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CIRCLED SURFACE

Other name: cyclic surface.
See bogomolov-lab.ru/AG2012/Talks/Skopenkov_talk.pdf |

A *circled* surface is a surface generated by the
movement of a circle (the radius of which can vary).

Examples:

- the envelopes
of spheres, (see NSCs below),
with the special cases of surfaces
of revolution and the cyclides.

- the tubes
(with variable section or not)

- the cyclotomic
surfaces

- the inverse
of a ruled surface by an inversion
the center of which is not on the surface (and if the ruled surface is
developable, the inverse is the envelope of spheres)

- the pedal
surfaces of a curve

- the
quadrics
(even those that are not of revolution), except the hyperbolic paraboloid

- the circled
helicoids

- the Bohemian
dome

- the skew
catenoid, only circled minimal surface

- a model of cross-cap.

- the sea-shells.

Here are various NSCs for a surface to be the envelope
of spheres:

1) Circled surface the circles of which are curvature
lines

2) Circled surface the circles of which are in a principal
direction at any of their points

3) Circled surface one of the focals
of which is a curve

Simple example of a circled surface that is not the envelope
of spheres: a non circular elliptic cone.

Tori, and their inverses the Dupin
cyclides, are fourfold circled surfaces (by any point passes four circles,
two of which are Villarceau circles). But a compact surface other than
the sphere can not be more than sixfold circled (Takeuchi
theorem, 1995), as are, for example, the Darboux
cyclides.

Circled surfaces made by Robert March's students:

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© Robert FERRÉOL 2017