107:) attached to a given curve as that curve rolls without slipping, along a second given curve that is fixed. More precisely, given a curve attached to a plane which is moving so that the curve rolls, without slipping, along a given curve attached to a fixed plane occupying the same space, then a point attached to the moving plane describes a curve, in the fixed plane called a roulette.
1321:
1042:
665:
If, instead of a single point being attached to the rolling curve, another given curve is carried along the moving plane, a family of congruent curves is produced. The envelope of this family may also be called a roulette.
655:
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rolls along an equal blue parabola which remains fixed. The generator is the vertex of the rolling parabola and describes the roulette, shown in red. In this case the roulette is the
2128:
377:
84:
173:
at a point of contact that moves with the same speed when taken along either curve (another way to express this constraint is that the point of contact of the two curves is the
505:
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477:
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253:
2121:
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231:
2114:
1316:{\displaystyle f(t)+(p-r(t)){f'(t) \over r'(t)}=t-i+{p-\sinh(t)+i(1+p\sinh(t)) \over \cosh(t)}=t-i+(p+i){1+i\sinh(t) \over \cosh(t)}.}
553:
691:
2138:
793:
191:
31:
17:
382:
138:, the curve described by a point attached to a given curve as it slides along two (or more) given curves.
123:
of the fixed curve. If the rolling curve is a circle and the fixed curve is a line then the roulette is a
669:
Roulettes in higher spaces can certainly be imagined but one needs to align more than just the tangents.
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482:
174:
70:. On a basic level, it is the path traced by a curve while rolling on another curve without slipping.
2028:
2002:
2098:
2266:
1998:
1911:
328:
147:
2099:
Eine einheitliche
Darstellung von ebenen, verallgemeinerten Rollbewegungen und deren Anwendungen
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2017:
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8:
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1338:) and the roulette is a horizontal line. An interesting application of this is that a
1037:{\displaystyle |f'(t)|={\sqrt {1^{2}+\sinh ^{2}(t)}}={\sqrt {\cosh ^{2}(t)}}=|r'(t)|.}
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could roll without bouncing on a road that is a matched series of catenary arcs.
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Historical Math
Monographs, originally published by Deighton, Bell & Co.
1948:
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of the congruence transformation). The resulting roulette is formed by the
99:
Roughly speaking, a roulette is the curve described by a point (called the
63:
181:
of the generator subjected to the same set of congruence transformations.
2198:
1921:
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59:
51:
127:. If, in this case, the point lies on the circle then the roulette is a
2164:
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47:
1693:
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135:
119:
and the generator is a point on the line, the roulette is called an
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83:
27:
Mathematical curves generated by rolling other curves together
39:
2136:
1901:
650:{\displaystyle t\mapsto f(t)+(p-r(t)){f'(t) \over r'(t)}.}
2018:"Roulette with straight fixed curve" on www.mathcurve.com
2091:
Base, roulante et roulettes d'un mouvement plan sur plan
780:{\displaystyle f(t)=t+i(\cosh(t)-1)\qquad r(t)=\sinh(t)}
883:{\displaystyle f'(t)=1+i\sinh(t)\qquad r'(t)=\cosh(t).}
1334:
the expression has a constant imaginary part (namely −
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transformation such that at all times the curves are
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893:The parameterization of the line is chosen so that
1849:Equal parabola parameterized in opposite direction
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223:{\displaystyle r,f:\mathbb {R} \to \mathbb {C} }
1939:
184:Modeling the original curves as curves in the
2122:
2129:
2115:
1971:"Delaunay's roulette" on www.mathcurve.com
1997:
1987:"Delaunay's roulette" on www.2dcurves.com
493:
216:
208:
115:In the case where the rolling curve is a
82:
1960:"Sturm's roulette" on www.mathcurve.com
14:
2289:
2004:The applications of elliptic functions
1047:Applying the formula above we obtain:
146:Formally speaking, the curves must be
78:
2110:
2060:
452:{\displaystyle |r'(t)|=|f'(t)|\neq 0}
2029:"Centered trochoid" on mathcurve.com
1953:
1345:
141:
24:
2079:
660:
111:Special cases and related concepts
25:
2308:
2049:Notes on Roulettes and Glissettes
500:{\displaystyle p\in \mathbb {C} }
841:
746:
32:differential geometry of curves
2199:Pedal & Contrapedal curves
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2011:
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547:is then given by the mapping:
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1949:"Cissoid" on www.2dcurves.com
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479:. The roulette of generator
7:
2137:Differential transforms of
1890:
681:and the rolling curve is a
372:{\displaystyle r'(0)=f'(0)}
10:
2313:
672:
175:instant centre of rotation
2265:
2241:
2222:
2188:
2145:
318:{\displaystyle r(0)=f(0)}
232:natural parameterizations
2086:Roulette at 2dcurves.com
2007:. Macmillan. p. 88.
