mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph ...

s, given by the equation :

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

The bullet curve has three double points in the

real projective plane In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...

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

genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...

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 Quartic curves {{algebraic-geometry-stub