# 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.[1]

## Formula

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

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

with the cylinder centered at ${\displaystyle (a,0,0)}$ of radius ${\displaystyle a}$ given by

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

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

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

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