1932:
677:If the fixed curve is a
1912:Superposition principle
158:is kept invariant; the
134:A related concept is a
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2269:on a family of curves
2226:defined by two points
1570:Rectangular hyperbola
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2204:Negative pedal curve
1834:Point on the circle
1828:of a quarter of the
1809:Point on the circle
1784:Point on the circle
1740:Point on the circle
1715:Point on the circle
1663:Point on the circle
1581:Rectangular elastica
1462:Center of the conic
1442:Point on the circle
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1907:Locus (mathematics)
1558:Hyperbolic catenary
79:Informal definition
2245:defined by a point
2192:defined by a point
2062:Weisstein, Eric W.
2054:Cornell University
1859:Cissoid of Diocles
1803:of a third of the
1377:Point on the line
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281:curves, such that
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162:is subjected to a
97:
93:cissoid of Diocles
2297:Roulettes (curve)
2284:
2283:
2243:Binary operations
1888:
1887:
1623:Centered trochoid
1577:of the hyperbola
1554:of the hyperbola
1535:Elliptic catenary
1489:Delaunay roulette
1359:Generating point
1346:List of roulettes
1308:
1231:
1136:
999:
970:
642:
540:{\displaystyle f}
520:{\displaystyle r}
472:{\displaystyle t}
272:{\displaystyle f}
248:{\displaystyle r}
142:Formal definition
16:(Redirected from
2304:
2224:Unary operations
2190:Unary operations
2147:Unary operations
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1968:
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1855:of the parabola
1508:of the parabola
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234:of the rolling (
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2080:Further reading
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2016:
2012:
1996:
1992:
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1976:
1969:
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1958:
1954:
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1927:Rosetta (orbit)
1893:
1531:of the ellipse
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661:Generalizations
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152:Euclidean plane
144:
113:
81:
76:
42:, generalizing
28:
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22:
15:
12:
11:
5:
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2175:Parallel curve
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1466:Sturm roulette
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1356:Rolling curve
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218:
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197:
150:curves in the
148:differentiable
143:
140:
112:
109:
80:
77:
75:
72:
26:
18:Roulette curve
9:
6:
4:
3:
2:
2309:
2298:
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2292:
2277:
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2255:
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2240:
2234:
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2228:
2225:
2221:
2215:
2212:
2210:
2209:Pursuit curve
2207:
2205:
2202:
2200:
2197:
2196:
2194:
2191:
2187:
2181:
2178:
2176:
2173:
2171:
2170:Inverse curve
2168:
2166:
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2161:
2158:
2156:
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2148:
2144:
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2100:
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2055:
2051:
2050:
2045:
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2041:
2030:
2025:
2019:
2014:
2006:
2005:
2000:
1999:Greenhill, G.
1994:
1988:
1983:
1981:
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1972:
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1961:
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1733:
1730:
1728:
1725:Outside of a
1724:
1723:
1720:
1717:
1714:
1712:
1709:of identical
1708:
1705:
1703:
1700:Outside of a
1699:
1698:
1695:
1692:
1690:
1687:
1685:
1682:of identical
1681:
1678:
1676:
1673:Outside of a
1672:
1671:
1668:
1665:
1662:
1660:
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1652:Outside of a
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1485:of the conic
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1478:Conic section
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186:complex plane
182:
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160:rolling curve
157:
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94:
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71:
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64:hypotrochoids
61:
57:
53:
49:
45:
41:
38:is a kind of
37:
33:
19:
2251:
2139:plane curves
2068:
2047:
2044:W. H. Besant
2024:
2013:
2003:
1993:
1966:
1955:
1875:
1819:Inside of a
1794:Inside of a
1773:Inside of a
1767:Hypotrochoid
1761:
1750:Inside of a
1734:of half the
1688:
1640:
1617:
1593:Cyclocycloid
1580:
1557:
1534:
1488:
1465:
1420:
1398:
1393:
1367:
1353:Fixed curve
1340:square wheel
1335:
1331:
1327:
1325:
1046:
892:
676:
668:
664:
183:
159:
155:
145:
133:
114:
104:
100:
98:
60:epitrochoids
52:hypocycloids
35:
29:
2102:(in German)
2094:(in French)
1922:Tusi couple
1788:Hypocycloid
1646:Epitrochoid
685:, we have:
230:be the two
156:fixed curve
48:epicycloids
2267:Operations
2165:Dual curve
2065:"Roulette"
2038:References
1917:Spirograph
1667:Epicycloid
257:and fixed
167:congruence
164:continuous
74:Definition
2233:Strophoid
2070:MathWorld
1547:Hyperbola
1368:Any curve
1362:Roulette
1296:
1279:
1240:−
1219:
1199:
1166:
1160:−
1145:−
1079:−
988:
959:
866:
830:
766:
738:−
726:
585:−
561:↦
490:∈
444:≠
213:→
136:glissette
101:generator
68:involutes
56:trochoids
2291:Category
2276:Envelope
2252:Roulette
2160:Involute
2001:(1892).
