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Viviani's Curve


VivianisCurveIntersection

As defined by Gray (1997, p. 201), Viviani's curve, sometimes also called Viviani's window, is the space curve giving the intersection of the cylinder of radius a and center (a,0)

 (x-a)^2+y^2=a^2
(1)

and the sphere

 x^2+y^2+z^2=4a^2
(2)

with center (0,0,0) and radius 2a. This curve was studied by Viviani in 1692 (Teixeira 1908-1915, pp. 311-320; Struik 1988, pp. 10-11; Gray 1997, p. 201).

VivianisCurve

Solving directly for x and y as a function of z gives

x=2a-(z^2)/(2a)
(3)
y=+/-z/2sqrt(4-(z^2)/(a^2)).
(4)

This curve is given by the parametric equations

x=a(1+cost)
(5)
y=asint
(6)
z=2asin(1/2t)
(7)

for t in (-2pi,2pi) (Gray 1997, p. 201).

VivianisCurveSections

From the parametric equations, it can be immediately seen that views of the curve from the front, top, and left are given by a lemniscate-like curve, circle, and parabolic segment, respectively. The lemniscate-like figure has parametric equations

x=sint
(8)
y=2sin(1/2t),
(9)

which can be written in Cartesian coordinates as the quartic curve

 4x^2+y^4=4y^2.
(10)

Viviani's curve has arc length

 s=8sqrt(2)aE(1/2sqrt(2))
(11)

where E(k) is a complete elliptic integral of the second kind.

The arc length function, curvature, and torsion of Viviani's curve are given by

s(t)=2asqrt(2)E(1/2t,1/2sqrt(2))
(12)
kappa(t)=(sqrt(13+3cost))/(a(3+cost)^(3/2))
(13)
tau(t)=(6cos(1/2t))/(a(13+3cost))
(14)

(Gray 1997, p. 202), where E(x,k) is an incomplete elliptic integral of the second kind.


See also

Cylinder, Cylinder-Sphere Intersection, Devil's Curve, Lemniscate, Sphere, Steinmetz Solid

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References

Gray, A. "Viviani's Curve." §8.6 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 201-202, 1997.Kenison, E. and Bradley, H. C. Descriptive Geometry. New York: Macmillan, p. 284, 1935.Struik, D. J. Lectures on Classical Differential Geometry. New York: Dover, 1988.Teixeira, F. G. Traité des courbes spéciales remarquables plane et gauches, Vol. 2. Coimbra, Portugal, 1908-1915. Reprinted New York: Chelsea, 1971 and Paris: Gabay.von Seggern, D. CRC Standard Curves and Surfaces. Boca Raton, FL: CRC Press, p. 270, 1993.

Cite this as:

Weisstein, Eric W. "Viviani's Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VivianisCurve.html

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