Author: 藍邦偉　老師

$α=%v$

$β=%v$

$O$

$H$

$B$

$A$

$M_2$

$L=%v$

單位向量

座標軸

Start

x

y

z

$ O $

$Step=%v$

$C$

$D$

$E_2$

$E_1$

$E_3$

$E_4$

$E$

$M_1$

$α$

$β$

$作正方形ABCD$

$分別以\overline{AB},\overline{BC},\overline{CD},\overline{DA}為邊,作正三角$
$形\Delta ABE_2,\Delta BCE_3, \Delta CDE_4, \Delta DAE_1$

$摺成一個正四角錐$

$是真的正四角錐$

$取\overline{AE}的中點M_1$

$連\overline{M_1D},\overline{M_1B}與\overline{BD}⇒\overline{M_1B}⊥\overline{AE},\overline{M_1D}⊥\overline{AE}$
$α=∠DM_1B為平面ABE與平面ADE的兩面角$

$\Delta BDM_1中,\overline{BD}=2\sqrt{2},\overline{BM_1}=\overline{DM_1}=\sqrt{3}$
$由餘弦定理知\;cosα=-\frac{1}{3}$

$過E作\overline{EH}垂直平面ABCD, H為垂足$

$取\overline{AB}之中點M_2$

$連\overline{HM_2},\overline{EM_2},此時$
$\overline{EM_2}⊥\overline{AB}且\overline{HM_2}⊥\overline{AB}$

$β=∠EM_2H為平面ABE與$
$平面ABCD的兩面角$

$此時\overline{HM_2}=1,\overline{EM_2}=\sqrt{3}⇒\overline{EH}=\sqrt{2}=\overline{AH}$
$sinβ=\frac{\sqrt{2}}{\sqrt{3}}=\frac{\sqrt{6}}{3}$

