2023年2月9日 星期四

Python 量子運算（一八）：和差角公式

Python 量子運算（一八）：和差角公式

 2023/02/04

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Fig. 18.1. Angle sum and difference formulas.

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代碼 18.1


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# Program 18.1：Angle sum and difference formulas
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt


# defined by angle
class PointT1:
    def __init__(self, t, r):
        self.t = t  # theta

        self.r = r  # radius
        self.x = r * np.cos(t)
        self.y = r * np.sin(t)


# defined by points
class PointT2:
    def __init__(self, x, y):
        self.x = x
        self.y = y


def Subplot_1():
    ax = plt.subplot(221)

    # circle
    c = np.linspace(0, np.pi*2, 100)
    r = 1
    c_x = r * np.cos(c)
    c_y = r * np.sin(c)
    ax.plot(c_x, c_y, 'g', linestyle='--')

    # points
    t_A = (31/24) * np.pi                # theta A setting
    PA = PointT1(t_A, r)                 # A
    PC = PointT1(t_A-np.pi, r)           # C, determined by A
    ax.plot([PA.x, PC.x], [PA.y, PC.y], 'purple', linewidth=5.0)  # AC
    ax.text(PA.x-0.1, PA.y-0.1, 'A')
    ax.text(PC.x+0.05, PC.y+0.05, 'C')

    PB = PointT2(PC.x, PA.y)             # B, determined by A, C
    ax.plot([PA.x, PB.x], [PA.y, PA.y], 'r', linestyle='--')  # AB
    ax.plot([PB.x, PB.x], [PB.y, PC.y], 'purple', linestyle='--')  # BC
    ax.text(PB.x+0.05, PB.y-0.1, 'B')

    PG = PointT2(PA.x, PC.y)             # G, determined by A, C
    ax.plot([PA.x, PG.x], [PA.y, PG.y], 'purple', linestyle='--')  # AG
    ax.plot([PC.x, PG.x], [PC.y, PG.y], 'purple', linestyle='--')  # CG
    ax.text(PG.x-0.1, PG.y+0.05, 'G')

    t_E = (47/24) * np.pi                # theta E setting
    PE = PointT1(t_E, r)                 # E
    ax.plot([PA.x, PE.x], [PA.y, PE.y], 'r', linewidth=5.0)  # AE
    ax.plot([PC.x, PE.x], [PC.y, PE.y], 'b', linewidth=5.0)  # CE
    ax.text(PE.x+0.05, PE.y, 'E')

    PD = PointT2(PE.x, PC.y)             # D, determined by C, E
    ax.plot([PC.x, PD.x], [PC.y, PD.y], 'b', linestyle='--')  # CD
    ax.plot([PD.x, PE.x], [PD.y, PE.y], 'b', linestyle='--')  # DE
    ax.text(PD.x+0.05, PD.y+0.05, 'D')

    PF = PointT2(PE.x, PB.y)             # F, determined by B, E
    ax.plot([PB.x, PF.x], [PB.y, PF.y], 'r', linestyle='--')  # BF
    ax.plot([PE.x, PF.x], [PE.y, PF.y], 'r', linestyle='--')  # EF
    ax.text(PF.x+0.05, PF.y-0.1, 'F')

    # angles
    ax.text(PA.x+0.25, PA.y+0.20, r'$\alpha$')
    ax.text(PA.x+0.35, PA.y+0.05, r'$\beta$')
    ax.text(PE.x-0.08, PE.y+0.25, r'$\beta$')
    ax.text(PC.x-0.35, PC.y-0.10, r'$\alpha+\beta$')

