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:

© Robert FERRÉOL 2017