Roulette

Description

Roulette (Latin, round, to run, roll) is a method to generate new curves. Curves generated this way are also called roulette. It is the trace of a point (or a line) attached to a curve, while this curve rolls on another curve without slipping. The resulting curve is called a point-roulette or line-roulette respectively. A special class of point-roulette is rolling a circle on a line or another circle. These are known as cycloidal curves. Many of the famous curves, including the ellipse, can be generated this way. (See: curve family tree)

Glissette (meaning glide or slide) is the locus of a point or envelope of a line attached to a curve, which slides along two fixed curves. It can be shown that any glissette may also be defined as a roulette. [J. Dennis Lawrence] The most popular example of glissette is the trammel of Archimedes, used to generate astroid and ellipse.

History

Formula

Properties

Curve relations by roulette

Fixed Curve c1 Rolling Curve c2 Tracing Point Roulette
any curve line on line involute
line circle on circum. cycloid
circle circle any point epitrochoid, hypotrochoid
parabola equal parabola vertex cissoid of Diocles
line parabola focus catenary
line ellipse focus elliptic catenary?
line hyperbola focus hyperbolic catenary?
line equiangular spiral any point? line
line hyperbolic spiral pole tractrix
line involute of circle center parabola?
line cycloid center ellipse?

Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

Robert Yates: Curves and Their Properties.

Roulette (curve).

2006-05

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Plane Curves

Ancient

  1. Conic Sections
  2. Parabola
  3. Hyperbola
  4. Ellipse
  5. Cissoid
  6. Conchoid
  7. Quadratrix
  8. Archimedean Spiral
  9. Equiangular Spiral
  10. Lituus
  11. Cornu Spiral

Cyclodal

  1. Epitrochoid
  2. Hypotrochoid
  3. Epicycloid and Hypocycloid
  4. Rose Curve
  5. Astroid
  6. Deltoid
  7. Nephroid
  8. Cardioid
  9. Trochoid
  10. Cycloid

Calculus Era

  1. Cassinian Oval
  2. Cross Curve
  3. Folium of Descartes
  4. Piriform
  5. Semicubic Parabola
  6. Tractrix
  7. Trisectrix
  8. Trisectrix of Maclaurin
  9. Lemniscate of Bernoulli
  10. Lemniscate of Gerono
  11. Limacon Of Pascal
  12. Witch of Agnesi
  13. Sine Curve
  14. Catenary
  15. Bezier Curve

Methods

  1. Caustics
  2. Cissoid
  3. Conchoid
  4. Envelope
  5. Evolute
  6. Involute
  7. Geometric Inversion
  8. Orthoptic
  9. Parallel Curve
  10. Pedal Curve
  11. Radial Curve
  12. Roulette

Math of Curves

  1. Geometry: Coordinate Systems for Plane Curves
  2. Coordinate Transformation
  3. Vectors
  4. Naming and Classification of Curves
  1. Cusp
  2. Curvature