Lemniscate of Gerono

From Wikipedia, the free encyclopedia
Jump to: navigation, search
The lemniscate of Gerono

In algebraic geometry, the lemniscate of Gerono, or lemniscate of Huygens, or figure-eight curve, is a plane algebraic curve of degree four and genus zero and is a lemniscate curve shaped like an symbol, or figure eight. It has equation

It was studied by Camille-Christophe Gerono.

Because the curve is of genus zero, it can be parametrized by rational functions; one means of doing that is

Another representation is

which reveals that this lemniscate is a special case of a lissajous figure.

The dual curve (see Plücker formula), pictured below, has therefore a somewhat different character. Its equation is

Dual to the lemniscate of Gerono

References[edit]

  • J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. p. 124. ISBN 0-486-60288-5. 

Notes[edit]

External links[edit]