    # edges
    ax.text((PA.x+PE.x)/2-0.2, (PA.y+PE.y)/2,
            r'$\cos\alpha$', color='r')
    ax.text((PA.x+PF.x)/2-0.1, (PA.y+PF.y)/2,
            r'$\cos\alpha\cos\beta$', color='r')
    ax.text((PE.x+PF.x)/2, (PE.y+PF.y)/2,
            r'$\cos\alpha\sin\beta$', color='r')

    ax.text((PC.x+PE.x)/2-0.2, (PC.y+PE.y)/2,
            r'$\sin\alpha$', color='b')
    ax.text((PD.x+PE.x)/2, (PD.y+PE.y)/2,
            r'$\sin\alpha\cos\beta$', color='b')
    ax.text((PC.x+PD.x)/2-0.15, (PC.y+PD.y)/2,
            r'$\sin\alpha\sin\beta$', color='b')

    ax.text((PC.x+PG.x)/2-0.2, (PC.y+PG.y)/2,
            r'$\cos(\alpha+\beta)$', color='purple')
    ax.text((PA.x+PG.x)/2-0.4, (PA.y+PG.y)/2,
            r'$\sin(\alpha+\beta)$', color='purple')

    ax.text(0, -1.3, '(a)')
    ax.set_aspect(1)  # height : width
    ax.set_axis_off()

    return


def Subplot_2():
    ax = plt.subplot(222)

    # string setting
    s1 = r'$\overline{AC}=1$'
    s2_1 = r'$\sin(\alpha+\beta)$'
    s2_2 = r'$=\overline{AG}=\overline{DE}+\overline{EF}$'
    s2_3 = r'$=\sin\alpha\cos\beta+\cos\alpha\sin\beta$'
    s3_1 = r'$\cos(\alpha+\beta)$'
    s3_2 = r'$=\overline{CG}=\overline{AF}-\overline{BF}$'
    s3_3 = r'$=\cos\alpha\cos\beta-\sin\alpha\sin\beta$'

    # string output
    ax.text(0.10, 0.95, s1, fontsize=24)

    ax.text(0.20, 0.80, s2_1, color='purple', fontsize=40)
    ax.text(0.10, 0.65, s2_2, fontsize=24)
    ax.text(0.10, 0.50, s2_3, fontsize=40)

    ax.text(0.20, 0.30, s3_1, color='purple', fontsize=40)
    ax.text(0.10, 0.15, s3_2, fontsize=24)
    ax.text(0.10, 0.00, s3_3, fontsize=40)

    ax.text(0.5, -0.09, '(b)')
    ax.set_axis_off()

    return


def Subplot_3():
    ax = plt.subplot(223)

    # circle
    c = np.linspace(0, np.pi*2, 100)
    r = 1
    c_x = r * np.cos(c)
    c_y = r * np.sin(c)
    ax.plot(c_x, c_y, 'g', linestyle='--')

    t_A = (31/24) * np.pi                # theta A setting
    PA = PointT1(t_A, r)                 # A
    PC = PointT1(t_A-np.pi, r)           # C, determined by A
    ax.plot([PA.x, PC.x], [PA.y, PC.y])  # AC
    ax.plot([PA.x, PC.x], [PA.y, PC.y], 'purple', linewidth=5.0)  # AC
    ax.text(PA.x-0.1, PA.y-0.1, 'A')
    ax.text(PC.x+0.05, PC.y+0.05, 'C')

    PB = PointT2(PC.x, PA.y)             # B, determined by A, C
    ax.plot([PA.x, PB.x], [PA.y, PA.y])  # AB
    ax.plot([PB.x, PB.x], [PB.y, PC.y])  # BC
    ax.plot([PA.x, PB.x], [PA.y, PA.y], 'r', linewidth=5.0)  # AB
    ax.plot([PB.x, PB.x], [PB.y, PC.y], 'b', linewidth=5.0)  # BC
    ax.text(PB.x+0.05, PB.y-0.1, 'B')

    t_E = (47/24) * np.pi                # theta E setting
    PE = PointT1(t_E, r)                 # E
    ax.plot([PA.x, PE.x], [PA.y, PE.y], 'purple', linestyle='--')  # AE
    ax.plot([PC.x, PE.x], [PC.y, PE.y], 'purple', linestyle='--')  # CE
    ax.text(PE.x+0.05, PE.y, 'E')

