thumb|[[Cycloid - curve generated by a rotating point on a wheel]]

thumb|[[Epitrochoid - Wheel rotating around a wheel ]]

In the differential geometry of curves, a roulette is a kind of kinematic curve, generalizing cycloids, epicycloids, hypocycloids, trochoids, epitrochoids, hypotrochoids, and involutes. On a basic level, it is the path traced by a curve while rolling on another curve without slipping.

Definition

Informal definition

right|frame|A green [[parabola 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 cissoid of Diocles.

|-

| Line

| Conic section

| Focus of the conic

| Delaunay roulette

|-

| Line

| Parabola

| Focus of the parabola

| Catenary

|-

| Line

| Ellipse

| Focus of the ellipse

| Elliptic catenary

|-

| Line

| Cyclocycloid

| Center

| Ellipse

|-

| Circle

| Circle

| Any

| Centered trochoid

|-

|Outside of a circle

|Circle

|Any

|Epitrochoid

|-

|Outside of a circle

|Circle

|Point on the circle

|Epicycloid

|-

|Outside of a circle

|Circle of identical radius

|Any

|Limaçon

|-

|Outside of a circle

|Circle of identical radius

|Point on the circle

|Cardioid

|-

|Outside of a circle

|Circle of half the radius

|Point on the circle

|Nephroid

|-

|Inside of a circle

|Circle

|Any

|Hypotrochoid

|-

|Inside of a circle

|Circle

|Point on the circle

|Hypocycloid

|-

|Inside of a circle

|Circle of a third of the radius

|Point on the circle

|Deltoid

|-

|Inside of a circle

|Circle of a quarter of the radius

|Point on the circle

|Astroid

|-

| Parabola

| Equal parabola parameterized in opposite direction

| Vertex of the parabola

| Cissoid of Diocles

|-

| Catenary

| Line

| See example above

| Line

|}

See also

  • Rolling
  • Gear
  • Locus (mathematics)
  • Superposition principle
  • Spirograph
  • Tusi couple
  • Rosetta (orbit)

Notes

References

  • W. H. Besant (1890) Notes on Roulettes and Glissettes from Cornell University Historical Math Monographs, originally published by Deighton, Bell & Co.

Further reading

  • Roulette at 2dcurves.com
  • Base, roulante et roulettes d'un mouvement plan sur plan
  • Eine einheitliche Darstellung von ebenen, verallgemeinerten Rollbewegungen und deren Anwendungen