next surface | previous surface | 2D curves | 3D curves | surfaces | fractals | polyhedra |

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:

next surface | previous surface | 2D curves | 3D curves | surfaces | fractals | polyhedra |

© Robert FERRÉOL 2017