평면 방정식

n = (a, b, c)

X - X_0 = (x - x_0, y - y_0, z - z_0)

0 = <n, X - X_0>

0 = a (x - x_0) + b (y - y_0) + c (z - z_0)

0 = a x + b y + c z - a x_0 - b y_0 - c z_0

0 = a x + b y + c z + d

세점으로 부터 평면 구하기

Overscript[n,^] = (a, b, c, d)

Overscript[X,^] _0 = (x_0, y_0, z_0, 1)

Overscript[X,^] _1 = (x_1, y_1, z_1, 1)

Overscript[X,^] _2 = (x_2, y_2, z_2, 1)

a x_0 + b y_0 + c z_0 + d·1 = 0 ⇒ <Overscript[n,^], Overscript[X,^] _0> = 0

a x_1 + b y_1 + c z_1 + d·1 = 0 ⇒ <Overscript[n,^], Overscript[X,^] _1> = 0

a x_2 + b y_2 + c z_2 + d·1 = 0 ⇒ <Overscript[n,^], Overscript[X,^] _2> = 0

Overscript[n,^] = Overscript[X,^] _0∧Overscript[X,^] _1∧Overscript[X,^] _2

Wedge Product

Overscript[n,^] = Overscript[X,^] _0∧Overscript[X,^] _1∧Overscript[X,^] _2 = ({{e_1, e_2, e_3, e_4}, {x_0, y_0, z_0, 1}, {x_1, y_1, z_1, 1}, {x_2, y_2, z_2, 1}})

세평면으로 부터 교점 구하기

Overscript[X,^] = (x, y, z, w)

Overscript[n,^] _0 = (a_0, b_0, c_0, d_0)

Overscript[n,^] _1 = (a_1, b_1, c_1, d_1)

Overscript[n,^] _2 = (a_2, b_2, c_2, d_2)

Overscript[X,^] = Overscript[n,^] _0∧Overscript[n,^] _1∧Overscript[n,^] _2