金字塔的內切球半徑
Author: 藍邦偉  老師

將金字塔切成四個全等的正三角錐與一個正立方錐，利用體積和，可求內切球半徑。

x

y

z

顯示xy平面

顯示單位向量

顯示座標軸

I

d = 0 (slider from 0 to 3, step 1)
O = (-1.4435335103035256, -2.0353796094361623)
A: (x((Corner[3.0] + 39.0 * Corner[4.0]) / 40.0), y((Corner[1.0] + 9.0 * Corner[4.0]) / 10.0))
B: (x((4.0 * Corner[3.0] + 16.0 * Corner[4.0]) / 20.0), y((Corner[1.0] + 9.0 * Corner[4.0]) / 10.0))
D: (3.0 * B + A) / 4.0
C: (3.0 * A + B) / 4.0
a = Segment(A, B)
E = Point(a)
E' = Rotate(Translate[E, Vector[C, D]], 90.0*°, D)
c = Circle(C, E)
c' = Translate(c, Vector[Vector[C, D]])
G = Point(c')
c'' = Rotate(c', 270.0*°, C)
F = Point(c'')
A' = Rotate(A, 270.0*°, C)
B' = Rotate(B, 270.0*°, C)
a' = Segment(A', B')
D' = Rotate(D, 270.0*°, C)
b = Segment(D', F)
E'' = Rotate(E', 270.0*°, C)
α = Angle(E'', D', F)
d_1 = Segment(D, G)
β = Angle(G, D, E')
K: C + (0.0, b * cos(β))
K' = Translate(K, Vector[Vector[C, O]])
w = Vector(O, K')
I: C + (b * cos(α), b * sin(α) * sin(β))
J: C + (b * cos(α + 90.0°), b * sin(α + 90.0°) * sin(β))
e = Segment(C, K)
f = Segment(C, I)
g = Segment(C, J)
I' = Translate(I, Vector[Vector[C, O]])
J' = Translate(J, Vector[Vector[C, O]])
L = 8 (slider from 1 to 10, step 1)
I'' = Dilate(I', L, O)
J'' = Dilate(J', L, O)
K'' = Dilate(K', L, O)
I''' = Dilate(I'', -1.0, O)
J''' = Dilate(J'', -1.0, O)
K''' = Dilate(K'', -1.0, O)
u' = Vector(I''', I'')
v' = Vector(J''', J'')
w' = Vector(K''', K'')
h = Segment(D', E'')
i = Segment(D, E')
軌跡1 = Locus(I, F)
L_{axis}: L
j = false
u = Vector(O, I')
v = Vector(O, J')
k = false
list1 = Sequence(CurveCartesian[x(O + t * u + s * v), y(O + t * u + s * v), t, -(L), L], s, -(L), L, 1.0)
list2 = Sequence(CurveCartesian[x(O + t * u + s * v), y(O + t * u + s * v), s, -(L), L], t, -(L), L, 1.0)
l = false
A_1: O + 4.0 * u + 4.0 * v - 2.0 * d * w
A_2: O + 4.0 * u - 4.0 * v - 2.0 * d * w
A_3: O - 4.0 * u - 4.0 * v - 2.0 * d * w
A_4: O - 4.0 * u + 4.0 * v - 2.0 * d * w
m = Segment(A_2, A_1)
n = Segment(A_1, A_4)
p = Segment(A_4, A_3)
q = Segment(A_3, A_2)
A_0: O + (sqrt(6.0) + sqrt(2.0)) / 2.0 * w - 2.0 * d * w
b_1 = Segment(A_0, A_1)
c_1 = Segment(A_0, A_4)
e_1 = Segment(A_0, A_3)
