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Difference between revisions of "Ribaucour curve"

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A planar curve with curvature radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081760/r0817601.png" /> at an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081760/r0817602.png" /> proportional to the length of the segment of the normal MP (see Fig.).
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A planar curve with curvature radius $R$ at an arbitrary point $M$ proportional to the length of the segment of the normal $MP$ (see Fig.).
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r081760a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/r081760a.gif" />
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The equation for the Ribaucour curve in Cartesian orthogonal coordinates is
 
The equation for the Ribaucour curve in Cartesian orthogonal coordinates 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/r/r081/r081760/r0817603.png" /></td> </tr></table>
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$$x=\int\limits_0^y\frac{dy}{\sqrt{(y/c)^{2n}-1}},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081760/r0817604.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081760/r0817605.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081760/r0817606.png" /> is any integer), then a parametric equation for the Ribaucour curve is
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where $n=MP/R$. If $n=1/h$ ($h$ is any integer), then a parametric equation for the Ribaucour 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/r/r081/r081760/r0817607.png" /></td> </tr></table>
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$$x=(m+1)C\int\limits_0^t\sin^{m+1}tdt,\quad y=C\sin^{m+1}t,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081760/r0817608.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081760/r0817609.png" />, the Ribaucour curve is a circle; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081760/r08176010.png" />, it is a [[Cycloid|cycloid]]; when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081760/r08176011.png" />, it is a [[Catenary|catenary]]; and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081760/r08176012.png" />, it is a [[Parabola|parabola]].
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where $m=-(n+1)n$. When $m=0$, the Ribaucour curve is a circle; when $m=1$, it is a [[Cycloid|cycloid]]; when $m=-2$, it is a [[Catenary|catenary]]; and when $m=-3$, it is a [[Parabola|parabola]].
  
 
The length of an arc of the curve is
 
The length of an arc of the 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/r/r081/r081760/r08176013.png" /></td> </tr></table>
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$$l=(m+1)C\int\limits_0^t\sin^mtdt;$$
  
 
and the curvature radius is
 
and the curvature radius 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/r/r081/r081760/r08176014.png" /></td> </tr></table>
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$$R=-(m+1)C\sin^mt.$$
  
 
This curve was studied by A. Ribaucour in 1880.
 
This curve was studied by A. Ribaucour in 1880.

Latest revision as of 20:35, 7 April 2015

A planar curve with curvature radius $R$ at an arbitrary point $M$ proportional to the length of the segment of the normal $MP$ (see Fig.).

Figure: r081760a

The equation for the Ribaucour curve in Cartesian orthogonal coordinates is

$$x=\int\limits_0^y\frac{dy}{\sqrt{(y/c)^{2n}-1}},$$

where $n=MP/R$. If $n=1/h$ ($h$ is any integer), then a parametric equation for the Ribaucour curve is

$$x=(m+1)C\int\limits_0^t\sin^{m+1}tdt,\quad y=C\sin^{m+1}t,$$

where $m=-(n+1)n$. When $m=0$, the Ribaucour curve is a circle; when $m=1$, it is a cycloid; when $m=-2$, it is a catenary; and when $m=-3$, it is a parabola.

The length of an arc of the curve is

$$l=(m+1)C\int\limits_0^t\sin^mtdt;$$

and the curvature radius is

$$R=-(m+1)C\sin^mt.$$

This curve was studied by A. Ribaucour in 1880.

References

[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
[2] P.K. Rashevskii, "A course of differential geometry" , Moscow (1956) (In Russian)


Comments

References

[a1] F. Gomes Teixeira, "Traité des courbes" , 1–3 , Chelsea, reprint (1971)
How to Cite This Entry:
Ribaucour curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ribaucour_curve&oldid=15711
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article