Spiric Section

Ring torus spiric sections
Horn torus spiric sections
Spindle torus spiric sections

The equation of the curve of intersection of a torus with a plane perpendicular to both the midplane of the torus and to the plane x=0. (The general intersection of a torus with a plane is called a toric section). Let the tube of a torus have radius a, let its midplane lie in the z=0 plane, and let the center of the tube lie at a distance c from the origin. Now cut the torus with the plane y=r. The equation of the torus with y=r gives the equation


The above plots show a series of spiric sections for the ring torus, horn torus, and spindle torus, respectively. When r=0, the curve consists of two circles of radius a whose centers are at (c,0) and (-c,0). If r=c+a, the curve consists of one point (the origin), while if r>c+a, no point lies on the curve.

The spiric extensions are an extension of the conic sections constructed by Menaechmus around 150 BC by cutting a cone by a plane, and were first considered around 50 AD by the Greek mathematician Perseus (MacTutor).


If r=a, then (3) simplifies to


which is the equation of Cassini ovals. Cassini ovals are therefore spiric sections. Furthermore, the surface having these curves as cross sections is the Cassini surface illustrated above, with the modification that the vertical component is squared instead of to the fourth power.

See also

Toric Section, Torus

Explore with Wolfram|Alpha


MacTutor History of Mathematics Archive. "Spiric Sections."

Referenced on Wolfram|Alpha

Spiric Section

Cite this as:

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

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