Toric Section

A toric section is a curve obtained by slicing a torus (generally a horn torus) with a plane. A spiric section is a special case of a toric section in which the slicing plane is perpendicular to both the midplane of the torus and to the plane x=0.

Toric sections along z-axis

Consider a torus with tube radius a. For a cutting plane parallel to the xy-plane, the toric section is either a single circle (for |z|=a) or two concentric circles (for 0<=|z|<=a). For planes containing the z-axis, the section is two equal circles.

Oblique toric sections

Toric sections at oblique angles can be more complicated, passing from a crescent shape, through a U-shape, and into two disconnected kidney-shaped curves.


Certain toric sections with cutting plane tangent to the torus along the circumference of its central hole, as illustrated above, resemble (but are not equivalent to) a lemniscate curve. However, in the special case of a torus with parameters c^'=2a^', the toric section when the cutting plane is tangent to the torus along the circumference of its central hole becomes exactly a lemniscate with half-width


See also

Cassini Ovals, Conic Section, Cross Section, Cylindric Section, Ellipsoidal Section, Lemniscate, Spheric Section, Spheroidal Section, Spiric Section, Torus

Explore with Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Toric Section." From MathWorld--A Wolfram Web Resource.

Subject classifications