In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, the lemniscate of Gerono, or lemniscate of Huygens, or figure-eight curve, is a
plane algebraic curve of degree four and
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
zero and is a
lemniscate
In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves. The word comes from the Latin "''lēmniscātus''" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons",. or which alternative ...
curve shaped like an
symbol, or figure eight. It has equation
:
It was studied by
Camille-Christophe Gerono.
Parameterization
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.
Dual curve
The
dual curve (see
Plücker formula), pictured below, has therefore a somewhat different character. Its equation is
:
References
*
External links
*{{MacTutor, class=Curves, id=Eight, title=Figure Eight Curve
Algebraic curves
Christiaan Huygens