Viviani's curve

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.

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

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

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