Viviani's curve

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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]


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

with the cylinder centered at of radius given by

The resulting curve of intersection, , can be parameterized by to give the parametric equation of Viviani's curve:

This is a clelie with , where .

See also[edit]


  1. ^ Costa, Luisa Rossi; Marchetti, Elena (2005), "Mathematical and Historical Investigation on Domes and Vaults", in Weber, Ralf; Amann, Matthias Albrecht, Aesthetics and architectural composition : proceedings of the Dresden International Symposium of Architecture 2004, Mammendorf: Pro Literatur, pp. 73–80 .

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