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. Before Viviani this curve was studied by Simon de La Loubère and Gilles de Roberval.

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

• Berger, Marcel: Geometry. II. Translated from the French by M. Cole and S. Levy. Universitext. Springer-Verlag, Berlin, 1987.
• Berger, Marcel: Geometry. I. Translated from the French by M. Cole and S. Levy. Universitext. Springer-Verlag, Berlin, 1987. xiv+428 pp. ISBN 3-540-11658-3
• "Viviani curve", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Weisstein, Eric W. "Viviani's Curve". MathWorld.