Viviani window: intersection of a sphere with a cylinder in contact
The light blue semi- spherical surface is quadrierbar
A vivian window or vivian curve , named after the Italian mathematician and physicist Vincenzo Viviani , is an 8-shaped curve on a sphere that can be created as the intersection of the sphere (radius ) and a cylinder with a radius that touches the sphere . (See picture).
In 1692 Viviani set the task of cutting two windows out of a hemisphere (radius ) so that the rest of the hemisphere can be "squared". Squarable means: You can construct a square with the same area using a compass and ruler. It turns out (see below) that the area in question is.
It is easy to check that this curve not only lies on the sphere, but also satisfies the cylinder equation. However, this curve is only one half (red) of the Viviani curve, namely the part from the bottom left to the top right. The other part (green, from bottom right to top left) is obtained from the relationship
Viviani's task can easily be solved with the help of this parameter representation.
Squarability of the remaining area
The content of the upper right quarter of the Vivian window (see picture) is obtained by means of a surface integral :
The total area of the area enclosed by the Vivian curve is thus and
the content of the hemisphere surface ( ) without the content of the Vivian window is equal to the square of the diameter of the sphere.