    # finding Point G for learning
    slope_AE = (PE.y - PA.y) / (PE.x - PA.x)

    # finding Point D
    slope_CE = (PE.y - PC.y) / (PE.x - PC.x)  # -(1/slope_AE)
    slope_BD = slope_CE
    x1 = (slope_AE * PA.x - slope_BD * PB.x + PB.y - PA.y)
    x2 = (slope_AE - slope_BD)
    PD_x = x1 / x2
    PD_y = slope_AE * (PD_x - PA.x) + PA.y
    PD = PointT2(PD_x, PD_y)
    ax.plot([PB.x, PD.x], [PB.y, PD.y], 'r', linestyle='--')  # BD
    ax.text(PD.x, PD.y+0.05, 'D')

    # finding Point F
    PF_x = PE.x - PD.x + PB.x
    PF_y = PE.y - PD.y + PB.y
    PF = PointT2(PF_x, PF_y)
    ax.plot([PB.x, PF.x], [PB.y, PF.y], 'b', linestyle='--')  # BF
    ax.plot([PE.x, PF.x], [PE.y, PF.y], 'r', linestyle='--')  # EF
    ax.text(PF.x+0.05, PF.y-0.1, 'F')

    # angles
    ax.text(PA.x+0.20, PA.y+0.20, r'$\alpha-\beta$')
    ax.text(PA.x+0.35, PA.y+0.05, r'$\beta$')
    ax.text(PC.x+0.02, PC.y-0.30, r'$\beta$')

    # edges
    ax.text((PA.x+PB.x)/2, (PA.y+PB.y)/2+0.01,
            r'$\cos\alpha$', color='r')
    ax.text((PA.x+PD.x)/2, (PA.y+PD.y)/2+0.10,
            r'$\cos\alpha\cos\beta$', color='r')
    ax.text((PB.x+PD.x)/2-0.20, (PB.y+PD.y)/2,
            r'$\cos\alpha\sin\beta$', color='r')

    ax.text((PB.x+PC.x)/2-0.21, (PB.y+PC.y)/2,
            r'$\sin\alpha$', color='b')
    ax.text((PB.x+PF.x)/2, (PB.y+PF.y)/2-0.05,
            r'$\sin\alpha\sin\beta$', color='b')

    ax.text((PC.x+PE.x)/2, (PC.y+PE.y)/2,
            r'$\sin(\alpha-\beta)$', color='purple')

    ax.text(0, -1.38, '(c)')
    ax.set_aspect(1)  # height : width
    ax.set_axis_off()

    return


def Subplot_4():
    ax = plt.subplot(224)

    # string setting
    s1 = r'$\overline{AC}=1$'
    s2_1 = r'$\sin(\alpha-\beta)$'
    s2_2 = r'$=\overline{CE}=\overline{CF}-\overline{EF}$'
    s2_3 = r'$=\sin\alpha\cos\beta-\cos\alpha\sin\beta$'
    s3_1 = r'$\cos(\alpha-\beta)$'
    s3_2 = r'$=\overline{AE}=\overline{AD}+\overline{DE}$'
    s3_3 = r'$=\cos\alpha\cos\beta+\sin\alpha\sin\beta$'

    # string output
    ax.text(0.10, 0.95, s1, fontsize=24)

    ax.text(0.20, 0.80, s2_1, color='purple', fontsize=40)
    ax.text(0.10, 0.65, s2_2, fontsize=24)
    ax.text(0.10, 0.50, s2_3, fontsize=40)

    ax.text(0.20, 0.30, s3_1, color='purple', fontsize=40)
    ax.text(0.10, 0.15, s3_2, fontsize=24)
    ax.text(0.10, 0.00, s3_3, fontsize=40)

    ax.text(0.5, -0.09, '(d)')
    ax.set_axis_off()

    return


mpl.rcParams['text.usetex'] = True
mpl.rcParams['text.latex.preamble'] = r'\usepackage{{amsmath}}'
mpl.rcParams['font.size'] = 20
fig = plt.figure(figsize=(16, 16))

Subplot_1()
Subplot_2()
Subplot_3()
Subplot_4()

# plt.savefig('/content/drive/My Drive/pqc/0018_001.png')
plt.show()

解說：

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Python 量子運算（目錄）

https://mandhistory.blogspot.com/2022/01/quantum-computing.html

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