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Approximating the Length of Viviani's Curve

​
points
10
sphere and cylinder
The approximate length of Viviani's curve
Γ
r
is 7.54482 r.
The positive part of Viviani's curve is the intersection of the hemisphere
2
x
+
2
y
+
2
z
=4
2
r
,
z≥0
, and the cylinder
2
x
+
2
y
=2rx
. Using a partition of the circumference
2
x
+
2
y
=2rx
,
z=0,
we construct a polygon that approximates the length of the curve. The exact value is
4
2
∫
0
r
2
d(4-
2
z
)
d z
+
2
dz
4-
2
z
d z
+
2
d(2z)
dz
dz=4
2
rE
1
2
≈7.6404r
, where
E(m)
is the complete elliptic integral function.
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