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

-----

張貼者： 於 沒有留言:
標籤：

Python 量子運算（一六）：和角

Python 量子運算（一六）：和角

2023/01/24

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Fig. 1. Angle sum.

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和角 angle sum

和角公式 angle sum formula

差角 angle difference

差角公式 angle difference formula

「Along with angle difference formulas, we have angle sum formulas as well.」[1]。

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代碼 16.1


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# Program 16.1：Angle sum
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(111)

    # 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.set_aspect(1)  # height : width
    ax.set_axis_off()

    return


mpl.rcParams['text.usetex'] = True
mpl.rcParams['text.latex.preamble'] = r'\usepackage{{amsmath}}'
mpl.rcParams['font.size'] = 40
fig = plt.figure(figsize=(16, 16))

Subplot_1()

# plt.savefig('/content/drive/My Drive/pqc/0016_001.png')
plt.show()

解說：


-----

References


[1] What are Angle Difference Formulas? Examples

https://www.cuemath.com/angle-difference-formula/


[2] 7.2: Sum and Difference Identities - Mathematics LibreTexts

https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(OpenStax)/07%3A_Trigonometric_Identities_and_Equations/7.02%3A_Sum_and_Difference_Identities


[3] 三角函數的和差角公式

http://www.mathland.idv.tw/fun/triiden.htm


# dashed

[4] How to use linestyles in Matplotlib

https://www.educative.io/answers/how-to-use-linestyles-in-matplotlib


# grey

[5] matplotlib.pyplot.plot — Matplotlib 3.6.3 documentation

https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.plot.html


[6] 如何在 Matplotlib 圖例中設定線的線寬 | D棧 - Delft Stack

https://www.delftstack.com/zh-tw/howto/matplotlib/how-to-change-line-width-of-lines-of-in-matplotlib-legend/


# class、object、__init__()、method、self、del

[7] Python Classes

https://www.w3schools.com/python/python_classes.asp


# inheritance、parent class、child class、super()

[8] Python Inheritance

https://www.w3schools.com/python/python_inheritance.asp

-----

Python 量子運算（目錄）

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

-----

張貼者： 於 沒有留言:
標籤：

Python 量子運算（一五）：內積

 Python 量子運算（一五）：內積

2023/01/11

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Fig. 15.1. Inner product.

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代碼 15.1


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# Program 15.1：Inner product
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt


class Vec:
    def __init__(self, t1, t2, v_r, a_r):
        self.t1 = t1    # theta 1
        self.t2 = t2    # theta 2

        self.v_r = v_r  # vector radius
        self.v_x = v_r * np.cos(t2)
        self.v_y = v_r * np.sin(t2)

        self.a = np.linspace(t1, t2, 100)  # arc
        self.a_r = a_r                     # arc radius
        self.a_x = a_r * np.cos(self.a)
        self.a_y = a_r * np.sin(self.a)


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

    # coordinates
    ax.plot([-0.1, 1], [0, 0], 'k')
    ax.plot([0, 0], [-0.1, 1], 'k')

    # vector setting
    v1 = Vec(0, np.pi/3, 1, 0.6)
    v2 = Vec(0, np.pi/12, 1, 0.5)
    v3 = Vec(np.pi/12, np.pi/3, 1, 0.4)

    # vector x
    plt.quiver(0, 0, v1.v_x, v1.v_y, scale=1.25, color='r')
    ax.plot([0, v1.v_x], [v1.v_y, v1.v_y], 'r', linestyle=':')
    ax.plot([v1.v_x, v1.v_x], [0, v1.v_y], 'r', linestyle=':')
    ax.text(v1.v_x+0.05, v1.v_y+0.05, r"$\mathbf{x}$", color='r', fontsize=32)
    ax.text(v1.v_x-0.05, -0.1, r"$x1$", color='r', fontsize=32)
    ax.text(-0.15, v1.v_y, r"$x2$", color='r', fontsize=32)

