A **toric section** is an intersection of a plane with a torus, just as a conic section is the intersection of a plane with a cone. Special cases have been known since antiquity, and the general case was studied by Jean Gaston Darboux.^{ [1] }

In general, toric sections are fourth-order (quartic) plane curves ^{ [1] } of the form

A special case of a toric section is the spiric section, in which the intersecting plane is parallel to the rotational symmetry axis of the torus. They were discovered by the ancient Greek geometer Perseus in roughly 150 BC.^{ [2] } Well-known examples include the hippopede and the Cassini oval and their relatives, such as the lemniscate of Bernoulli.

Another special case is the Villarceau circles, 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.^{ [3] }

More complicated figures such as an annulus can be created when the intersecting plane is perpendicular or oblique to the rotational symmetry axis.

In topology, a branch of mathematics, the **Klein bottle** is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a Klein bottle has no boundary.

In mathematics, a **parabola** is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several other superficially different mathematical descriptions, which can all be proved to define exactly the same curves.

A **sphere** is a geometrical object in three-dimensional space that is the surface of a ball.

In mathematics and abstract algebra, **group theory** studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

In geometry, a **torus** is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.

In geometry, a **hyperboloid of revolution**, sometimes called a **circular hyperboloid**, is the surface generated by rotating a hyperbola around one of its principal axes. A **hyperboloid** is the surface obtained from a hyperboloid of revolution by deforming it by means of directional scalings, or more generally, of an affine transformation.

A **surface of revolution** is a surface in Euclidean space created by rotating a curve around an axis of rotation.

In non-Euclidean geometry, the **Poincaré half-plane model** is the upper half-plane, denoted below as **H**, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.

In mathematics, a **Dupin cyclide** or **cyclide of Dupin** is any geometric inversion of a standard torus, cylinder or double cone. In particular, these latter are themselves examples of Dupin cyclides. They were discovered by Charles Dupin in his 1803 dissertation under Gaspard Monge. The key property of a Dupin cyclide is that it is a channel surface in two different ways. This property means that Dupin cyclides are natural objects in Lie sphere geometry.

In algebraic geometry, a **toric variety** or **torus embedding** is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be normal. Toric varieties form an important and rich class of examples in algebraic geometry, which often provide a testing ground for theorems. The geometry of a toric variety is fully determined by the combinatorics of its associated fan, which often makes computations far more tractable. For a certain special, but still quite general class of toric varieties, this information is also encoded in a polytope, which creates a powerful connection of the subject with convex geometry. Familiar examples of toric varieties are affine space, projective spaces, products of projective spaces and bundles over projective space.

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 the plane parallel to the equatorial plane of the torus. Another is perpendicular to it. The other two are Villarceau circles. They are named after the French astronomer and mathematician Yvon Villarceau (1813–1883). Mannheim (1903) showed that the Villarceau circles meet all of the parallel circular cross-sections of the torus at the same angle, a result that he said a Colonel Schoelcher had presented at a congress in 1891.

In geometry, a **hippopede** is a plane curve determined by an equation of the form

In mathematics, **enumerative geometry** is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.

In algebraic geometry, a **lemniscate** is any of several figure-eight or ∞-shaped curves. The word comes from the Latin "lēmniscātus" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons", or which alternatively may refer to the wool from which the ribbons were made.

**Perseus** was an ancient Greek geometer, who invented the concept of spiric sections, in analogy to the conic sections studied by Apollonius of Perga.

In geometry, a **spiric section**, sometimes called a **spiric of Perseus**, is a quartic plane curve defined by equations of the form

In mathematics, a **conic section** 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 special case of the ellipse, though historically it was sometimes called a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.

In geometry, an object has **symmetry** if there is an operation or transformation that maps the figure/object onto itself. Thus, a symmetry can be thought of as an immunity to change. For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable. A circle is thus said to be *symmetric under rotation* or to have *rotational symmetry*. If the isometry is the reflection of a plane figure about a line, then the figure is said to have reflectional symmetry or line symmetry; it is also possible for a figure/object to have more than one line of symmetry.

In differential geometry, the **Angenent torus** is a smooth embedding of the torus into three-dimensional Euclidean space, with the property that it remains self-similar as it evolves under the mean curvature flow. Its existence shows that, unlike the one-dimensional curve-shortening flow, the two-dimensional mean-curvature flow has embedded surfaces that form more complex singularities as they collapse.

In differential geometry **Dupin's theorem**, named after the French mathematician Charles Dupin, is the statement:

- 1 2 Sym, Antoni (2009), "Darboux's greatest love",
*Journal of Physics A: Mathematical and Theoretical*,**42**(40): 404001, doi:10.1088/1751-8113/42/40/404001 . - ↑ Brieskorn, Egbert; Knörrer, Horst (1986), "Origin and generation of curves",
*Plane algebraic curves*, Basel: Birkhäuser Verlag, pp. 2–65, doi:10.1007/978-3-0348-5097-1, ISBN 3-7643-1769-8, MR 0886476 . - ↑ Schoenberg, I. J. (1985), "A direct approach to the Villarceau circles of a torus",
*Simon Stevin*,**59**(4): 365–372, MR 0840858 .

- "The toric section: intersection of a torus with a plane" at
*"worlds of math and physics"*

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