Pedal of a Cycloid.

Mathematica Notebook for This Page.

radial evolute negative pedal pedal caustic parallel inversion derivative cissoid wrt a line conchoid strophoid



Cycloid (tautochrone, brachistochrone) is a member of cycloidal family of curves. (See: Curve Family Index) Prolate (extended) or curtate (contracted) cycloids are also known as trochoids. In this page, we use the narrowest definition of the term cycloid, defined as the trace of a point on the circumsference of a circle rolling on a line without slipping.

In the right figure, c is the rolling circle. P is the tracing point. A is the point of contact with line. PA is the normal at P. E is a reflection of P through A. The locus of E is the evolute of the cycloid.

cycloid cycloidGen
Tracing a Cycloid
cycloid Tangent Construct
Tracing a Cycloid by Tangent




The catacaustic of a cycloid with respect to parallel rays coming beneath its arc are two smaller cycloids. (Or, the diacaustic of the cycloid with rays coming from above.)

cycloid cycloid
Catacaustic with vertical rays

Evolute and Involute

The evolute of a cycloid is another cycloid. The first figure show succesive evolutes of a cycloid. The second connect points on the curve with their center of osculating circles.

cycloid cycloid
Left: The green cycloid is the evolute of the red cycloid is the evolute of the blue cycloid. Right: The yellow cycloid's normals (green) are draw up to the center of osculating circle. The endings form its evolute curve, which is another cycloid. Constructing Evolute of a Cycloid

The involute of a cycloid is also a cycloid. Both evolute and involute properties are easily proved by a direct application of the formula and simplify the result.


The radial of a cycloid is a circle.

cycloid cycloidRadial
Generating Radial Radial of a Cycloid

Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

Robert Yates: Curves and Their Properties.

The MacTutor History of Mathematics archive

Brombacher Aarnout, with GSP and QuickTime movies.

Joseph Portney. A somewhat detailed exposition on the property of cycloid being the curve of fastest decent. (called brachistochrone)

If you have a question, put $5 at patreon and message me.

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