# Semi-cubic parabola

From Encyclopedia of Mathematics

2020 Mathematics Subject Classification: *Primary:* 53A04 [MSN][ZBL]

A third-order algebraic curve in the plane whose equation in Cartesian coordinates is

$$y=ax^{3/2}.$$

The origin is a cusp (see Fig.). The length of the arc from the origin equals

$$l=\frac{1}{27a^2}[(4+9a^2x)^{2/3}-8];$$

and the curvature equals

$$k=\frac{6a}{\sqrt x(4+9a^2x)^{3/2}}.$$

A semi-cubic parabola is sometimes called a Neil parabola, after W. Neil who found its arc length in 1657.

Figure: s084040a

#### References

[1] | A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian) |

[2] | A.S. Smogorzhevskii, E.S. Stolova, "Handbook of the theory of planar curves of the third order" , Moscow (1961) (In Russian) |

#### Comments

#### References

[a1] | J.D. Lawrence, "A catalog of special plane curves" , Dover, reprint (1972) |

[a2] | F. Gomes Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971) |

**How to Cite This Entry:**

Semi-cubic parabola.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Semi-cubic_parabola&oldid=43172

This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article