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
) and a cylinder touching the ball with a radius
can create. [first] [2] (S. Picture).
In 1692 Viviani provided the task of a hemisphere (radius
) 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
is.
In order to be able to show the quadrability as easily as possible, it is assumed here that
- the Bullet By equation is described and
- the Cylinder vertical stands and the equation enough.
The cylinder touches the ball in the point
Basic, impact and side cracks [ Edit | Edit the source text ]
Through elimination of
or.
or.
The equations follow:
The orthogonal projection of the curve on the
- – -Level is that Circle With the equation
- – -Level the parabola With the equation
- – -Bain the algebraic curve with the equation
Parameter presentation [ Edit | Edit the source text ]
If you put the ball with spherical coordinates
it and sets
you get the curve
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
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:
The entire area of the area enclosed by the Vivian curve is therefore
and
- The contents of the hemispherical surface ( ) without the content of the Vivian window , 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
If you subtract the cylinder equation from the ball equation 2 × and carry out square addition, you get the equation
This equation describes a vertical circular cone with the tip at the point
, 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
- as well as when cutting
- b) of the cylinder with this cone.
- ↑ Kuno flat: Analytical geometry of special areas and space curves. Spring-Pictite, 2013, Wingge 3322853659, 9783328653, 9763228653, P.CRI393, 97.
- ↑ K. Strubecker: Lectures of the performing geometry. Vandenhoeck & Ruprecht, Göttingen 1967, p. 250.
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