重新排列

Straighten the Arc

2π \ell

2πr

2 π r

2π\ell

\ell

Straighten the Arc

求面积

\ell

将曲面面积分割并重新排列。当分割份数增加，图形会趋近什么？

Straighten the Arc

2 π r

2\; π\; r

2\; π\; r

h

r

Press the red point

圆锥的曲面面积是什么？

按平曲面看看。

\text{试以}π\text{、 }r \text{ 和 }\ell\text{ 表示圆锥的曲面面积。}

What is the area of the curved surface of the cone? 

Flatten the curved surface to see.

Rearrange the dissections. Increase the number of dissections.
What shape does the figure tend to?

\text{What is the area of the curved surface in terms of }\pi\text{, }r\text{ and }l\text{ ?} 

v = 1 (slider from 0 to 1, step 0.01)
O = (0, 1)
t = 0 (slider from 0 to 2, step 0.01)
Q = Point(CircleArc[(0, 1), (1, 1), (0, 0)])
s = If(y(Q) > 0, CircleArc[(0, 1), (0, 0), Q], 6.283185307179586)
r: s / ((2 * π))
h: sqrt(1 - r^(2))
A: O + (0, 0, h)
α' = If(y(Q) > 0, Angle[(0, 0), (0, 1), Q], 6.283185307179586)
n = 3 (slider from 3 to 40, step 1)
θ: α' / ((2 * n))
list1 = Sequence((sin((i * θ)), 1 - cos((i * θ))), i, 0, (2 * n))
l: (2 * sin(θ / 2))
O_1 = (0, 0)
E = (2, -0.5)
k = If(t ≤ 1, t, 1)
k' = If(t ≥ 1, t - 1, 0)
φ: ((1 - k) * θ / 2)
list2 = Sequence(If[k < 1, (x(O_1) + ((l * sin((i * φ))) * cos((i * φ)) / sin(φ)), y(O_1) + ((l * sin((i * φ))) * sin((i * φ)) / sin(φ))), (x(O_1) + (l * i), y(O_1))], i, 0, (2 * n))
list3 = Sequence(Sector[Rotate[Translate[Circle[(x(O_1) + 0, y(O_1) + 1), Element[list1, i]], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], Rotate[Translate[Element[list1, i], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], Rotate[Translate[Element[list1, i + 1], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]]], i, 1, n)
list4 = Sequence(Sector[Rotate[Rotate[Translate[Circle[(x(O_1) + 0, y(O_1) + 1), Element[list1, i]], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O_1) + (n * l) - sin(θ / 2) / 2, y(O_1) + cos(θ / 2) / 2)], Rotate[Rotate[Translate[Element[list1, i], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O_1) + (n * l) - sin(θ / 2) / 2, y(O_1) + cos(θ / 2) / 2)], Rotate[Rotate[Translate[Element[list1, i + 1], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O_1) + (n * l) - sin(θ / 2) / 2, y(O_1) + cos(θ / 2) / 2)]], i, n + 1, (2 * n))
m = 0 (slider from 0 to 1, step 0.01)
L: (sin(((1 - m) * s)) / (1 - m), (1 - cos(((1 - m) * s))) / (1 - m))
b = CircleArc((0, 1 / (1 - m)), O_1, L)
d = Segment((0, 0), (s, 0))
B' = Midpoint(d)
c = CircleArc((0, 1), Q, (0, 0))
e = Segment(O, (x(O), y(O) - 1))
G = Midpoint(e)
g = Circle((0, 1), 1)
dissect = false
H = Midpoint(Q, O)
j = 0 (slider from 0 to 1, step 0.1)
I: (x(Corner[3] - j), y(Corner[3] - j))
P: (v * A) + ((1 - v) * O)
a = Segment(O, A)
p = Circle(P, 1, yAxis)
D, C = Intersect(xOyPlane, p)
B: O - (r, 0)
α: ((2 * π) * x(B) / x(D))
F = Rotate(O_1, α, a)
f(x) = (((-z(P))) / abs(x(D)) * (x - abs(x(D))))
γ: π - α / 2
o = Surface((u * cos(v)), (u * sin(v)) + 1, f(u), u, 0, abs(x(D)), v, -1.5708, (3 * π / 2) - (2 * γ))
D_1: (0, 1 - abs(x(D)), 0)
D' = Rotate(D_1, α, a)
q = CircumcircleArc(D_1, C, D')
c_1 = Circle(O, D_1, a)
i = Segment(P, D')
a_1 = Segment(P, D_1)
J = Point(b)
b_1 = Segment(O, D_1)
u = Vector(A + (0, 0, 0.3), A + (0, 0, 0.06))
β = Angle(D_1, O, A)
lang = 1
text21: "" + (If[lang ≟ 1, "按平曲面", "Flatten the surface"]) + ""
text9_1: "" + ((2 * n)) + " " + (If[lang ≟ 1, "份", "dissections"]) + ""