Viviani's curve


In mathematics, Viviani's curve, also known as Viviani's window, is a figure eight shaped space curve named after the Italian mathematician Vincenzo Viviani. It is 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.
In 1692 Viviani tackled the task: Cut out of a half sphere two windows, such that the remaining surface can be squared, i.e. a square with the same area can be constructed using only compasses and ruler. His solution has an area of .

Equations

In order to keep the proof for squaring simple,
and
The cylinder has radius and is tangent to the sphere at point

Properties of the curve

Floor plan, elevation and side plan

Elimination of , , respectively yields:
The orthogonal projection of the intersection curve onto the

Parametric representation

Representing the sphere by
and setting yields the curve
One easily checks, that the spherical curve fulfills the equation of the cylinder. But the boundaries allow only the red part of Viviani's curve. The missing second half has the property
With help of this parametric representation it is easy to proof the statement: The area of the half sphere minus the area of the two windows is :

Squaring

The area of the upper right part of Viviani's window can be calculated by an integration:
Hence the total area of the spherical surface included by Viviani's curve is and
Subtracting 2× the cylinder equation from the sphere's equation and applying completing the square leads to the equation
which describes a right circular cone with its apex at
, the double point of Viviani's curve. Hence