Viviani's curve

Viviani's curve as intersection of a sphere and a cylinder

In mathematics, particularly geometry, Viviani's curve, also known as Viviani's window, is a figure eight shaped space curve named after the Italian mathematician Vincenzo Viviani, the intersection of a sphere with a cylinder that is tangent to the sphere and passes through the center of the sphere.

The projection of Viviani's curve onto a plane perpendicular to the line through the crossing point and the sphere center is the lemniscate of Gerono.

Formula

The curve can be obtained by intersecting a sphere of radius $2a$ centered at the origin,

$x^{2}+y^{2}+z^{2}=4a^{2}\,$

with the cylinder centered at $(a,0,0)$ of radius $a$ given by

$(x-a)^{2}+y^{2}=a^{2}.\,$

The resulting curve of intersection, $V$, can be parameterized by $t$ to give the parametric equation of Viviani's curve:

$V(t)=\left\langle a(1+\cos(t)),a\sin(t),2a\sin \left({\frac {t}{2}}\right)\right\rangle .$

This is a clelie with $m=1$, where $\theta ={\frac {t-\pi }{2}}$.

See also

• Sphere-cylinder intersection

External links

• "Viviani curve", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Viviani's Curve". MathWorld.