Difference between revisions of "Roulette"
From Encyclopedia of Mathematics
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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]]. | ||
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====References==== | ====References==== | ||
<table> | <table> | ||
| − | <TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> A.A. Savelov, "Planar curves" , Moscow (1960) (In Russian)</TD></TR> |
| − | <TR><TD valign="top">[a2]</TD> <TD valign="top"> | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Berger, "Geometry" , '''I''' , Springer (1987)</TD></TR> |
| − | <TR><TD valign="top">[a3]</TD> <TD valign="top"> | + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)</TD></TR> |
| − | <TR><TD valign="top">[a4]</TD> <TD valign="top"> | + | <TR><TD valign="top">[a3]</TD> <TD valign="top"> M.P. Do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976) pp. 145</TD></TR> |
| + | <TR><TD valign="top">[a4]</TD> <TD valign="top"> J.D. Lawrence, "A catalog of special plane curves" , Dover (1972) {{ISBN|0-486-60288-5}} {{ZBL|0257.50002}}</TD></TR> | ||
</table> | </table> | ||
Latest revision as of 20:51, 23 November 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=42495
Roulette. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Roulette&oldid=42495
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article