Lemon Surface
A surface of revolution defined by Kepler. It consists of less than half of a circular arc rotated about an axis passing through the endpoints of the arc. The equations of the upper and lower boundaries in the xz plane are
| z_+/-=+/-sqrt(R^2-(x+r)^2) |
(1)
|
for R>r and x in [-(R-r),R-r]. The cross section of a lemon is a lens. The lemon is the inside surface of a spindle torus. The American football is shaped like a lemon.
Two other lemon-shaped surfaces are given by the sextic surface
| a^4(x^2+z^2)+(y-a)^3y^3=0 |
(2)
|
called the "citrus" (or zitrus) surface by Hauser (left figure), and the sextic surface
| x^2-y^3z^3=0, |
(3)
|
whose upper and lower portions resemble two halves of a lemon, called the limão surface by Hauser (right figure).
The citrus surface had bounding box ((-a/8,a/8),(0,a),(-a/8,a/8)), centroid at (0,a/2,0), volume
| V_(citrus)=1/(140)pia^3, |
(4)
|
and a moment of inertia tensor
![画像: I=[(1445)/(5148)Ma^2 0 0; 0 5/(858)Ma^2 0; 0 0 (1445)/(5148)Ma^2]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/LemonSurface/NumberedEquation5.svg){kind=link}
for a uniform density solid citrus with mass M.
See also
Apple Surface, Lens, Oval, Prolate Spheroid, Spindle TorusExplore with Wolfram|Alpha
More things to try:
References
Hauser, H. "Gallery of Singular Algebraic Surfaces: Zitrus." https://homepage.univie.ac.at/herwig.hauser/gallery.html.JavaView. "Classic Surfaces from Differential Geometry: Football/Barrel." http://www.javaview.de/demo/surface/common/PaSurface_FootballBarrel.html.Referenced on Wolfram|Alpha
Lemon SurfaceCite this as:
Weisstein, Eric W. "Lemon Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LemonSurface.html