Elliptic Torus
Am elliptic torus is a surface of revolution which is a generalization of the ring torus. It is produced by rotating an ellipse embedded in the xz-plane having horizontal semi-axis a, vertical semi-axis b, and located a distance c away from the z-axis about the z-axis. It is given by the parametric equations
for u,v in [0,2pi).
This gives first fundamental form coefficients of
second fundamental form coefficients of
The Gaussian curvature and mean curvature are
![画像:(4ab^2cosv)/((c+acosv)[a^2+b^2+(b^2-a^2)cos(2v)]^2)](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/EllipticTorus/Inline37.svg){kind=link}
![画像:-(b[4ac+(5a^2+3b^2)cosv+(b^2-a^2)cos(3v)])/(2sqrt(2)cosc+acosv[a^2+b^2+(b^2-a^2)cos(2v)]^(3/2)).](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/EllipticTorus/Inline40.svg){kind=link}
By Pappus's centroid theorems, the surface area and volume are
where E(k) is a complete elliptic integral of the second kind and
| [画像: e=sqrt(1-(b^2)/(a^2)) ] |
(16)
|
is the eccentricity of the ellipse cross section.
See also
Ring Torus, Surface of Revolution, TorusExplore with Wolfram|Alpha
More things to try:
References
Gray, A. "Tori." §11.4 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 210 and 304-305, 1997.Referenced on Wolfram|Alpha
Elliptic TorusCite this as:
Weisstein, Eric W. "Elliptic Torus." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EllipticTorus.html