# 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)