最近更新于 2025-08-15 20:13

1 心形线

\begin{array}{l}
\rho=a(1-\cos\theta)（a>0，\theta\in[0,2\pi]） \\
\\
x^2+y^2+ax=a\sqrt{x^2+y^2}（a\gt0）\\
\\
\begin{cases}
x=a(1-\cos\theta)\cos\theta \\
y=a(1-\cos\theta)\sin\theta
\end{cases}
（a>0，\theta\in[0,2\pi]）
\end{array}

import matplotlib.pyplot as plt
import numpy as np

alpha = 1
theta = np.linspace(0, 2 * np.pi, 100)

rho = alpha * (1 - np.cos(theta))

plt.polar(theta, rho, color='r')

---

\begin{array}{l}
\rho=a(1+\cos\theta)（a>0，\theta\in[0,2\pi]） \\
\\
x^2+y^2-ax=a\sqrt{x^2+y^2}（a\gt0）\\
\\
\begin{cases}
x=a(1+\cos\theta)\cos\theta \\
y=a(1+\cos\theta)\sin\theta
\end{cases}
（a>0，\theta\in[0,2\pi]）
\end{array}

import matplotlib.pyplot as plt
import numpy as np

alpha = 1
theta = np.linspace(0, 2 * np.pi, 100)

rho = alpha * (1 + np.cos(theta))

plt.polar(theta, rho, color='r')

---

\begin{array}{l}
\rho=a(1-\sin\theta)（a>0，\theta\in[0,2\pi]） \\
\\
x^2+y^2+ay=a\sqrt{x^2+y^2}（a\gt0）\\
\\
\begin{cases}
x=a(1-\sin\theta)\cos\theta \\
y=a(1-\sin\theta)\sin\theta
\end{cases}
（a>0，\theta\in[0,2\pi]）
\end{array}

import matplotlib.pyplot as plt
import numpy as np

alpha = 1
theta = np.linspace(0, 2 * np.pi, 100)

rho = alpha * (1 - np.sin(theta))

plt.polar(theta, rho, color='r')

---

\begin{array}{l}
\rho=a(1+\sin\theta)（a>0，\theta\in[0,2\pi]） \\
\\
x^2+y^2-ay=a\sqrt{x^2+y^2}（a\gt0）\\
\\
\begin{cases}
x=a(1+\sin\theta)\cos\theta \\
y=a(1+\sin\theta)\sin\theta
\end{cases}
（a>0，\theta\in[0,2\pi]）
\end{array}

import matplotlib.pyplot as plt
import numpy as np

alpha = 1
theta = np.linspace(0, 2 * np.pi, 100)

rho = alpha * (1 + np.sin(theta))

plt.polar(theta, rho, color='r')

2 阿基米德螺线

\begin{array}{l}
r=a\theta（r\gt0，\theta\ge0）(a\gt0，\theta\in[0,+\infty)) \\
\\
\begin{cases}
x=a\theta\cos\theta \\
y=a\theta\sin\theta
\end{cases}
(a\gt0，\theta\in[0,+\infty))
\end{array}

import matplotlib.pyplot as plt
import numpy as np

alpha = 1
theta = np.linspace(0, 2 * np.pi, 100)
rho = alpha * theta

x = rho * np.cos(theta)
y = rho * np.sin(theta)

plt.axis('equal')
plt.plot(x, y, color='r')
plt.axhline(y=0, color='k')
plt.axvline(x=0, color='k')

4 伯努利双扭线

\begin{array}{l}
r^2=a^2\cos2\theta（\theta\in[-\frac{\pi}{4},\frac{\pi}{4}]\cup[\frac{3\pi}{4},\frac{5\pi}{4}]，a\gt0）\\
\\
(x^2+y^2)^2=2a^2(x^2-y^2)（a\gt0）
\end{array}

import numpy as np
import matplotlib.pyplot as plt

alpha = 1
theta = np.concatenate((
    np.linspace(-np.pi / 4, np.pi / 4, 100),
    np.linspace(3 * np.pi / 4, 5 * np.pi / 4, 100)
))

rho = alpha * np.sqrt(np.abs(np.cos(2 * theta)))

plt.polar(theta, rho, color='r')

---

\begin{array}{l}
r^2=a^2\sin2\theta（\theta\in[0,\frac{\pi}{2}]\cup[\pi,\frac{3\pi}{2}]，a\gt0）
\\
\\
(x^2+y^2)^2=2a^2xy（a\gt0）
\end{array}

import numpy as np
import matplotlib.pyplot as plt

alpha = 1
theta = np.concatenate((
    np.linspace(0, np.pi / 2, 100),
    np.linspace(np.pi, 3 * np.pi / 2, 100)
))

rho = alpha * np.sqrt(np.abs(np.sin(2 * theta)))

plt.polar(theta, rho, color='r')

5 摆线

\begin{cases}
x=a(\theta-\sin\theta) \\
y=a(1-\cos\theta)
\end{cases}
（a\gt0）

import numpy as np
import matplotlib.pyplot as plt

alpha = 1
theta = np.linspace(0, 4 * np.pi, 100)

x = alpha * (theta - np.sin(theta))
y = alpha * (1 - np.cos(theta))

plt.axis('equal')
plt.plot(x, y, color='r')
plt.axhline(y=0, color='k')
plt.axvline(x=0, color='k')
plt.axhline(y=2 * alpha, color='b')
plt.axvline(x=np.pi * alpha, color='b')