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[[Image:Viviani curve.png|thumb|right|200px|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 [[Lemniscate|figure eight shaped]] space [[curve]] named after the Italian mathematician [[Vincenzo Viviani]], the intersection of a [[sphere]] with a [[cylinder (geometry)|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]].<ref>{{citation|first1=Luisa Rossi|last1=Costa|first2=Elena|last2=Marchetti|contribution=Mathematical and Historical Investigation on Domes and Vaults|pages=73–80|title=Aesthetics and architectural composition : proceedings of the Dresden International Symposium of Architecture 2004|year=2005|location=Mammendorf|publisher=Pro Literatur|editor1-last=Weber|editor1-first=Ralf|editor2-last=Amann|editor2-first=Matthias Albrecht}}.</ref>

== Formula ==
The curve can be obtained by intersecting a sphere of radius <math>2a</math> centered at the origin,

:<math>x^2+y^2+z^2=4a^2 \, </math>

with the cylinder centered at <math>(a,0,0)</math> of radius <math>a</math> given by

:<math>(x-a)^2+y^2=a^2. \, </math>

The resulting curve of intersection, <math>V</math>, can be parameterized by <math>t</math> to give the parametric equation of Viviani's curve:

:<math>V(t)= \left\langle a( 1+\cos(t) ), a\sin(t), 2a\sin\left(\frac{t}{2}\right) \right\rangle. </math>

This is a [[clelies|clelie]] with <math>m=1</math>, where <math>\theta=\frac{t-\pi}{2}</math>.

==See also==
*[[Sphere-cylinder intersection]]

==References==
{{reflist}}

==External links==
* {{springer|title=Viviani curve|id=p/v096800}}
* {{mathworld|urlname=VivianisCurve|title=Viviani's Curve}}

[[Category:Curves]]


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