    # vector y
    plt.quiver(0, 0, v2.v_x, v2.v_y, scale=1.2, color='b')
    ax.plot([0, v2.v_x], [v2.v_y, v2.v_y], 'b', linestyle=':')
    ax.plot([v2.v_x, v2.v_x], [0, v2.v_y], 'b', linestyle=':')
    ax.text(v2.v_x+0.05, v2.v_y+0.05, r"$\mathbf{y}$", color='b', fontsize=32)
    ax.text(v2.v_x-0.05, -0.1, r"$y1$", color='b', fontsize=32)
    ax.text(-0.15, v2.v_y, r"$y2$", color='b', fontsize=32)

    # arc
    ax.plot(v1.a_x, v1.a_y, 'r')
    ax.text(0.36, 0.36, r"$\alpha$", color='r', fontsize=32)
    ax.plot(v2.a_x, v2.a_y, 'b')
    ax.text(0.41, 0.03, r"$\beta$", color='b', fontsize=32)
    ax.plot(v3.a_x, v3.a_y, 'k')
    ax.text(0.11, 0.11, r"$\alpha - \beta$", color='k', fontsize=32)

    ax.text(0.5, -0.34, '(a)', fontsize=20)
    ax.set_axis_off()

    return


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

    # string setting
    s1_1 = r'$\mathbf{x}=(x_1,\cdots,x_n),$'
    s1_2 = r'$\mathbf{y}=(y_1,\cdots,y_n).$'

    s2_1 = r'$\mathbf{x} \cdot \mathbf{y}$'
    s2_2 = r'$=x_1y_1+ \cdots +x_ny_n$'
    s2_3 = r'$=\sum_{i=1}^n x_iy_i.$'

    s3_1 = r'$\mathbf{x}^T\mathbf{y}$'
    s3_2 = r'$=\begin{bmatrix}x_1 \cdots x_n\end{bmatrix}$'\
           r'$\begin{bmatrix}y_1\\ \vdots \\y_n\end{bmatrix}$'
    s3_3 = r'$=\sum_{i=1}^n x_iy_i$.'

    # string output
    ax.text(0.1, 0.75, s1_1)
    ax.text(0.1, 0.63, s1_2)

    ax.text(0.1, 0.45, s2_1)
    ax.text(0.3, 0.45, s2_2)
    ax.text(0.3, 0.33, s2_3)

    ax.text(0.1, 0.15, s3_1)
    ax.text(0.3, 0.15, s3_2)
    ax.text(0.3, 0.03, s3_3)

    ax.text(0.5, -0.15, '(b)', fontsize=20)
    ax.set_axis_off()

    return


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

    # string setting
    s1 = r'$\|\mathbf{x}\|=\sqrt{x_1^2+x_2^2},$'
    s2_1 = r'$\langle \mathbf{x}, \mathbf{y} \rangle$'
    s2_2 = r'$=\|\mathbf{x}\| \cos(\alpha - \beta) \|\mathbf{y}\|$'
    s2_3 = r'$=x_1y_1+x_2y_2.$'

    # string output
    ax.text(0.10, 0.75, s1)

    ax.text(0.20, 0.55, s2_1)
    ax.text(0.10, 0.35, s2_2)
    ax.text(0.10, 0.15, s2_3)

    ax.text(0.5, -0.15, '(c)', fontsize=20)
    ax.set_axis_off()

    return


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

    # string setting
    s1_1 = r'$\cos (\alpha - \beta)$'
    s1_2 = r'$=\cos\alpha\cos\beta+\sin\alpha\sin\beta$'
    s1_3 = r'$=\frac{x_1}{\|\mathbf{x}\|}\frac{y_1}{\|\mathbf{y}\|}$'\
           r'$+\frac{x_2}{\|\mathbf{x}\|}\frac{y_2}{\|\mathbf{y}\|}$'
    s1_4 = r'$=\frac{x_1y_1+x_2y_2}{\|\mathbf{x}\|\|\mathbf{y}\|}.$'

