Author: 藍邦偉　老師

單位向量

座標軸

x

y

z

空間中的面

積投影公式

A(

B(

C(

1

2

3

x1

x2

x3

y1

y2

y3

z1

z2

z3

,

$ , $

$ ) $

,

,

)

,

,

)

O = (11.310158479473479, 1.6199708204419094)
β = 1.3962634015954636 rad (slider from 0 to 6.283185307179586, step 0.017453292519943295)
Z_{unit}: O + (0, sin(β))
w = Vector(O, Z_{unit})
α = 4.276056667386108 rad (slider from 180*° to 270*°, step 0.01*°)
X_{unit}: O + (cos(α), sin(α) * cos(β))
u = Vector(O, X_{unit})
Y_{unit}: O + (-1 * sin(α), cos(α) * cos(β))
v = Vector(O, Y_{unit})
L = 7 (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)
c = false
d = true
Q_2: 0.2 * Corner[2] + 0.8 * Corner[1]
Q_3: (x(Q_2), y(Corner[3]))
g = Segment(Q_2, Q_3)
poly1, h, q_1, q_4, i = Polygon(Corner[1], Q_2, Q_3, Corner[4])
x_1 = 2 (slider from 0 to 5, step 0.1)
x_2 = 3 (slider from 0 to 5, step 0.1)
x_3 = 4 (slider from 0 to 5, step 0.1)
y_1 = 4 (slider from 0 to 5, step 0.1)
y_2 = 1 (slider from 0 to 5, step 0.1)
y_3 = 5 (slider from 0 to 5, step 0.1)
z_1 = 5 (slider from 0 to 5, step 0.1)
z_2 = 3 (slider from 0 to 5, step 0.1)
A: O + x_1 * u + y_1 * v + z_1 * w
B: O + x_2 * u + y_2 * v + z_2 * w
z_3 = 4 (slider from 0 to 5, step 0.1)
C: O + x_3 * u + y_3 * v + z_3 * w
Z_3 = 3
A_1: O + x_1 * u + y_1 * v
B_1: O + x_2 * u + y_2 * v
C_1: O + x_3 * u + y_3 * v
A_2: O + y_1 * v + z_1 * w
B_2: O + y_2 * v + z_2 * w
C_2: O + y_3 * v + z_3 * w
A_3: O + x_1 * u + z_1 * w
B_3: O + x_2 * u + z_2 * w
C_3: O + x_3 * u + z_3 * w
Tri, c_1, a_1, b_1 = Polygon(B, A, C)
Tri_1, c_2, a_2, b_2 = Polygon(A_1, B_1, C_1)
Tri_2, c_3, a_3, b_3 = Polygon(A_2, B_2, C_2)
Tri_3, c_4, a_4, b_4 = Polygon(A_3, B_3, C_3)
e = false
f = false
j = false
l_1 = Segment(A, A_1)
l_2 = Segment(B, B_1)
l_3 = Segment(C, C_1)
l_4 = Segment(A, A_2)
l_5 = Segment(B, B_2)
l_6 = Segment(C, C_2)
l_7 = Segment(A, A_3)
l_8 = Segment(B, B_3)
l_9 = Segment(C, C_3)
Squ1, o, k, l, m = Polygon(O, O + (L - 1) * u, O + (L - 1) * u + (L - 1) * v, O + (L - 1) * v)
Squ2, o_1, n, p, q = Polygon(O, O + (L - 1) * v, O + (L - 1) * v + (L - 1) * w, O + (L - 1) * w)
Squ3, o_2, r, s, t = Polygon(O, O + (L - 1) * w, O + (L - 1) * w + (L - 1) * u, O + (L - 1) * u)
inputBox1 = Textfield(x_1)
inputBox2 = Textfield(x_2)
inputBox3 = Textfield(x_3)
inputBox4 = Textfield(y_1)
inputBox5 = Textfield(y_2)
inputBox6 = Textfield(y_3)
inputBox7 = Textfield(z_1)
inputBox8 = Textfield(z_2)
inputBox9 = Textfield(z_3)
T1: 0.5 * abs(Determinant[{{x_1, y_1, 1}, {x_2, y_2, 1}, {x_3, y_3, 1}}])
T2: 0.5 * abs(Determinant[{{y_1, z_1, 1}, {y_2, z_2, 1}, {y_3, z_3, 1}}])
T3: 0.5 * abs(Determinant[{{z_1, x_1, 1}, {z_2, x_2, 1}, {z_3, x_3, 1}}])
T_A: 0.5 * sqrt(((y_2 - y_1) * (z_3 - z_1) - (z_2 - z_1) * (y_3 - y_1))^(2) + ((z_2 - z_1) * (x_3 - x_1) - (x_2 - x_1) * (z_3 - z_1))^(2) + ((x_2 - x_1) * (y_3 - y_1) - (y_2 - y_1) * (x_3 - x_1))^(2))
text19: "投影在xy平面上的三角形面積" + (LaTeX[T1]) + ""
text20: "投影在yz平面上的三角形面積" + (LaTeX[T2]) + ""
text21: "投影在zx平面上的三角形面積" + (LaTeX[T3]) + ""
text18: "三角形面積為" + (LaTeX[T_A]) + "=\sqrt{" + (LaTeX[T1]) + "²+" + (LaTeX[T2]) + "²+" + (LaTeX[T3]) + "²}"