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Viviani's curve: intersection of a sphere with a tangent cylinder
The light blue part of the half sphere can be squared
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 (radius ) two windows, such that the remaining surface (of the half sphere) 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 (see below).
In order to keep the proof for squaring simple,
the sphere has the equation
the cylinder is upright with equation .
The cylinder has radius and is tangent to the sphere at point
For parametric representation and the determination of the area
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 (see diagram) of Viviani's curve. The missing second half (green) has the property
With help of this parametric representation it is easy to proof the statement: The area of the half sphere (containing Viviani's curve) minus the area of the two windows is :
The area of the upper right part of Viviani's window (see diagram) can be calculated by a integration:
Hence the total area of the spherical surface included by Viviani's curve is and
the area of the half sphere () minus the area of Viviani's window is , the area of a square with the sphere's diameter as the length of an edge.
^Kuno Fladt: Analytische Geometrie spezieller Flächen und Raumkurven. Springer-Verlag, 2013, ISBN3322853659, 9783322853653, p. 97.
^K. Strubecker: Vorlesungen der Darstellenden Geometrie. Vandenhoeck & Ruprecht, Göttingen 1967, p. 250.
^Costa, Luisa Rossi; Marchetti, Elena (2005), "Mathematical and Historical Investigation on Domes and Vaults", in Weber, Ralf; Amann, Matthias Albrecht (eds.), Aesthetics and architectural composition : proceedings of the Dresden International Symposium of Architecture 2004, Mammendorf: Pro Literatur, pp. 73-80.
Berger, Marcel: Geometry. II. Translated from the French by M. Cole and S. Levy. Universitext. Springer-Verlag, Berlin, 1987.
Berger, Marcel: Geometry. I. Translated from the French by M. Cole and S. Levy. Universitext. Springer-Verlag, Berlin, 1987. xiv+428 pp. ISBN3-540-11658-3