Toric section

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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]

Contents

Mathematical formulae

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

Spiric sections

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.

Villarceau circles

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]

General toric sections

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

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Hippopede plane curve

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

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

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

Conic section Curve obtained by intersecting a cone and a plane

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.

Symmetry (geometry) geometrical property and transformation

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.

Dupins theorem

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

References

  1. 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 .
  2. 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 .
  3. Schoenberg, I. J. (1985), "A direct approach to the Villarceau circles of a torus", Simon Stevin, 59 (4): 365–372, MR   0840858 .