球的體積
Author: 藍邦偉  老師

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單位向量

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球的體積

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$ O $

$ V=\pi\int_{-r}^{r}{ (\sqrt{r^2-x^2})²dx} $

$ =2\pi\int_{0}^{r}{ (r²-x²)dx} $

$ =2\pi\left\cdot  [ r²x-\frac{x³}{3} \right ]_0^r  $

$ =2\pi\left ( r³-\frac{r³}{3} \right )  $

$ =\frac{4\pi r³}{3} $

Start

α = 0 rad (slider from 0*° to 360*°, step 0.1*°)
β = 1.5707963267948966 rad (slider from 0 to 6.283185307179586, step 5*°)
O = (12.278903856981975, 4.077433687353318)
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 = 5 (slider from 1 to 10, step 1)
yy = Vector(O - L * v, O + L * v)
zz = Vector(O - L * w, O + L * w)
xx = Vector(O - L * u, O + L * u)
A_1: O + L * u
B_1: O + L * v
c = false
d = true
Q_2: 0.42 * Corner[2] + 0.58 * Corner[1]
Q_3: (x(Q_2), y(Corner[3]))
多邊形1, e, q_2, q_3, f = Polygon(Corner[1], Q_2, Q_3, Corner[4])
g = Segment(Q_2, Q_3)
r = 3.3 (slider from 1 to 4.5, step 0.1)
n = -18 (slider from -18 to 18, step 1)
m = 0 (slider from 0 to 24, step 1)
list1 = Sequence(CurveCartesian[x(O + r * cos(s) * cos(-t) * w + r * cos(s) * sin(t) * v - r * sin(s) * u), y(O + r * cos(s) * cos(-t) * w + r * cos(s) * sin(t) * v - r * sin(s) * u), s, -1.5708, π / 2 * n / 18], t, 0, m / 24 * 2 * π, 0.2618)
list2 = Sequence(CurveCartesian[x(O + r * cos(s) * cos(-t) * w + r * cos(s) * sin(t) * v - r * sin(s) * u), y(O + r * cos(s) * cos(-t) * w + r * cos(s) * sin(t) * v - r * sin(s) * u), t, 0, 2 * m * π / 24], s, -1.5708, π / 2 * n / 18, 0.17453)
Step = 0 (slider from 0 to 5, step 1)

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')