Man and History: Euler's Formula

2022年5月10日星期二

Euler's Formula

2022/01/19

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https://pixabay.com/zh/illustrations/pay-numbers-infinity-digits-937884/

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◎ 說明：

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Fig. 1. Euler's Formula and Euler's identity [3].

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Fig. 2. Euler's Formula [3].

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整數次方是整數倍數的膨脹。開根號是整數倍數的收縮。i 次方是將實數在複數平面上逆時針旋轉 90 度 [4]。

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Euler's Formula：複數平面上的單位圓。

Euler's identity：在 180 度的地方值為 -1。

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Fig. 4. 歐拉公式的螺旋線呈現 [8]。

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代碼一：

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

n = 1000 # points of the helix
fig = plt.figure(figsize=(16,16))
ax = fig.add_subplot(111, projection='3d')

# Plot a helix along the x-axis (e.g. time)
theta_max = 6 * np.pi # number of the semicircle
theta = np.linspace(0, theta_max, n)
x = theta
z =  np.sin(theta)
y =  np.cos(theta)
ax.plot(x, y, z, 'r', lw=2)

# An line through the centre of the helix
ax.plot((-theta_max*0.2, theta_max * 1.2), (0,0), (0,0), color='k', lw=2)
# sin/cos components of the helix (e.g. phase and amplitude
# components of a circularly-polarized Eular's Formula
ax.plot(x, y, 0, color='b', lw=1, alpha=0.5)
ax.plot(x, [0]*n, z, color='m', lw=1, alpha=0.5)

# Remove axis planes, ticks and labels
ax.set_axis_off()

# plt.savefig('/content/drive/My Drive/helix.png')
# plt.show()