$正四角錐的性質$

$顯示文字說明$

t10 = 0 (slider from 0 to 1, step 0.01)
α = 0 rad (slider from 0 to 6.283185307179586, step 0.1*°)
β = 0 rad (slider from 0 to 6.283185307179586, step 0.1*°)
O = (13.76306914361355, 3.8855265737233293)
H: O
Y_{unit}: O + (-1 * sin(α), cos(α) * cos(β))
v = Vector(O, Y_{unit})
X_{unit}: O + (cos(α), sin(α) * cos(β))
u = Vector(O, X_{unit})
B: O - u + v
A: O + u + v
M_2 = Midpoint(A, B)
Z_{unit}: O + (0, sin(β))
w = Vector(O, Z_{unit})
L = 3 (slider from 1 to 10, step 1)
yy = Vector(O - L * v, O + L * v)
zz = Vector(O - L * w, O + L * w)
xx = Vector(O - L * u, O + L * u)
c = false
d = false
Q_2: 0.3 * Corner[2] + 0.7 * Corner[1]
Q_3: (x(Q_2), y(Corner[3]))
poly1_1, q_3, q_1, q_4, q_2 = Polygon(Corner[1], Q_2, Q_3, Corner[4])
n = 0 (slider from 0 to 12, step 1)
C: O - u - v
D: O + u - v
a_1 = Segment(A, B)
a_2 = Segment(C, B)
a_3 = Segment(C, D)
a_4 = Segment(A, D)
E_2: O + (1 + sqrt(3)) * v
E_1: O + (1 + sqrt(3)) * u
E_3: O - (1 + sqrt(3)) * u
E_4: O - (1 + sqrt(3)) * v
b_1 = Segment(A, E_1)
b_2 = Segment(D, E_1)
b_3 = Segment(A, E_2)
b_4 = Segment(B, E_2)
b_5 = Segment(B, E_3)
b_6 = Segment(C, E_3)
b_7 = Segment(C, E_4)
b_8 = Segment(D, E_4)
θ3 = 0 rad (slider from 0 to 125.26438968*°, step 0.1*°)
F_1: O + u + sqrt(3) * (cos(θ3) * u + sin(θ3) * w)
F_3: O - u + sqrt(3) * (-cos(θ3) * u + sin(θ3) * w)
F_2: O + v + sqrt(3) * (cos(θ3) * v + sin(θ3) * w)
F_4: O - v + sqrt(3) * (-cos(θ3) * v + sin(θ3) * w)
c_1 = Segment(A, F_2)
c_2 = Segment(B, F_2)
c_3 = Segment(B, F_3)
c_4 = Segment(C, F_3)
c_5 = Segment(C, F_4)
c_7 = Segment(D, F_1)
c_6 = Segment(A, F_1)
c_8 = Segment(D, F_4)
E: O + sqrt(2) * w
d_1, d_2 = SDLine(E, A, D, B)
d_3, d_4 = SDLine(E, B, A, C)
d_5, d_6 = SDLine(E, C, B, D)
d_7, d_8 = SDLine(E, D, C, A)
d_9, d_{10} = SDLine(A, B, C, E)
d_{11}, d_{12} = SDLine(B, C, D, E)
d_{13}, d_{14} = SDLine(C, D, A, E)
d_{15}, d_{16} = SDLine(D, A, B, E)
M_1 = Midpoint(A, E)
t6 = 0 (slider from 0 to 1, step 0.01)
G_1: M_1 + t6 * Vector[M_1, D]
G_2: D + t6 * Vector[D, B]
G_3: B + t6 * Vector[B, M_1]
e_1 = Segment(M_1, G_1)
e_2 = Segment(D, G_2)
e_3 = Segment(B, G_3)
γ = Angle(B, M_1, D)
Trig_1, z_b, z_a, z_c = Polygon(B, M_1, D)
t8 = 0 (slider from 0 to 1, step 0.01)
H_1: E + t8 * Vector[E, H]
f_1 = Segment(E, H_1)
H_2: H + t10 * Vector[H, M_2]
f_2 = Segment(H, H_2)
H_3: E + t10 * Vector[E, M_2]
H3 = 0
F = Point(Circle[E, H3])
f_3 = Segment(E, H_3)
Trig_2, z_d, z_e, z_f = Polygon(E, M_2, H)
δ = Angle(E, M_2, H)
a = true
poly1 = Poly_Y3(E, B, C)
poly2 = Poly_Y3(E, C, D)
poly3 = Poly_Y3(E, A, B)
poly4 = Poly_Y3(E, D, A)
Square1 = Poly_G4(A, D, C, B)

Tool: SDLine
Input: A, B, C, D
Output: g, h

b: sgn(sin(Angle[D, B, A]))
a: sgn(sin(Angle[A, B, C]))
c: sgn(a + b + 0.5)
d: sqrt(c)
e: sqrt(-c)
E = Midpoint(A, B)
A' = Dilate(A, d, E)
B' = Dilate(B, d, E)
g = Segment(A', B')
A'_1 = Dilate(A, e, E)
B'_1 = Dilate(B, e, E)
h = Segment(A'_1, B'_1)

Tool: Poly_B3
Input: A, B, C
Output: poly1'

α = Angle(C, B, A)
d: sqrt(sgn(sin(α)))
A' = Dilate(A, d, A)
B' = Dilate(B, d, A)
C' = Dilate(C, d, A)
poly1', c', a', b' = Polygon(A', B', C')

Tool: Poly_Y3
Input: A, B, C
Output: poly1

poly1 = Poly_B3(A, B, C)

Tool: Poly_G4
Input: A, B, C, D
Output: poly1'

α = Angle(C, B, A)
a: sqrt(sgn(sin(α)))
A' = Dilate(A, a, B)
B' = Dilate(B, a, B)
C' = Dilate(C, a, B)
D' = Dilate(D, a, B)
poly1', a', b', c', d' = Polygon(A', B', C', D')