Viviani’s curve

3 years ago by in Featured, Geometry, Geometry

In mathematics, particularly geometry, Viviani’s curve, also known as Viviani’s window, is a figure eight shaped space curve named after the Italian mathematicianVincenzo 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.

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}.

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