無理數的作圖(4)
Author: 藍邦偉老師

$A$

$ 無理數的作圖 $

1

$ 顯示控制滑竿 $

A_1 = (1.5650623885918007, -1.977963176064444)
c_1 = Circle(A_1, 1)
A = Point(c_1)
A_2 = Rotate(A, 90*°, A_1)
a_a = Segment(A, A_1)
a_b = Segment(A_1, A_2)
a_c = Segment(A, A_2)
A_3 = Rotate(A_1, 135*°, A_2)
a_d = Segment(A_3, A_2)
a_e = Segment(A_3, A)
L_4 = OrthogonalLine(A_3, a_e)
c_2 = Circle(A_3, A_2)
A_4 = Intersect(c_2, L_4, 2)
a_f = Segment(A_3, A_4)
a_g = Segment(A, A_4)
A_5 = Intersect(c_2, a_g, 2)
L_7 = OrthogonalLine(A_4, a_g)
c_3 = Circle(A_4, A_3)
A_6 = Intersect(c_3, L_7, 1)
a_h = Segment(A_6, A_4)
a_i = Segment(A_6, A)
L_a = OrthogonalLine(A_6, a_i)
c_4 = Circle(A_6, A_4)
A_7 = Intersect(c_4, L_a, 2)
a_k = Segment(A_7, A)
L_c = OrthogonalLine(A_7, a_k)
c_5 = Circle(A_7, A_6)
A_8 = Intersect(c_5, L_c, 2)
a_j = Segment(A_6, A_7)
a_l = Segment(A_7, A_8)
a_m = Segment(A_8, A)
L_1 = OrthogonalLine(A_8, a_m)
c_6 = Circle(A_8, A_7)
A_9 = Intersect(c_6, L_1, 2)
a_n = Segment(A_8, A_9)
a_o = Segment(A_9, A)
L_5 = OrthogonalLine(A_9, a_o)
c_7 = Circle(A_9, A_8)
L_8 = OrthogonalLine(A_9, a_o)
A_{10} = Intersect(c_7, L_8, 2)
a_p = Segment(A_{10}, A_9)
a_q = Segment(A_{10}, A)
c_8 = Circle(A_{10}, A_9)
A_{11} = Intersect(c_8, a_q, 1)
c_9 = Circle(A_{11}, A_{10})
A_{12} = Intersect(c_9, a_q, 2)
L_9 = OrthogonalLine(A_{10}, a_q)
c_{10} = Circle(A_{10}, A_{11})
A_{13} = Intersect(c_{10}, L_9, 2)
a_r = Segment(A_{10}, A_{13})
a_s = Segment(A_{13}, A)
L_2 = OrthogonalLine(A_{13}, a_s)
c_{11} = Circle(A_{13}, A_{10})
A_{14} = Intersect(c_{11}, L_2, 2)
a_t = Segment(A_{13}, A_{14})
a_u = Segment(A, A_{14})
L_b = OrthogonalLine(A_{14}, a_u)
c_{12} = Circle(A_{14}, A_{13})
A_{15} = Intersect(c_{12}, L_b, 1)
a_v = Segment(A_{14}, A_{15})
a_w = Segment(A_{15}, A)
L_d = OrthogonalLine(A_{15}, a_w)
c_{13} = Circle(A_{15}, A_{14})
A_{16} = Intersect(c_{13}, L_d, 2)
a_x = Segment(A_{15}, A_{16})
a_y = Segment(A_{16}, A)
L_e = OrthogonalLine(A_{16}, a_y)
c_{14} = Circle(A_{16}, A_{15})
A_{17} = Intersect(c_{14}, L_e, 2)
a_z = Segment(A_{16}, A_{17})
a_0 = Segment(A_{17}, A)
L_3 = OrthogonalLine(A_{17}, a_0)
c_{15} = Circle(A_{17}, A_{16})
A_{18} = Intersect(c_{15}, L_3, 2)
a_2 = Segment(A_{17}, A_{18})
a_3 = Segment(A_{18}, A)
L_6 = OrthogonalLine(A_{18}, a_3)
c_{16} = Circle(A_{18}, A_{17})
A_{19} = Intersect(c_{16}, L_6, 2)
a_1 = Segment(A, A_{19})
a_4 = Segment(A_{19}, A_{18})
c_{20} = Circle(A_{19}, 1)
A_{21} = Midpoint(A, A_{19})
A_{20} = Midpoint(A_{21}, A_{19})
A_{22} = Midpoint(A_{21}, A)
n = 0 (slider from -5 to 5, step 0.1)
t = 0 (slider from 0 to 40, step 0.01)
A1 = 0
A2 = 1
A3: 2 * A2 - A1
A4: 2 * A3 - A2
A5: 2 * A4 - A3
A6: 2 * A5 - A4
A7: 2 * A6 - A5
A8: 2 * A7 - A6
A9: 2 * A8 - A7
