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Difference between revisions of "Cesàro curve"

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A plane curve whose radius of curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021350/c0213501.png" /> at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021350/c0213502.png" /> is proportional to the segment of the normal cut off by the polar (line) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021350/c0213503.png" /> with respect to a certain circle. The [[Natural equation|natural equation]] of a Cesàro curve is
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A plane curve whose radius of curvature $R$ at any point $M$ is proportional to the segment of the normal cut off by the polar (line) of $M$ with respect to a certain circle. The [[Natural equation|natural equation]] of a Cesàro curve is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021350/c0213504.png" /></td> </tr></table>
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$$s=\int\frac{dR}{(R/b)^m-1},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021350/c0213505.png" /> is a constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021350/c0213506.png" /> is a real number. Investigated by E. Cesàro [[#References|[1]]].
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where $b$ is a constant and $m$ is a real number. Investigated by E. Cesàro [[#References|[1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Cesàro,  "Vorlesungen über natürliche Geometrie" , Teubner  (1901)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Savelov,  "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Cesàro,  "Vorlesungen über natürliche Geometrie" , Teubner  (1901)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.A. Savelov,  "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR></table>

Latest revision as of 09:01, 3 October 2014

A plane curve whose radius of curvature $R$ at any point $M$ is proportional to the segment of the normal cut off by the polar (line) of $M$ with respect to a certain circle. The natural equation of a Cesàro curve is

$$s=\int\frac{dR}{(R/b)^m-1},$$

where $b$ is a constant and $m$ is a real number. Investigated by E. Cesàro [1].

References

[1] E. Cesàro, "Vorlesungen über natürliche Geometrie" , Teubner (1901)
[2] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
How to Cite This Entry:
Cesàro curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ces%C3%A0ro_curve&oldid=15152
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article