92指考一題
Author: 藍邦偉 老師

單位向量

座標軸

92指考數學一題

x

y

z

A

B

C

D

P

M

參考線

α = 0.6981317007977318 rad (slider from 0 to 6.283185307179586, step 0.01*°)
β = 1.3089969389957472 rad (slider from 0 to 6.283185307179586, step 0.01*°)
O = (8.928865875994504, 1.8768822491523323)
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 = 3 (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 = false
h = 1 (slider from 0.5 to 3, step 0.05)
H: O + h * w
i = Segment(O, H)
A_1: O + 4 * sqrt(3) / 3 * u
A_2: O - 2 * sqrt(3) / 3 * u + 2 * v
A_3: O - 2 * sqrt(3) / 3 * u - 2 * v
A_4: O + h * w
p_1, p_0 = SDLine(A_4, A_1, A_3, A_2)
p_2, p_3 = SDLine(A_4, A_2, A_1, A_3)
p_4, p_5 = SDLine(A_4, A_3, A_2, A_1)
p_6, p_7 = SDLine(A_3, A_1, A_2, A_4)
p_8, p_9 = SDLine(A_1, A_2, A_3, A_4)
p_a, p_b = SDLine(A_2, A_3, A_1, A_4)
e = Ray(A_1, A_4)
M = Midpoint(A_2, A_3)
f = OrthogonalLine(M, e)
P = Intersect(e, f)
g = Segment(P, M)
Q_2: 0.8 * Corner[1] + 0.2 * Corner[2]
Q_3: (x(Q_2), y(Corner[3]))
poly1, j, q_2, q_3, k = Polygon(Corner[1], Q_2, Q_3, Corner[4])
p_c = Segment(A_1, M)
l = true

Tool: Poly_B3
Input: A, B, C
Output: poly1'

α = Angle(C, B, A)
d: sqrt(sgn(sin(α)))
A' = Dilate(A, d, A)
B' = Dilate(B, d, A)
C' = Dilate(C, d, A)
poly1', c', a', b' = Polygon(A', B', C')

Tool: Poly_B5
Input: A, B, C, D, E
Output: poly1'

poly1, a, b, c, d, e = Polygon(A, B, C, D, E)
F = Midpoint(b)
G = Midpoint(d)
α = Angle(G, F, A)
f: sqrt(sgn(sin(α)))
A' = Dilate(A, f, A)
B' = Dilate(B, f, A)
C' = Dilate(C, f, A)
D' = Dilate(D, f, A)
E' = Dilate(E, f, A)
poly1', a', b', c', d', e' = Polygon(A', B', C', D', E')

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)