{t, 2 Cos[t] +t} by the envelope of its tangents.

Mathematica Notebook for This Page.


From Robert Yates:

Leibnitz (1694) and Taylor (1715) were the first to encounter singular solutions of differential equations. Their geometrical significance was first indicated by Lagrange in 1774. Particular studies were made by Cayley in 1872 and Hill in 1888 and 1918.


Envelope can be thought of as a way of deriving a new curve based on a set of curves. The envelope of a set of curves is a curve C such that C is tangent to every member of the set. (Two curves are tangent to each other iff both curves share a commen tangent at a common point.)

The concept of envelope is easily understood by looking at its graph. When a family of curves are drawn together, their envelope takes shape.

Cycloid, formed by the envelope of its tangents, and osculating circles.



More examples of envelope can be found in: astroid, cardioid, caustics, deltoid, evolute, lemniscate of Bernoulli, nephroid.

Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

Robert Yates: Curves and Their Properties.

J W Wilson. Envelopes of Lines and Circles. http://jwilson.coe.uga.edu/Texts.Folder/Envel/envelopes.html

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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