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Apple Surface

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A surface of revolution defined by Kepler. It consists of more 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 x-z plane are

z_+/-=+/-sqrt(R^2-(x-r)^2)
(1)

for R>r and x in [-(r+R),r+R]. It is the outside surface of a spindle torus.

It is also a quartic surface given by Cartesian equation

(r^2-R^2+x^2+y^2+z^2)^2=4r^2(x^2+y^2)
(2)

or

r^4-2r^2(R^2+x^2+y^2-z^2)+(-R^2+x^2+y^2+z^2)^2=0.
(3)

See also

Bubble, Lemon Surface, Oblate Spheroid, Snake, Sphere-Sphere Intersection, Spindle Torus

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Cite this as:

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

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