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]

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 $\theta=\frac{t-\pi}{2}$.

References

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.