Vivian window – Wikipedia

A Vivian window or Vivian curve , named after the Italian mathematician and physicist Vincenzo Viviani, is an 8-shaped curve on a ball that is as an intersection of the ball (radius

${displaystyle r}$

) and a cylinder touching the ball with a radius

${displaystyle r/2}$

can create. ^{[first] [2]} (S. Picture).

In 1692 Viviani provided the task of a hemisphere (radius

${displaystyle r}$

) to cut out two windows so that the rest of the hemisphere can be “squared”. Squarable means: With circle and ruler you can construct a square of the same area. It turns out (see below) that the area in question

${displaystyle 4r^{2}}$

is.

In order to be able to show the quadrability as easily as possible, it is assumed here that

the Bullet By equation

${displaystyle ;x^{2}+y^{2}+z^{2}=r^{2};}$

is described and

the Cylinder vertical stands and the equation

${displaystyle ;x^{2}+y^{2}-rx=0;}$

enough.

The cylinder touches the ball in the point

${displaystyle (r,0,0) .}$

Basic, impact and side cracks [ Edit | Edit the source text ]

Through elimination of

${displaystyle x}$

or.

${displaystyle y}$

or.

${displaystyle with}$

The equations follow:
The orthogonal projection of the curve on the

${displaystyle x}$

–

${displaystyle y}$

-Level is that Circle With the equation

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

${displaystyle x}$

–

${displaystyle with}$

-Level the parabola With the equation

${displaystyle ;x=-{tfrac {1}{r}}z^{2}+r;.}$

${displaystyle y}$

–

${displaystyle with}$

-Bain the algebraic curve with the equation

${displaystyle ;z^{4}+r^{2}(y^{2}-z^{2})=0;.}$

Parameter presentation [ Edit | Edit the source text ]

If you put the ball with spherical coordinates

${Displaytle {begin{splay} {cll}x&= ({RCDOT COSDA CDOT COS DIT COCOS \Z&RCDOT NO LOVE \Z&RCDOT NO LOVE \Z&RcDOT NO MAN {pi {pi {2}} , -pi Leq Varphi Leq Pi ;,End{array}}}}$

it and sets

${displaystyle ;varphi =theta ,;}$

you get the curve

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

  

It is easily checked that this curve is not only on the ball, but also fulfills the cylinder equation. However, this curve is only half (red) of the Viviani curve, namely the part from left at the bottom right. The other part (green, from the bottom right to the top left) you get through the relationship

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

With the help of this parameter presentation, Viviani’s task can be easily solved.

Quadrability of the remaining area [ Edit | Edit the source text ]

The content of the right upper quarter of the Vivian window (see picture) is obtained using a surface integral:

${Displaystyle Int _{O_{KUG}}R^{2}COS Theta ,mathrm {D} Theta Mathrm {D} Varph =r^{2}int _{0}^{pi /2} int _{0} ^{theta }Cos theta ,mathrm {d} varph mathrm {d} Theta =r^{2}({frac {pi }{2}-1) .}$

The entire area of ​​the area enclosed by the Vivian curve is therefore

${displaystyle 2pi r^{2}-4r^{2}}$

and

• The contents of the hemispherical surface (
  ${displaystyle 2pi r^{2}}$

  ) without the content of the Vivian window

  ${displaystyle ;4r^{2};}$

  , so the square of the ball diameter.

• The Outline (see above) is a Lemniskate from Gerono.
• The Vivian curve is a special case of a clelia curve. With a clelia curve is
  ${displaystyle; varphi = c; Theeta;.}$

  

If you subtract the cylinder equation from the ball equation 2 × and carry out square addition, you get the equation

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

This equation describes a vertical circular cone with the tip at the point

${displaystyle ;(r,0,0);}$

, the colon of the Vivian curve. So applies

• The Vivian curve also results in both cutting

a) the ball with the Kegel With the equation

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

as well as when cutting

b) of the cylinder with this cone.

1. ↑ Kuno flat: Analytical geometry of special areas and space curves. Spring-Pictite, 2013, Wingge 3322853659, 9783328653, 9763228653, P.CRI393, 97.
2. ↑ K. Strubecker: Lectures of the performing geometry. Vandenhoeck & Ruprecht, Göttingen 1967, p. 250.