mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a bullet-nose curve is a unicursal quartic curve with three

inflection point In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of ...

s, given by the equation :

a^2y^2-b^2x^2=x^2y^2 \,

The bullet curve has three

double point In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied. Algebraic curves in the plane Algebraic curv ...

s in the

real projective plane In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has bas ...

, at and , and , and and , and is therefore a unicursal (rational) curve of

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

zero. If :

f(z) = \sum_^  z^ = z+2z^3+6z^5+20z^7+\cdots

then :

y = f\left(\frac\right)\pm 2b\

are the two branches of the bullet curve at the origin.

References

* Plane curves Algebraic curves {{algebraic-geometry-stub