As defined by Gray (1997, p. 201), Viviani's curve, sometimes also called Viviani's window, is the space curve giving the intersection
of the cylinder of radius 
and center 
 |
(1)
|
and the sphere
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(2)
|
with center 
and radius 
.
This curve was studied by Viviani in 1692 (Teixeira 1908-1915, pp. 311-320;
Struik 1988, pp. 10-11; Gray 1997, p. 201).
Solving directly for 
and 
as a function of 
gives
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(3)
|
 |  |  |
(4)
|
This curve is given by the parametric equations
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(5)
|
 |  |  |
(6)
|
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(7)
|
for 
(Gray 1997, p. 201).
From the parametric equations, it can be immediately seen that views of the curve from the front, top, and left are given by a lemniscate-like curve, circle, and parabolic segment, respectively. The lemniscate-like figure has parametric equations
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(8)
|
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(9)
|
which can be written in Cartesian coordinates as the quartic curve
 |
(10)
|
Viviani's curve has arc length
 |
(11)
|
where 
is a complete elliptic integral
of the second kind.
The arc length function, curvature, and torsion of Viviani's curve are given by
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(12)
|
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(13)
|
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(14)
|
(Gray 1997, p. 202), where 
is an incomplete elliptic
integral of the second kind.
