Difference between revisions of "Semi-cubic parabola"
From Encyclopedia of Mathematics
(Importing text file) |
(TeX) |
||
| Line 1: | Line 1: | ||
| + | {{TEX|done}} | ||
A third-order algebraic curve in the plane whose equation in Cartesian coordinates is | A third-order algebraic curve in the plane whose equation in Cartesian coordinates is | ||
| − | + | $$y=ax^{3/2}.$$ | |
The origin is a [[Cusp(2)|cusp]] (see Fig.). The length of the arc from the origin equals | The origin is a [[Cusp(2)|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 | and the curvature equals | ||
| − | + | $$k=\frac{6a}{\sqrt x(4+9a^2x)^{3/2}}.$$ | |
A semi-cubic parabola is sometimes called a Neil parabola. | A semi-cubic parabola is sometimes called a Neil parabola. | ||
Revision as of 11:41, 29 June 2014
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.
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=32340
Semi-cubic parabola. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-cubic_parabola&oldid=32340
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article