f_1 = Segment(A_0, A_2)
A_{11}: O + 4.0 * u + 4.0 * v + 2.0 * d * u
A_{21}: O + 4.0 * u - 4.0 * v + 2.0 * d * u
A_{51}: O + 4.0 * sqrt(2.0) * w + 2.0 * d * u
A_{02}: O + (sqrt(6.0) + sqrt(2.0)) / 2.0 * w + 2.0 * d * u
r = Segment(A_{51}, A_{21})
s = Segment(A_{21}, A_{11})
t = Segment(A_{11}, A_{51})
a_1 = Segment(A_{02}, A_{51})
g_1 = Segment(A_{02}, A_{21})
h_1 = Segment(A_{02}, A_{11})
A_{03}: O + (sqrt(6.0) + sqrt(2.0)) / 2.0 * w + 2.0 * d * v
A_{12}: O + 4.0 * u + 4.0 * v + 2.0 * d * v
A_{42}: O - 4.0 * u + 4.0 * v + 2.0 * d * v
A_{52}: O + 4.0 * sqrt(2.0) * w + 2.0 * d * v
i_1 = Segment(A_{52}, A_{12})
j_1 = Segment(A_{12}, A_{42})
k_1 = Segment(A_{42}, A_{52})
l_1 = Segment(A_{03}, A_{12})
m_1 = Segment(A_{03}, A_{42})
n_1 = Segment(A_{03}, A_{52})
A_{04}: O + (sqrt(6.0) + sqrt(2.0)) / 2.0 * w - 2.0 * d * u
A_{01}: O + (sqrt(6.0) + sqrt(2.0)) / 2.0 * w - 2.0 * d * v
A_{53}: O + 4.0 * sqrt(2.0) * w - 2.0 * d * u
A_5: O + 4.0 * sqrt(2.0) * w - 2.0 * d * v
A_{43}: O - 4.0 * u + 4.0 * v - 2.0 * d * u
A_{33}: O - 4.0 * u - 4.0 * v - 2.0 * d * u
p_1 = Segment(A_{53}, A_{43})
q_1 = Segment(A_{43}, A_{33})
r_1 = Segment(A_{33}, A_{53})
s_1 = Segment(A_{53}, A_{04})
t_1 = Segment(A_{04}, A_{33})
a_2 = Segment(A_{43}, A_{04})
A_{22}: O + 4.0 * u - 4.0 * v - 2.0 * d * v
A_{35}: O - 4.0 * u - 4.0 * v - 2.0 * d * v
b_2 = Segment(A_5, A_{35})
c_2 = Segment(A_{35}, A_{22})
d_2 = Segment(A_{22}, A_5)
e_2 = Segment(A_5, A_{01})
f_2 = Segment(A_{01}, A_{35})
g_2 = Segment(A_{01}, A_{22})
多邊形1, a_3, a_4, a_5, a_6 = Polygon(A_2, A_3, A_4, A_1)
多邊形2, a_{35}, a_7, a_{22} = Polygon(A_5, A_{22}, A_{35})
多邊形3, a_{43}, a_{53}, a_{33} = Polygon(A_{53}, A_{33}, A_{43})
多邊形4, a_{12}, a_{52}, a_{42} = Polygon(A_{52}, A_{42}, A_{12})
多邊形5, a_{21}, a_{51}, a_{11} = Polygon(A_{51}, A_{11}, A_{21})
O_{G2}: 1.0 / 3.0 * (A_5 + A_{22} + A_{35})
O_{G3}: 1.0 / 3.0 * (A_{51} + A_{21} + A_{11})
O_{G4}: 1.0 / 3.0 * (A_{53} + A_{33} + A_{43})
O_{G5}: 1.0 / 3.0 * (A_{12} + A_{42} + A_{52})
j_2 = Segment(A_{02}, O_{G3})
k_2 = Segment(A_{04}, O_{G4})
i_2 = Segment(O_{G5}, A_{03})
l_2 = Segment(A_{01}, O_{G2})
O_{G1}: 0.5 * (A_1 + A_3)
h_2 = Segment(A_0, O_{G1})