A toric section is an intersection of a

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...

with a

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...

, just as a

conic section In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...

is the intersection of a

with a

cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...

. Special cases have been known since antiquity, and the general case was studied by

Jean Gaston Darboux Jean-Gaston Darboux FAS MIF FRS FRSE (14 August 1842 – 23 February 1917) was a French mathematician. Life According this birth certificate he was born in Nîmes in France on 14 August 1842, at 1 am. However, probably due to the midnigh ...

Mathematical formulae

In general, toric sections are fourth-order ( quartic)

plane curve In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic pla ...

s of the form :

\left( x^2 + y^2 \right)^2 + a x^2 + b y^2 + cx + dy + e = 0.

Spiric sections

A special case of a toric section is the

spiric section In geometry, a spiric section, sometimes called a spiric of Perseus, is a quartic plane curve defined by equations of the form :(x^2+y^2)^2=dx^2+ey^2+f. \, Equivalently, spiric sections can be defined as bicircular quartic curves that are symme ...

, in which the intersecting plane is parallel to the rotational symmetry axis of the

. They were discovered by the ancient Greek geometer

Perseus In Greek mythology, Perseus (Help:IPA/English, /ˈpɜːrsiəs, -sjuːs/; Greek language, Greek: Περσεύς, Romanization of Greek, translit. Perseús) is the legendary founder of Mycenae and of the Perseid dynasty. He was, alongside Cadmus ...

in roughly 150 BC. Well-known examples include the

hippopede In geometry, a hippopede () is a plane curve determined by an equation of the form :(x^2+y^2)^2=cx^2+dy^2, where it is assumed that and since the remaining cases either reduce to a single point or can be put into the given form with a rotation. ...

and the

Cassini oval In geometry, a Cassini oval is a quartic plane curve defined as the locus (mathematics), locus of points in the plane (geometry), plane such that the Product_(mathematics), product of the distances to two fixed points (Focus (geometry), foci) is ...

and their relatives, such as the

lemniscate of Bernoulli In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the ∞ symbol. I ...

Villarceau circles

Another special case is the

Villarceau circles In geometry, Villarceau circles () are a pair of circles produced by cutting a torus obliquely through the center at a special angle. Given an arbitrary point on a torus, four circles can be drawn through it. One is in a plane parallel to the e ...

, in which the intersection is a circle despite the lack of any of the obvious sorts of symmetry that would entail a circular cross-section..

General toric sections

More complicated figures such as an

annulus Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to: Human anatomy * ''Anulus fibrosus disci intervertebralis'', spinal structure * Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...

can be created when the intersecting plane is

perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...

oblique Oblique may refer to: * an alternative name for the character usually called a slash (punctuation) ( / ) * Oblique angle, in geometry *Oblique triangle, in geometry *Oblique lattice, in geometry * Oblique leaf base, a characteristic shape of the b ...

to the rotational symmetry axis.

References

External links

"The toric section: intersection of a torus with a plane"
at ''"worlds of math and physics"'' Algebraic curves {{geometry-stub