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As the distance to the axis of revolution decreases, the ring torus becomes a spindle torus and then degenerates into a sphere.

The standard tori are the three classes of tori characterized by the extent of their self-intersection. In this general context, a torus is any surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, with no restriction on whether the axis touches the circle.

A torus can be defined parametrically by:


u, v are in the interval [0, 2π),

R is the distance from the center of the tube to the center of the torus, and

r is the radius of the tube.

The three different classes of standard tori correspond to the three possible relative sizes of r and R. When R > r, the surface will be the familiar ring torus. The case R = r corresponds to the horn torus, which in effect is a torus with no "hole". The case R < r describes a self-intersecting surface called a spindle torus. When used by itself, the word torus always refers to the ring torus.

Kepler called the external portion of the spindle torus the "apple", and defined the "lemon" as the interior portion of the surface. The two surfaces have the same equations for their cross-sections in the x-z plane

but for the apple, x is restricted to the to the interval [ − (r + R),(r + R)], and for the lemon, x is restricted to [ − (r − R),(r − R)].

Bottom halves and cross sections

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