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The name of a planar curve considered as the trajectory of a point which is rigidly connected with some curve rolling upon another fixed curve. In case a circle rolls upon a straight line, the roulette is a [[Cycloid|cycloid]]; if a circle rolls upon another circle it is a [[Cycloidal curve|cycloidal curve]]; if a hyperbola, an ellipse or a parabola rolls upon a straight line it is a Sturm curve (cf. [[Sturm curves|Sturm curves]]). The trajectory of an ellipse rolling upon another ellipse is called an epi-ellipse. Each planar curve can be considered as a roulette in many ways; for example, any curve can be formed by rolling a straight line upon its [[Evolute|evolute]].
 
The name of a planar curve considered as the trajectory of a point which is rigidly connected with some curve rolling upon another fixed curve. In case a circle rolls upon a straight line, the roulette is a [[Cycloid|cycloid]]; if a circle rolls upon another circle it is a [[Cycloidal curve|cycloidal curve]]; if a hyperbola, an ellipse or a parabola rolls upon a straight line it is a Sturm curve (cf. [[Sturm curves|Sturm curves]]). The trajectory of an ellipse rolling upon another ellipse is called an epi-ellipse. Each planar curve can be considered as a roulette in many ways; for example, any curve can be formed by rolling a straight line upon its [[Evolute|evolute]].
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Savelov,  "Planar curves" , Moscow  (1960)  (In Russian)</TD></TR></table>
 
 
 
 
====Comments====
 
 
  
 
====References====
 
====References====
 
<table>
 
<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">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''I''' , Springer  (1987)</TD></TR>
 
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Berger,  "Geometry" , '''I''' , Springer  (1987)</TD></TR>
 
<TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1963)</TD></TR>
 
<TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Introduction to geometry" , Wiley  (1963)</TD></TR>

Revision as of 16:52, 8 April 2023

The name of a planar curve considered as the trajectory of a point which is rigidly connected with some curve rolling upon another fixed curve. In case a circle rolls upon a straight line, the roulette is a cycloid; if a circle rolls upon another circle it is a cycloidal curve; if a hyperbola, an ellipse or a parabola rolls upon a straight line it is a Sturm curve (cf. Sturm curves). The trajectory of an ellipse rolling upon another ellipse is called an epi-ellipse. Each planar curve can be considered as a roulette in many ways; for example, any curve can be formed by rolling a straight line upon its evolute.

References

[1] A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)
[a1] M. Berger, "Geometry" , I , Springer (1987)
[a2] H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)
[a3] M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 145
[a4] J.D. Lawrence, "A catalog of special plane curves" , Dover (1972) ISBN 0-486-60288-5 Zbl 0257.50002
How to Cite This Entry:
Roulette. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Roulette&oldid=53687
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article