Semi-cubic parabola

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2010 Mathematics Subject Classification: Primary: 53A04 [MSN][ZBL]

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


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


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


[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)



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