Vivian window

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). ${\displaystyle r}$ ${\displaystyle r/2}$

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. ${\displaystyle r}$ ${\displaystyle 4r^{2}}$

Vertical cylinder

Analytical description

In order to be able to show the squarability as simply as possible, it is assumed here that

the sphere is described by the equation and ${\displaystyle \;x^{2}+y^{2}+z^{2}=r^{2}\;}$

the cylinder is vertical and the equation is sufficient. ${\displaystyle \;x^{2}+y^{2}-rx=0\;}$

The cylinder touches the ball at the point ${\displaystyle (r,0,0)\ .}$

Properties of the curve

Floor, elevation and side plans

Ground, top and side elevation

By eliminating or respectively from the equations, the orthogonal projection of the curve on the ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle z}$

${\displaystyle x}$- -plane is the circle with the equation ${\displaystyle y}$ ${\displaystyle \;(x-{\tfrac {r}{2}})^{2}+y^{2}=({\tfrac {r}{2}})^{2}\ .}$

${\displaystyle x}$- -plane the parabola with the equation ${\displaystyle z}$ ${\displaystyle \;x=-{\tfrac {1}{r}}z^{2}+r\;.}$

${\displaystyle y}$- -plane the algebraic curve with the equation ${\displaystyle z}$ ${\displaystyle \;z^{4}+r^{2}(y^{2}-z^{2})=0\;.}$

Parametric representation

For parameter display and content determination

If you put the sphere with spherical coordinates

${\displaystyle {\begin{array}{cll}x&=&r\cdot \cos \theta \cdot \cos \varphi \\y&=&r\cdot \cos \theta \cdot \sin \varphi \\z&=&r\cdot \sin \theta \qquad \qquad -{\tfrac {\pi }{2}}\leq \theta \leq {\tfrac {\pi }{2}}\ ,\ -\pi \leq \varphi \leq \pi \;,\end{array}}}$

and you get the curve ${\displaystyle \;\varphi =\theta ,\;}$

• ${\displaystyle {\begin{array}{cll}x&=&r\cdot \cos \theta \cdot \cos \theta \\y&=&r\cdot \cos \theta \cdot \sin \theta \\z&=&r\cdot \sin \theta \qquad \qquad -{\tfrac {\pi }{2}}\leq \theta \leq {\tfrac {\pi }{2}}\ .\end{array}}}$

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 ${\displaystyle \;\color {green}\varphi =-\theta \;.}$

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 :

${\displaystyle \iint _{O_{Kug}}r^{2}\cos \theta \,\mathrm {d} \theta \mathrm {d} \varphi =r^{2}\int _{0}^{\pi /2}\int _{0}^{\theta }\cos \theta \,\mathrm {d} \varphi \mathrm {d} \theta =r^{2}({\frac {\pi }{2}}-1)\ .}$

The total area of ​​the area enclosed by the Vivian curve is thus and ${\displaystyle 2\pi r^{2}-4r^{2}}$

• the content of the hemisphere surface ( ) without the content of the Vivian window is equal to the square of the diameter of the sphere. ${\displaystyle 2\pi r^{2}}$ ${\displaystyle \;4r^{2}\;}$

Relationship to other curves

• The elevation (see above) is a lemniscate by Gerono .
• The Vivian curve is a special case of a Clelia curve . At a Clelia curve is ${\displaystyle \;\varphi =c\;\theta \;.}$

Vivian curve as the intersection of the sphere with a cone (pink)

If one subtracts the cylinder equation 2 × from the spherical equation and executes the quadratic addition, the equation is obtained

${\displaystyle (x-r)^{2}+y^{2}=z^{2}\;.}$

This equation describes a perpendicular circular cone with the apex at the point , the colon of the Vivian curve. So it applies ${\displaystyle \;(r,0,0)\;}$

• The Vivian curve also results from the cut

a) the sphere with the cone with the equation ${\displaystyle \;(x-r)^{2}+y^{2}=z^{2}\;}$

as well as the cut

b) the cylinder with this cone.

Individual evidence

1. ↑ Kuno Fladt: Analytical geometry of special surfaces and space curves. Springer-Verlag, 2013, ISBN 3322853659 , 9783322853653, p. 97.
2. ↑ K. Strubecker : Lectures of the Descriptive Geometry. Vandenhoeck & Ruprecht, Göttingen 1967, p. 250.