    # string output
    ax.text(0.20, 0.75, s1_1)
    ax.text(0.10, 0.55, s1_2)
    ax.text(0.10, 0.35, s1_3)
    ax.text(0.10, 0.15, s1_4)

    ax.text(0.5, -0.15, '(d)', fontsize=20)
    ax.set_axis_off()

    return


mpl.rcParams['text.usetex'] = True
mpl.rcParams['text.latex.preamble'] = r'\usepackage{{amsmath}}'
mpl.rcParams['font.size'] = 40
fig = plt.figure(figsize=(16, 16))

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

#plt.savefig('/content/drive/My Drive/pqc/0015_001.png')
plt.show()

解說：

-----

References


[1] Inner Product -- from Wolfram MathWorld

https://mathworld.wolfram.com/InnerProduct.html


[2] 內積的定義 | 線代啟示錄

https://ccjou.wordpress.com/2010/01/27/%E5%85%A7%E7%A9%8D%E7%9A%84%E5%AE%9A%E7%BE%A9/


[3] python 兩點連線matplotlib_yanni0616的博客-CSDN博客_python兩點之間連線

https://blog.csdn.net/yanni0616/article/details/99696020


[4] Python Classes

https://www.w3schools.com/python/python_classes.asp


# norm

# \|

[5] How to write norm symbol in LaTeX like ||a||?

https://www.physicsread.com/latex-norm-symbol/


# 向量

[6] 「LaTeX」LaTeX 中三種向量表示：粗體1，粗體2，箭頭向量- 嗶哩嗶哩

https://www.bilibili.com/read/cv3599113

-----

Python 量子運算（目錄）

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

-----

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標籤：

Python 量子運算（一四）：點積

Python 量子運算（一四）：點積

2023/01/03

-----


Fig. 14.1. Dot product.

-----

代碼 14.1


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# Program 14.1：Dot product
import numpy as np
import matplotlib as mpl
import matplotlib.pyplot as plt


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

    # basic setting
    theta = np.linspace(0, np.pi/2, 100)
    axis = np.linspace(0, 1, 100)
    zero = axis * 0
    radius = 0.7
    x1 = radius * np.cos(theta)
    y1 = radius * np.sin(theta)

    # basic plotting
    ax.plot(axis, zero, 'k')  # y = 0(x_axis)
    ax.plot(zero, axis, 'k')  # x = 0(y_axis)
    ax.plot(x1, y1, 'k')      # cicle

    # vector setting
    coordinates = [0, 0]      # original point
    splt_radian = np.pi / 4   # subplot

    # vector plotting
    v_p1 = splt_radian
    v_p2 = 0
    vector_1 = [radius*np.cos(v_p1), radius*np.sin(v_p1)]  # direction
    vector_2 = [radius*np.cos(v_p2), radius*np.sin(v_p2)]  # direction
    plt.quiver(coordinates[0], coordinates[1], vector_1[0], vector_1[1],
               scale=1.15, color='r')
    plt.quiver(coordinates[0], coordinates[1], vector_2[0], vector_2[1],
               scale=1.1, color='b')
    plt.quiver(coordinates[0], coordinates[1]-0.05, vector_1[0], vector_2[1],
               scale=1.15, color='k')

    # red line
    r_line_y = np.linspace(0, radius*np.sin(1.05*splt_radian), 100)
    r_line_x = r_line_y * 0 + radius * np.cos(1.05*splt_radian)
    ax.plot(r_line_x, r_line_y, 'r', linestyle=':')

