In 1692, Viviani solved the following task: Cut out of a hemisphere (radius ) two windows, such that the remaining surface (of the hemisphere) can be squared; that is, a square with the same area can be constructed using only ruler and compass. His solution has an area of (see below).
Equations
With the cylinder upright.
In order to keep the proof for squaring simple, suppose that the sphere and cylinder have the equations
and
respectively. The cylinder has radius and is tangent to the sphere at point
Properties of the curve
Floor plan, elevation, and side plan
Floor plan, elevation and side plan
Elimination of , , and respectively yields the orthogonal projections of the intersection curve onto the:
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 prove that the area of the hemisphere 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 an integration:
Hence the total area of the spherical surface included by Viviani's curve is , and the area of the hemisphere () minus the area of Viviani's window is , the area of a square with the sphere's diameter as the length of an edge.
Rational Bézier representation
The quarter of Viviani's curve that lies in the all-positive octant of 3D space cannot be represented exactly by a regular Bézier curve of any degree. However, it can be represented exactly by a 3D rational Bézier segment of degree 4, and there is an infinite family of rational Bézier control points generating that segment. One possible solution is given by the following five control points:
Viviani's curve is a special Clelia curve. For a Clelia curve, the relation between the angles is
Viviani's curve (red) as intersection of the sphere and a cone (pink)
Subtracting twice the cylinder equation from the sphere's equation and completing the square leads to the equation
which describes a right circular cone with its apex at , the double point of Viviani's curve. Hence, Viviani's curve can be considered not only as the intersection curve of a sphere and a cylinder but also as the intersection of a sphere and a cone, and as the intersection of a cylinder and a cone.
↑ 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.
External links
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
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