Boundary formed by trochoids of different parameters.

Mathematica Notebook for This Page.



Trochoid describe a family of curves. (See: Curve Family Index) Trochoid is defined as the trace of a point fixed on a circle that rolls along a line. Sometimes the name trochoid is used to mean hypotrochoid and epitrochoid. (curve traced by rolling circle on another circle) More generally, trochoid is any curve that is the locus of a point fixed to a curve A, while A rolls on another curve B without slipping.

trochoid trochoidGen
Trochoid Tracing. r:=1, h:={.5, 1, 2.5}.


Let the radius of the rolling circle be r and the distance from the tracing point Q to the center of the circle be h.


Special Cases

Cyloid (blue), extended cycloid (green), contracted cycloid (red).
trochoid trochoidRadialVar
This animation shows the trochoid with parameters r:=1 and h increase from 0 to 3. The radial of the curve is also ploted as black dots. When r==h, it is a cycloid and its radial is a circle.

Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

Robert Yates: Curves and Their Properties.

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Plane Curves


  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


  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


  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