Lemon (geometry)

In

The apple and lemon together make up a spindle torus (or self-crossing torus or self-intersecting torus). The lemon forms the boundary of a convex set, while its surrounding apple is non-convex.^[1][2]

The ball in North American football has a shape resembling a geometric lemon. However, although used with a related meaning in geometry, the term "football" is more commonly used to refer to a surface of revolution whose Gaussian curvature is positive and constant, formed from a more complicated curve than a circular arc.^[3] Alternatively, a football may refer to a more abstract orbifold, a surface modeled locally on a sphere except at two points.^[4]

Area and volume

The lemon is generated by rotating an arc of radius ${\displaystyle R}$ and half-angle ${\displaystyle \phi _{m}}$ less than ${\displaystyle \pi /2}$ about its chord. Note that ${\displaystyle \phi }$ denotes latitude, as used in geophysics. The surface area is given by^[5]

${\displaystyle A=2\pi R^{2}\int _{-\phi _{m}}^{\phi _{m}}(\cos \phi -\cos \phi _{m})d\phi }$

The volume is given by

${\displaystyle V=\pi R^{3}\int _{-\phi _{m}}^{\phi _{m}}(\cos \phi -\cos \phi _{m})^{2}\cos \phi d\phi }$

These integrals can be evaluated analytically, giving

${\displaystyle A=4\pi R^{2}(\sin \phi _{m}-\phi _{m}\cos \phi _{m})}$

${\displaystyle V={\tfrac {4}{3}}\pi R^{3}\left[\sin ^{3}\phi _{m}-{\tfrac {3}{4}}\cos \phi _{m}(2\phi _{m}-\sin 2\phi _{m})\right]}$

The apple is generated by rotating an arc of half-angle ${\displaystyle \phi _{m}}$ greater than ${\displaystyle \pi /2}$ about its chord. The above equations are valid for both the lemon and apple.

See also

• Sears–Haack body
• List of shapes

References