Apple Surface

An apple surface is 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 - plane are

(1)

for and . It is the outside surface of a spindle torus.

It is the outer portion of the quartic surface given by Cartesian equation

(2)

or

(3)

For a generating arc of radius whose circle center is a distance from the rotation axis, the surface area and enclosed volume are (Lynch 2021, Verrall et al. 2023)

(4)

and

(5)

for .