A10: 2 * A9 - A8
A11: 2 * A10 - A9
A12: 2 * A11 - A10
A13: 2 * A12 - A11
A14: 2 * A13 - A12
A15: 2 * A14 - A13
A16: 2 * A15 - A14
A17: 2 * A16 - A15
A18: 2 * A17 - A16
A19: 2 * A18 - A17
A20: 2 * A19 - A18
A21: 2 * A20 - A19
A22: 2 * A21 - A20
A23: 2 * A22 - A21
A24: 2 * A23 - A22
A25: 2 * A24 - A23
A26: 2 * A25 - A24
A27: 2 * A26 - A25
A28: 2 * A27 - A26
A29: 2 * A28 - A27
A30: 2 * A29 - A28
A31: 2 * A30 - A29
A32: 2 * A31 - A30
A33: 2 * A32 - A31
f(x, n) = (abs(x - n) - abs(x - n - 1) + 1) / 2
B1: f(t, A1)
B2: f(t, A2)
B3: f(t, A3)
B4: f(t, A4)
B5: f(t, A5)
B6: f(t, A6)
B7: f(t, A7)
B8: f(t, A8)
B9: f(t, A9)
B10: f(t, A10)
B11: f(t, A11)
B12: f(t, A12)
B13: f(t, A13)
B14: f(t, A14)
B15: f(t, A15)
B16: f(t, A16)
B17: f(t, A17)
B18: f(t, A18)
B19: f(t, A19)
B20: f(t, A20)
B21: f(t, A21)
B22: f(t, A22)
B23: f(t, A23)
B24: f(t, A24)
B25: f(t, A25)
B26: f(t, A26)
B27: f(t, A27)
B28: f(t, A28)
B29: f(t, A29)
B30: f(t, A30)
B31: f(t, A31)
B32: f(t, A32)
B33: f(t, A33)
P_1: A_1 + B1 * Vector[A_1, A]
b_0 = Segment(A_1, P_1)
P_2: A_1 + B2 * Vector[A_1, A_2]
b_1 = Segment(A_1, P_2)
P_3: A_2 + B3 * Vector[A_2, A]
b_2 = Segment(A_2, P_3)
P_5: A_3 + B5 * Vector[A_3, A]
b_4 = Segment(A_3, P_5)
P_7: A_4 + B7 * Vector[A_4, A]
b_5 = Segment(A_4, P_7)
P_{11}: A_6 + B11 * Vector[A_6, A]
b_6 = Segment(A_6, P_{11})
P_{13}: A_7 + B13 * Vector[A_7, A]
b_7 = Segment(A_7, P_{13})
P_{15}: A_8 + B15 * Vector[A_8, A]
b_8 = Segment(A_8, P_{15})
P_{17}: A_9 + B17 * Vector[A_9, A]
b_9 = Segment(A_9, P_{17})
P_{19}: A_{10} + B19 * Vector[A_{10}, A]
b_a = Segment(A_{10}, P_{19})
P_{24}: A_{13} + B24 * Vector[A_{13}, A]
b_b = Segment(A_{13}, P_{24})
P_{26}: A_{14} + B26 * Vector[A_{14}, A]
b_c = Segment(A_{14}, P_{26})
P_{28}: A_{15} + B28 * Vector[A_{15}, A]
b_d = Segment(P_{28}, A_{15})
P_{30}: A_{16} + B30 * Vector[A_{16}, A]
b_e = Segment(P_{30}, A_{16})
P_{32}: A_{17} + B32 * Vector[A_{17}, A]
b_f = Segment(P_{32}, A_{17})
A34: 2 * A33 - A32
A35: 2 * A34 - A33
A36: 2 * A35 - A34
A37: 2 * A36 - A35
A38: 2 * A37 - A36
A39: 2 * A38 - A37
B34: f(t, A34)
B35: f(t, A35)
B36: f(t, A36)
B37: f(t, A37)
B38: f(t, A38)
B39: f(t, A39)
P_{34}: A_{18} + B34 * Vector[A_{18}, A]
b_g = Segment(P_{34}, A_{18})
P_{36}: A_{19} + B36 * Vector[A_{19}, A]
b_3 = Segment(P_{36}, A_{19})
A40: 2 * A39 - A38
B40: f(t, A40)
P_4 = Rotate(A_1, B4 * Angle[A_1, A_2, A_3], A_2)
b_h = Segment(A_2, P_4)
P_6 = Rotate(A_2, B6 * Angle[A_2, A_3, A_4], A_3)
b_i = Segment(A_3, P_6)
P_8 = Rotate(A_3, B8 * Angle[A_3, A_4, A], A_4)
b_j = Segment(A_4, P_8)
P_9 = Rotate(A_4, B9 * Angle[A, 0.5 * (A_4 + A), A_4], P_8)
b_k = Segment(P_8, P_9)
P_{10} = Rotate(A_3, B10 * Angle[A_3, A_4, A_6], A_4)
b_l = Segment(A_4, P_{10})
