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Lemon Surface


lemon
LemonCrossSection

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.

Lemons

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

 I=[(1445)/(5148)Ma^2 0 0; 0 5/(858)Ma^2 0; 0 0 (1445)/(5148)Ma^2]
(5)

for a uniform density solid citrus with mass M.


See also

Apple Surface, Lens, Oval, Prolate Spheroid, Spindle Torus

Explore with Wolfram|Alpha

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-sfb288.math.tu-berlin.de/vgp/javaview/demo/surface/common/PaSurface_FootballBarrel.html.

Cite this as:

Weisstein, Eric W. "Lemon Surface." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LemonSurface.html

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