1891:See also
1866:Catenary
1845:Parabola
1744:Nephroid
1719:Cardioid
1512:Catenary
1501:Parabola
1426:Trochoid
1404:Cyclogon
1381:Involute
1123:′
1104:′
1014:′
913:′
847:′
802:′
679:catenary
629:′
610:′
459:for all
426:′
396:′
357:′
337:′
125:trochoid
121:involute
89:parabola
87:A green
44:cycloids
36:roulette
2257:Cissoid
2214:Caustic
2180:Isoptic
2155:Evolute
2046:(1890)
1897:Rolling
1878:example
1838:Astroid
1813:Deltoid
1694:Limaçon
1601:Ellipse
1597:Center
1524:Ellipse
1446:Cycloid
673:Example
171:tangent
129:cycloid
30:In the
1853:Vertex
1830:radius
1826:Circle
1821:circle
1805:radius
1801:Circle
1796:circle
1780:Circle
1775:circle
1757:Circle
1752:circle
1736:radius
1732:Circle
1727:circle
1711:radius
1707:Circle
1702:circle
1684:radius
1680:Circle
1675:circle
1659:Circle
1654:circle
1636:Circle
1631:circle
1613:Circle
1608:Circle
1575:Center
1438:Circle
1416:Circle
379:, and
188:, let
154:. The
66:, and
2052:from
1933:Notes
1884:Line
1880:above
1552:Focus
1529:Focus
1506:Focus
1483:Focus
179:locus
40:curve
1902:Gear
1876:See
1871:Line
1588:Line
1565:Line
1542:Line
1519:Line
1496:Line
1473:Line
1453:Line
1433:Line
1411:Line
1389:Line
1373:Line
1293:cosh
1276:sinh
1216:cosh
1196:sinh
1163:sinh
979:cosh
950:sinh
863:cosh
827:sinh
763:sinh
723:cosh
683:line
117:line
105:pole
34:, a
1762:Any
1689:Any
1641:Any
1618:Any
1421:Any
1399:Any
1394:Any
1330:= −
1326:If
507:as
103:or
2293::
2067:.
1977:^
1941:^
325:,
131:.
62:,
58:,
54:,
50:,
46:,
2130:e
2123:t
2116:v
2073:.
1336:i
1332:i
1328:p
1311:.
1305:)
1302:t
1299:(
1288:)
1285:t
1282:(
1273:i
1270:+
1267:1
1261:)
1258:i
1255:+
1252:p
1249:(
1246:+
1243:i
1237:t
1234:=
1228:)
1225:t
1222:(
1211:)
1208:)
1205:t
1202:(
1193:p
1190:+
1187:1
1184:(
1181:i
1178:+
1175:)
1172:t
1169:(
1157:p
1151:+
1148:i
1142:t
1139:=
1133:)
1130:t
1127:(
1120:r
1114:)
1111:t
1108:(
1101:f
1094:)
1091:)
1088:t
1085:(
1082:r
1076:p
1073:(
1070:+
1067:)
1064:t
1061:(
1058:f
1032:.
1028:|
1024:)
1021:t
1018:(
1011:r
1006:|
1002:=
997:)
994:t
991:(
983:2
973:=
968:)
965:t
962:(
954:2
946:+
941:2
937:1
931:=
927:|
923:)
920:t
917:(
910:f
905:|
878:.
875:)
872:t
869:(
860:=
857:)
854:t
851:(
844:r
839:)
836:t
833:(
824:i
821:+
818:1
815:=
812:)
809:t
806:(
799:f
775:)
772:t
769:(
760:=
757:)
754:t
751:(
748:r
744:)
741:1
735:)
732:t
729:(
720:(
717:i
714:+
711:t
708:=
705:)
702:t
699:(
696:f
645:.
639:)
636:t
633:(
626:r
620:)
617:t
614:(
607:f
600:)
597:)
594:t
591:(
588:r
582:p
579:(
576:+
573:)
570:t
567:(
564:f
558:t
535:f
515:r
494:C
487:p
467:t
447:0
440:|
436:)
433:t
430:(
423:f
418:|
414:=
410:|
406:)
403:t
400:(
393:r
388:|
367:)
364:0
361:(
354:f
350:=
347:)
344:0
341:(
334:r
313:)
310:0
307:(
304:f
301:=
298:)
295:0
292:(
289:r
279:)
267:f
259:(
255:)
243:r
217:C
209:R
205::
202:f
199:,
196:r
95:.
20:)
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