 四葉線 x=aρsinθ,y=aρcosθ,ρ=2sinsin2θx=a\rho sin\theta,y=a\rho cos\theta, \rho = \sqrt{2}sin{sin2\theta}x=aρsinθ,y=aρcosθ,ρ=2?sinsin2θ

函數(shù) 四葉線 x=aρsinθ,y=aρcosθ,ρ=2sin∣sin2θ∣x=a\rho sin\theta,y=a\rho cos\theta, \rho = \sqrt{2}sin{\vert sin2\theta \vert}x=aρsinθ,y=aρcosθ,ρ=2?sin∣sin2θ∣

總結