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
(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
(3)
| |||
(4)
|
This curve is given by the parametric equations
(5)
| |||
(6)
| |||
(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
(8)
| |||
(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
(12)
| |||
(13)
| |||
(14)
|
(Gray 1997, p. 202), where is an incomplete elliptic integral of the second kind.