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Viviani's Curve

The Space Curve giving the intersection of the Cylinder

\begin{displaymath}
(x-a)^2+y^2=a^2
\end{displaymath} (1)

and the Sphere
\begin{displaymath}
x^2+y^2+z^2=4^2.
\end{displaymath} (2)

It is given by the parametric equations
$\displaystyle x$ $\textstyle =$ $\displaystyle a(1+\cos t)$ (3)
$\displaystyle y$ $\textstyle =$ $\displaystyle a\sin t$ (4)
$\displaystyle z$ $\textstyle =$ $\displaystyle 2a\sin({\textstyle{1\over 2}}t).$ (5)

The Curvature and Torsion are given by
$\displaystyle \kappa(t)$ $\textstyle =$ $\displaystyle {\sqrt{13+3\cos t}\over a(3+\cos t)^{3/2}}$ (6)
$\displaystyle \tau(t)$ $\textstyle =$ $\displaystyle {6\cos({\textstyle{1\over 2}}t)\over a(13+3\cos t)}.$ (7)

See also Cylinder, Sphere, Steinmetz Solid


References

Gray, A. ``Viviani's Curve.'' §7.6 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 140-142, 1993.

von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 270, 1993.




© 1996-9 Eric W. Weisstein
1999-05-26