    # red arc
    phi = np.linspace(0, splt_radian, 100)
    x2 = 0.2 * np.cos(phi)
    y2 = 0.2 * np.sin(phi)
    ax.plot(x2, y2, 'r')

    ax.text(0.25, 0.1, r"$\theta=45^{\circ}$", fontsize='20')
    ax.text(radius+0.1, 0.05, r"$\vec u=(1,0)$", color='b')
    ax.text(radius*np.cos(v_p1)+0.1, radius*np.sin(v_p1),
            r"$\vec v=(\frac{\sqrt 2}{2},\frac{\sqrt 2}{2})$", color='r')
    ax.text(radius*np.cos(v_p1)+0.1, -0.15,
            r"$\vec w=(\frac{\sqrt 2}{2},0)$", color='k')
    ax.text(0.5, -0.28, '(a)', fontsize=20)
    ax.set_axis_off()

    return


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

    # string setting
    s1 = r'${\rm cos}\ \theta=\frac{\vert \vec w\vert}{\vert \vec v\vert}$'
    s2 = r'${\vert \vec w\vert}={\vert \vec v\vert}{\rm cos}\ \theta$'
    s3 = r'$\vec u \cdot \vec v={\vert \vec u\vert}{\vert \vec w\vert}=$'\
         r'${\vert \vec u\vert}{\vert \vec v\vert}{\rm cos}\ \theta$'

    # string output
    ax.text(0.10, 0.75, s1)
    ax.text(0.10, 0.45, s2)
    ax.text(0.10, 0.15, s3)
    ax.text(0.5, -0.15, '(b)', fontsize=20)
    ax.set_axis_off()

    return


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

    # string setting
    s1_1 = r'$\vec v\cdot \vec u$'
    s1_2 = r'$=\begin{bmatrix}\sqrt2/2\quad \sqrt2/2\end{bmatrix}$'\
           r'$\begin{bmatrix}1\\0\end{bmatrix}$'
    s2 = r'$=\frac{\sqrt 2}{2}*1+\frac{\sqrt 2}{2}*0$'
    s3 = r'$=\frac{\sqrt 2}{2}$'

    # string output
    ax.text(0.10, 0.75, s1_1)
    ax.text(0.30, 0.75, s1_2)
    ax.text(0.30, 0.45, s2)
    ax.text(0.30, 0.15, s3)
    ax.text(0.5, -0.15, '(c)', fontsize=20)
    ax.set_axis_off()

    return


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

    # string setting
    s1_1 = r'$\vec u\cdot \vec v$'
    s1_2 = r'$=\begin{bmatrix}1\quad 0\end{bmatrix}$'\
           r'$\begin{bmatrix}\sqrt2/2\\\sqrt2/2\end{bmatrix}$'
    s2 = r'$=1*\frac{\sqrt 2}{2}+0*\frac{\sqrt 2}{2}$'
    s3 = r'$=\frac{\sqrt 2}{2}$'

    # string output
    ax.text(0.10, 0.75, s1_1)
    ax.text(0.30, 0.75, s1_2)
    ax.text(0.30, 0.45, s2)
    ax.text(0.30, 0.15, s3)
    ax.text(0.5, -0.15, '(d)', fontsize=20)
    ax.set_axis_off()

    return


mpl.rcParams['text.usetex'] = True
mpl.rcParams['text.latex.preamble'] = r'\usepackage{{amsmath}}'
mpl.rcParams['font.size'] = 40
fig = plt.figure(figsize=(16, 16))

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

# plt.savefig('/content/drive/My Drive/pqc/0014_001.png')
plt.show()

解說：

-----

References


[1] Dot Product -- from Wolfram MathWorld

https://mathworld.wolfram.com/DotProduct.html


[2] Engineering Math | ShareTechnote

https://www.sharetechnote.com/html/Handbook_EngMath_Matrix_InnerProduct.html


[3] Dot Product of Two Vectors – GeoGebra

https://www.geogebra.org/m/e95pnpwk


[4] Dot Product vs Cross Product - Difference & Similarities in Tabular Form - Physics In My View

https://physicsinmyview.com/2020/10/dot-product-vs-cross-product.html

-----

Python 量子運算（目錄）

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

-----

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標籤：

2023年2月8日 星期三

內惟藝術中心

 內惟藝術中心

2023/02/08

-----



-----

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