# Semi-cubic parabola

2010 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)