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)

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.