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Difference between revisions of "Semi-cubic parabola"

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(historical note, from Neil parabola)
 
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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
  
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$$k=\frac{6a}{\sqrt x(4+9a^2x)^{3/2}}.$$
 
$$k=\frac{6a}{\sqrt x(4+9a^2x)^{3/2}}.$$
  
A semi-cubic parabola is sometimes called a Neil parabola.
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A semi-cubic parabola is sometimes called a Neil parabola, after W. Neil who found its arc length in 1657.
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s084040a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s084040a.gif" />
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,  "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.S. Smogorzhevskii,  E.S. Stolova,  "Handbook of the theory of planar curves of the third order" , Moscow  (1961)  (In Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,  "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  A.S. Smogorzhevskii,  E.S. Stolova,  "Handbook of the theory of planar curves of the third order" , Moscow  (1961)  (In Russian)</TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.D. Lawrence,  "A catalog of special plane curves" , Dover, reprint  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Gomes Teixeira,  "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.D. Lawrence,  "A catalog of special plane curves" , Dover, reprint  (1972)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Gomes Teixeira,  "Traité des courbes" , '''1–3''' , Chelsea, reprint  (1971)</TD></TR>
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</table>

Latest revision as of 16:04, 16 April 2018

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