Right Circular Conoid
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 = [画像: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
This is more difficult to get in closed form, but for the case a=h=1, the integral can be reduced to
= [画像: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 ConoidExplore with Wolfram|Alpha
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References
Sloane, N. J. A. Sequence A371923 in "The On-Line Encyclopedia of Integer Sequences."Referenced on Wolfram|Alpha
Right Circular ConoidCite this as:
Weisstein, Eric W. "Right Circular Conoid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RightCircularConoid.html