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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 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)

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))/(u-1)tanh^(-1)(sqrt((1-u)/(1+u^2)))du
(9)
approx 6.027212...
(10)

(OEIS A371923).

See also

Cone, Conoid, Right Conoid

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References

Sloane, N. J. A. Sequence A371923 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

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

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