正四面體的外接球與內切球
Author: 藍邦偉  老師

x

y

z

單位向量

座標軸

A

B

C

D

Equator

Meridian

外接球

內切球

正四面體

正四面體的外
接球與內切球

Reset

x

y

z

α = 4.363323129985824 rad (slider from 0 to 6.283185307179586, step 0.017)
β = 1.8325957145940461 rad (slider from 0 to 6.283185307179586, step 0.017)
O = (8.869707380618781, 4.0334440554002455)
X_{unit}: O + (cos(α), sin(α) * cos(β))
Y_{unit}: O + (-1 * sin(α), cos(α) * cos(β))
Z_{unit}: O + (0, sin(β))
u = Vector(O, X_{unit})
v = Vector(O, Y_{unit})
w = Vector(O, Z_{unit})
L = 5 (slider from 1 to 10, step 1)
a = Vector(O - L * v, O + L * v)
b = Vector(O - L * w, O + L * w)
z = Vector(O - L * u, O + L * u)
A_1: O + L * u
B_1: O + L * v
Z: O + L * w
c = false
d_1 = false
Q_2: 0.2 * Corner[2] + 0.8 * Corner[1]
Q_3: (x(Q_2), y(Corner[3]))
多邊形1, e, q_2, q_3, f = Polygon(Corner[1], Q_2, Q_3, Corner[4])
g = Segment(Q_2, Q_3)
r = 1 (slider from 0.5 to 1, step 0.01)
P_a: O + 3 * r * w
P_b: O + sqrt(2) * r * u - sqrt(6) * r * v - r * w
P_c: O + sqrt(2) * r * u + sqrt(6) * r * v - r * w
P_d: O - 2 * sqrt(2) * r * u - r * w
d = 0 (slider from 0 to 1.1, step 0.1)
O_1: O - d * Vector[O, P_d]
P_{a1}: P_a - d * Vector[O, P_d]
P_{b1}: P_b - d * Vector[O, P_d]
P_{c1}: P_c - d * Vector[O, P_d]
O_2: O - d * Vector[O, P_b]
P_{a_2}: P_a - d * Vector[O, P_b]
P_{a2}: P_a - d * Vector[O, P_b]
P_{c2}: P_c - d * Vector[O, P_b]
P_{d2}: P_d - d * Vector[O, P_b]
O_3: O - d * Vector[O, P_c]
P_{a3}: P_a - d * Vector[O, P_c]
P_{b3}: P_b - d * Vector[O, P_c]
P_{d3}: P_d - d * Vector[O, P_c]
O_4: O - d * Vector[O, P_a]
P_{b4}: P_b - d * Vector[O, P_a]
P_{c4}: P_c - d * Vector[O, P_a]
P_{d4}: P_d - d * Vector[O, P_a]
n_{41} = Segment(O_4, P_{b4})
n_{43} = Segment(O_4, P_{c4})
n_{42} = Segment(O_4, P_{d4})
n_{11} = Segment(O_1, P_{a1})
n_{13} = Segment(O_1, P_{b1})
n_{12} = Segment(O_1, P_{c1})
n_{21} = Segment(O_3, P_{a3})
n_{22} = Segment(O_3, P_{d3})
n_{23} = Segment(O_3, P_{b3})
n_1 = Segment(O_2, P_{a2})
n_23 = Segment(O_2, P_{c2})
n_2 = Segment(O_2, P_{d2})
多邊形2, p_{d4}, p_{b4}, p_{c4} = Polygon(P_{b4}, P_{c4}, P_{d4})
多邊形3, p_{c2}, p_{a2}, p_{d2} = Polygon(P_{a2}, P_{d2}, P_{c2})
多邊形4, p_{b1}, p_{a1}, p_{c1} = Polygon(P_{a1}, P_{c1}, P_{b1})
多邊形5, p_{b3}, p_{a3}, p_{d3} = Polygon(P_{a3}, P_{d3}, P_{b3})
p = Circle(O, 3 * r)
n = CurveCartesian(x(O + 3 * r * cos(θ) * u + 3 * r * sin(θ) * v), y(O + 3 * r * cos(θ) * u + 3 * r * sin(θ) * v), θ, 0, 6.28319)
o = CurveCartesian(x(O + 3 * r * cos(θ) * u + 3 * r * sin(θ) * w), y(O + 3 * r * cos(θ) * u + 3 * r * sin(θ) * w), θ, 0, 6.28319)
q_4 = CurveCartesian(x(O + 3 * r * cos(θ) * v + 3 * r * sin(θ) * w), y(O + 3 * r * cos(θ) * v + 3 * r * sin(θ) * w), θ, 0, 6.28319)
s = false
t = false
c_1 = Circle(O, r)
a_1 = false
b_1 = false
e_1 = CurveCartesian(x(O + r * cos(θ) * u + r * sin(θ) * v), y(O + r * cos(θ) * u + r * sin(θ) * v), θ, 0, 6.28319)
q = false
G_4: (P_{b4} + P_{c4} + P_{d4}) / 3
g_1 = Segment(O_4, G_4)
G_1: (P_{b1} + P_{c1} + P_{a1}) / 3
h_1 = Segment(G_1, O_1)
G_2: (P_{d2} + P_{c2} + P_{a2}) / 3
i_1 = Segment(O_2, G_2)
G_3: (P_{d3} + P_{b3} + P_{a3}) / 3
j_1 = Segment(G_3, O_3)
h, i = SDLine(P_a, P_d, P_b, P_c)
j, k = SDLine(P_a, P_b, P_c, P_d)
l, m = SDLine(P_a, P_c, P_d, P_b)
k_1, l_1 = SDLine(P_b, P_c, P_a, P_d)
m_1, p_1 = SDLine(P_c, P_d, P_a, P_b)
q_1, r_1 = SDLine(P_d, P_b, P_a, P_c)

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)