Approximating the Length of Viviani's Curve
Approximating the Length of Viviani's Curve
The positive part of Viviani's curve is the intersection of the hemisphere ++=4, , and the cylinder +=2rx. Using a partition of the circumference +=2rx, we construct a polygon that approximates the length of the curve. The exact value is , where is the complete elliptic integral function.
2
x
2
y
2
z
2
r
z≥0
2
x
2
y
2
x
2
y
z=0,
4r++dz=4≈7.6404r
2
∫
0
2
d(4-)
2
z
d z
2
dz
4-
2
z
d z
2
d(2z)
dz
2
rE1
2
E(m)