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Right Circular Conoid

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RightCircularConoid

The right circular conoid is the right conoid having a circle as its directrix. A right circular conoid is also known as the Wallis conocuneus or cono-cuneus. Wallis (1684) described the corresponding solid as the "shipwright's circular wedge." A right circular conoid with base in the xy-plane, narrowing along the positive z-axis, and with radius a and height h has parametric equations

x=acosu
(1)
y=a(1-v)sinu
(2)
z=hv
(3)

and cartesian equation

h^2(x^2+y^2)+x^2z^2=a^2(h-z)^2+2hx^2z.
(4)

The volume of the solid enclosed by capping the bottom is given by

V=4/hint_0^hint_0^azsqrt(a^2-x^2)dxdz
(5)
=1/2pia^2h.
(6)

At h/a=2lambda, where lambda is the Laplace limit, this is also the volume bounded by the critical catenoid spanning two equal coaxial circles of radius a, although the interior cross sections of the two solids are different (Anderson 2026).

The surface area (of the lateral portion only, thus excluding the base circle) is given by

S=4/hint_0^hint_0^asqrt((h^2(a^2-x^2)+(a^2-x^2)^2+x^2z^2)/(a^2-x^2))dxdz.
(7)

This is more difficult to get in closed form, but for the case a=h=1, the integral can be reduced to

S=2int_0^1int_0^1sqrt((1+u)/(1-u)+(z^2)/u)dzdu
(8)
=pi_3F_2(-1/2,1/4,3/4;1/2,1;-1)+int_0^1(sqrt(u)(u+1))/(1-u)tanh^(-1)(sqrt((1-u)/(1+u^2)))du
(9)
approx 6.027212...
(10)

(OEIS A371923).

See also

Cone, Conoid, Critical Catenoid, Right Conoid

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References

Anderson, T. "The Volume of the Critical Catenoid." Math. Intelligencer, 2026. https://doi.org/10.1007/s00283-026-10551-0.Wallis, J. Cono-Cuneus: or, The Shipwright's Circular Wedge. London: J. Playford for R. Davis, 1684. https://archive.org/details/bim_early-english-books-1641-1700_cono-cuneus-or-the-shi_wallis-john_1684.Sloane, N. J. A. Sequence A371923 in "The On-Line Encyclopedia of Integer Sequences."

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Right Circular Conoid

Cite this as:

Weisstein, Eric W. "Right Circular Conoid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RightCircularConoid.html

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