三元一次方程組的幾何意義
Author: 藍邦偉  老師

x

y

z

顯示座標軸

.

顯示xy平面

Δ≠0，三平面交於一點，恰一解

三元一次方程組的幾何意義

Δ=0, \Delta_x,Δ_y,Δ_z不全為0,無解,三平面兩兩相交於一直線，但三線不共點

Δ＝Δ_x=Δ_y=Δ_z=0,無限多解,三平面兩兩不重合，但交於一直線

Δ＝Δ_x=Δ_y=Δ_z=0，無限多解,
其中兩平面重合，與另一平面交於一直線

Δ=0,Δ_x,Δ_y,Δ_z不全為0,無解，三平面平行

Δ=Δ_x=Δ_y=Δ_z=0,無解,其中兩平面重合,與另一平面平行

Δ=0,Δ_x,Δ_y,Δ_z不全為0，
無解，其中兩平面平行，
另一平面與其交於兩平行線

Δ=Δ_x=Δ_y=Δ_z=0,無限多解,三平面重合

Reset

Start

α = 0 rad (slider from 0 to 6.283185307179586, step 0.1*°)
β = 4.537856055185257 rad (slider from 0 to 6.283185307179586, step 0.017453292519943295)
L = 1 (slider from 1 to 10, step 1)
O = (83.37451233732779, 11.141869284717645)
X_{unit}: O + (cos(α), sin(α) * cos(β))
Y_{unit}: O + (-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})
a = Vector(O - L * u, O + L * u)
b = Vector(O - L * v, O + L * v)
c = Vector(O - L * w, O + L * w)
多邊形1, d, e, f, g = Polygon(O + L * u + L * v, O + L * u - L * v, O - L * u - L * v, O - L * u + L * v)
X_{x}: O + L * u
Y_{y}: O + L * v
Z_{z}: O + L * w
h = false
i = false
n = 1 (slider from 1 to 8, step 1)
A_4: O + 4 * u - 4 * v
A_3: O - 4 * u - 4 * v
A_2: O - 4 * u + 4 * v
A_1: O + 4 * u + 4 * v
多邊形2, a_1, a_2, a_3, a_4 = Polygon(A_1, A_2, A_3, A_4)
B_1: O + 4 * u + 4 * v + 4 * w
B_2: O - 4 * u - 4 * v + 4 * w
B_4: O + 4 * u + 4 * v - 4 * w
B_3: O - 4 * u - 4 * v - 4 * w
多邊形3, b_1, b_2, b_3, b_4 = Polygon(B_1, B_2, B_3, B_4)
C_1: O + 4 * u - 4 * v + 4 * w
C_2: O + 4 * u - 4 * v - 4 * w
C_3: O - 4 * u + 4 * v - 4 * w
C_4: O - 4 * u + 4 * v + 4 * w
多邊形4, c_1, c_2, c_3, c_4 = Polygon(C_1, C_2, C_3, C_4)
j = Segment(A_2, A_4)
k = Segment(A_1, A_3)
A = Intersect(b_1, c_4)
B = Intersect(b_3, c_2)
l = Segment(A, B)
C = Intersect(j, k)
D_1: 0.2 * A_1 + 0.8 * A_4
D_2: 0.2 * A_2 + 0.8 * A_3
E_1: 0.2 * A_4 + 0.8 * A_1
E_2: 0.2 * A_3 + 0.8 * A_2
F_1: O + 4 * u + 3 * w
F_2: O - 4 * u + 3 * w
D = Dilate(D_1, 1.5, F_1)
E = Dilate(F_1, 1.5, D_1)
F = Dilate(F_2, 1.5, D_2)
G = Dilate(D_2, 1.5, F_2)
多邊形5, e_1, d_1, g_1, f_1 = Polygon(E, D, G, F)
m = Segment(D_1, D_2)
p = Segment(F_1, F_2)
H = Dilate(F_1, 1.5, E_1)
I = Dilate(F_2, 1.5, E_2)
J = Dilate(E_1, 1.5, F_1)
K = Dilate(E_2, 1.5, F_2)
多邊形6, k_1, i_1, h_1, j_1 = Polygon(K, I, H, J)
q = Segment(E_1, E_2)
M = Midpoint(A_4, A_1)
N = Midpoint(A_3, A_2)
r = Segment(M, N)
K_1: E_1 + 3 * w
K_2: E_2 + 3 * w
K_3: D_1 - 3 * w
K_4: D_2 - 3 * w
多邊形7, k_2, k_3, k_4, k_5 = Polygon(K_1, K_2, K_4, K_3)
H_1: D_1 + 3 * w
H_2: D_2 + 3 * w
H_3: E_2 - 3 * w
H_4: E_1 - 3 * w
多邊形8, h_2, h_3, h_4, h_5 = Polygon(H_1, H_2, H_3, H_4)
G_1: O + 2.5 * u + 2.5 * v
G_2: O - 2.5 * u + 2.5 * v
G_3: O - 2.5 * u - 2.5 * v
G_4: O + 2.5 * u - 2.5 * v
多邊形9, g_2, g_3, g_4, g_5 = Polygon(G_1, G_2, G_3, G_4)
M_1: A_1 + 2.5 * w
M_2: A_2 + 2.5 * w
M_3: A_3 + 2.5 * w
M_4: A_4 + 2.5 * w
多邊形10, m_1, m_2, m_3, m_4 = Polygon(M_1, M_2, M_3, M_4)
N_1: A_1 - 2.5 * w
N_2: A_2 - 2.5 * w
N_3: A_3 - 2.5 * w
N_4: A_4 - 2.5 * w
多邊形11, n_1, n_2, n_3, n_4 = Polygon(N_1, N_2, N_3, N_4)
Q_1: 0.8 * M_1 + 0.2 * M_4
Q_2: 0.8 * M_2 + 0.2 * M_3
P = Dilate(Q_1, 1.4, D_1)
Q = Dilate(D_2, 1.4, Q_2)
R = Dilate(D_1, 1.4, Q_1)
s = Segment(Q_1, Q_2)
t = Segment(D_1, D_2)
S = Dilate(Q_2, 1.4, D_2)
多邊形12, s_1, p_1, r_1, q_1 = Polygon(S, P, R, Q)
M_5: M_1 + 2 * v
M_6: M_2 + 2 * v
M_7: M_3 + 2 * v
M_8: M_4 + 2 * v
多邊形13, m_5, m_6, m_7, m_8 = Polygon(M_5, M_6, M_7, M_8)
R_1: O + 5.5 * u + 5.5 * v
R_2: O - 5.5 * u + 5.5 * v
R_3: O - 5.5 * u - 5.5 * v
R_4: O + 5.5 * u - 5.5 * v
多邊形14, r_2, r_3, r_4, r_5 = Polygon(R_1, R_2, R_3, R_4)
T = (69.65820100215001, 7.041724606030104)