P_{12} = Rotate(A_4, B12 * Angle[A_4, A_6, A_7], A_6)
b_m = Segment(P_{12}, A_6)
P_{14} = Rotate(A_6, B14 * Angle[A_6, A_7, A_8], A_7)
b_n = Segment(A_7, P_{14})
P_{16} = Rotate(A_7, B16 * Angle[A_7, A_8, A_9], A_8)
b_o = Segment(A_8, P_{16})
P_{18} = Rotate(A_8, B18 * Angle[A_8, A_9, A_{10}], A_9)
b_p = Segment(A_9, P_{18})
P_{20} = Rotate(A_9, B20 * Angle[A_9, A_{10}, A], A_{10})
b_q = Segment(A_{10}, P_{20})
P_{21} = Rotate(A_{10}, B21 * Angle[A_{10}, A_{11}, A_{12}], A_{11})
b_r = Segment(A_{11}, P_{21})
P_{22} = Rotate(A_{11}, B22 * Angle[A_{11}, A_{12}, A], A_{12})
b_s = Segment(A_{12}, P_{22})
P_{23} = Rotate(A_9, B23 * Angle[A_9, A_{10}, A_{13}], A_{10})
b_t = Segment(A_{10}, P_{23})
P_{25} = Rotate(A_{10}, B25 * Angle[A_{10}, A_{13}, A_{14}], A_{13})
b_u = Segment(A_{13}, P_{25})
P_{27} = Rotate(A_{13}, B27 * Angle[A_{13}, A_{14}, A_{15}], A_{14})
b_v = Segment(A_{14}, P_{27})
P_{29} = Rotate(A_{14}, B29 * Angle[A_{14}, A_{15}, A_{16}], A_{15})
b_w = Segment(A_{15}, P_{29})
P_{31} = Rotate(A_{15}, B31 * Angle[A_{15}, A_{16}, A_{17}], A_{16})
b_x = Segment(P_{31}, A_{16})
P_{33} = Rotate(A_{16}, B33 * Angle[A_{16}, A_{17}, A_{18}], A_{17})
b_y = Segment(A_{17}, P_{33})
P_{35} = Rotate(A_{17}, B35 * Angle[A_{17}, A_{18}, A_{19}], A_{18})
b_z = Segment(P_{35}, A_{18})
P_{37} = Rotate(A_{18}, B37 * Angle[A_{18}, A_{19}, A_{20}], A_{19})
d_0 = Segment(P_{37}, A_{19})
P_{38} = Rotate(A_{19}, B38 * Angle[A_{19}, A_{20}, A_{21}], A_{20})
d_1 = Segment(P_{38}, A_{20})
P_{39} = Rotate(A_{20}, B39 * Angle[A_{20}, A_{21}, A_{22}], A_{21})
d_2 = Segment(P_{39}, A_{21})
A41: 2 * A40 - A39
B41: f(t, A41)
P_{40} = Rotate(A_{21}, B40 * Angle[A_{21}, A_{22}, A], A_{22})
d_3 = Segment(A_{22}, P_{40})
α = Angle(A, A_1, A_2)
β = Angle(A, A_2, A_3)
γ = Angle(A, A_3, A_4)
δ = Angle(A, A_4, A_6)
ε = Angle(A, A_6, A_7)
ζ = Angle(A, A_7, A_8)
η = Angle(A, A_8, A_9)
θ = Angle(A, A_9, A_{10})
ι = Angle(A, A_{10}, A_{13})
κ = Angle(A, A_{13}, A_{14})
λ = Angle(A, A_{14}, A_{15})
μ = Angle(A, A_{15}, A_{16})
ν = Angle(A, A_{16}, A_{17})
ξ = Angle(A, A_{17}, A_{18})
ο = Angle(A, A_{18}, A_{19})
text1: "$ " + (LaTeX[If[t ≥ 34, "\sqrt{16}=4", If[(t ≥ 32) ∧ (t < 34), "\sqrt{15}", If[(t ≥ 30) ∧ (t < 32), "\sqrt{14}", If[(t ≥ 28) ∧ (t < 30), "\sqrt{13}", If[(t ≥ 26) ∧ (t < 28), "\sqrt{12}=2\sqrt{3}", If[(t ≥ 24) ∧ (t < 26), "\sqrt{11}", If[(t ≥ 22) ∧ (t < 24), "\sqrt{10}", If[(t ≥ 17) ∧ (t < 22), "\sqrt{9}=3", If[(t ≥ 15) ∧ (t < 17), "\sqrt{8}=2\sqrt{2}", If[(t ≥ 13) ∧ (t < 15), "\sqrt{7}", If[(t ≥ 11) ∧ (t < 13), "\sqrt{6}", If[(t ≥ 9) ∧ (t < 11), "\sqrt{5}", If[(t ≥ 5) ∧ (t < 9), "\sqrt{4}=2", If[(t ≥ 3) ∧ (t < 5), "\sqrt{3}", If[t < 3, "\sqrt{2}"]]]]]]]]]]]]]]]]) + " $"
a = true