A Hypotrochoid (red) and its evolute (black). Note that vertexes of a curve corresponds to its evolute cusps. The lines are radiuses of tangent circles. graphics code.

Mathematica Notebook for This Page.


Studied by Huygens in 1673.


Evolute is a method of deriving a new curve based on a given curve. It is the locus of the centers of tangent circles of the given curve.

Evolute of a ellipse.

evoluteEllipseF1 evoluteEllipseF2 evoluteEllipseF3 evoluteEllipseF4 evoluteEllipseF5 evoluteEllipseF6
Tangent circles of a ellipse. evoluteEllipse.gsp
evoluteEllipse1 evoluteEllipse2 evoluteEllipse3
Evolute of a curve can also be defined as the envelope of its normal.


Given a curve in parametric form {x[t], y[t]}, its evolute is

{x + (y'*(x'^2 + y'^2)) / (  y'*x''  - x'*y''),
 y + (x'*(x'^2 + y'^2)) / (-(y'*x'') + x'*y'')}


Parallels and Evolute

Theorem: The locus of Cusps of a curve C's parallel curves is the evolute of C. This is a alternative definition of evolute. See the Parallel page.

Evolute and Involute

If curve A is the involute of curve B, then curve B is the evolute of curve A. The converse is true locally, that is: If curve B is the evolute of curve A, then any part of curve A is the involute of some parts of B.

Curves relations by evolute and involute

Base Curve Evolute
cardioid cardioid scaled by 1/3
nephroid nephroid 1/2
astroid astroid 2
deltoid deltoid 3
epicycloid epicycloid
hypocycloid hypocycloid
cycloid cycloid
Cayley's sextic nephroid
parabola semicubic parabola
limacon of Pascal catacaustic of a circle
equiangular spiral equiangular spiral
tractrix catenary

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