Ribaucour curve
From Encyclopedia of Mathematics
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A planar curve with curvature radius at an arbitrary point 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
where . If ( is any integer), then a parametric equation for the Ribaucour curve is
where . When , the Ribaucour curve is a circle; when , it is a cycloid; when , it is a catenary; and when , it is a parabola.
The length of an arc of the curve is
and the curvature radius is
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=36400
Ribaucour curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ribaucour_curve&oldid=36400
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article