Witch of Agnesi

witch of Agnesi
Parallels of Witch of Agnesi.

Mathematica Notebook for This Page.


Studied by Maria Gaetana Agnesi (1718 to 1799) in 1748. Also studied by Fermat (1666), and Guido Grandi (1703). The name of this curve has a colorful history. Versaria is the name given by Grandi, meaning “turning in every direction”. In the course of time the word versariatook on another meaning. The Latin words adversaria, and by aphaeresis, versaria, signify a female that is contrary to God. Thus gradually the curve versaria is understood in English as the Witch.


Witch of Agnesi (Versiera) is defined as follows.

Step by step description:

  1. Let there be a circle of radius a with center at {0,a}.
  2. Let there be a horizontal line L passing through {0,2 a}.
  3. Draw a line passing the Origin and any point M on the circle. Let the intersection of this secant and line L be N.
  4. Witch of Agnesi is the locus of intersections of a horizontal line passing through M and a vertical line passing through N.
witch of Agnesi witchOfAgnesiGen
Tracing Witch of Agnesi Construction of the Witch


Let the construction circle be centered at {0,1} with radius 1, then:

The parametric equation {2*t, 2/(1+t^2)} is derived easily by starting with a equation of circle x^2+(y-1)^2==1 and a line y==t*x and solve the equation of circle and lines, and simplify the result by the replacement t→1/t. (or, start with y==1/t*x instead). Elimating t and we find the Cartesian equation.


Graphics Gallery

Normals, and Evolute of witch of Agnesi.

witch of Agnesi witch of Agnesi
Left: the Witch and its normals. Right: The Witch and its normals up to center of osculating circle.
witch of Agnesi witch of Agnesi
Left: Tangent circles of the Witch. Right: The Witch (blue) and its evolute (red).
witch of Agnesi witch of Agnesi
Artistic work based on the Conchoids of witch of Agnesi.
witchOfAgnesiConchoid witchOfAgnesiConchoid
Conhoids of Witch with respect to a moving point. Conchoid of the Witch
witch of Agnesi
Inversion curves of the Witch {Tan[t], Cos[t]^2} with respect to points {{0,-1}, {0, -.8},…,{0,1.6}} and radius of inversion 1, corresponding to curves with light to dark shades.
witch of Agnesi
Pedal curves of the Witch {Tan[t], Cos[t]^2} with respect to points {{0,-1}, {0, -.8},…,{0,1.6}}, corresponding to curves with light to dark shades.

Related Web Sites

See: Websites on Plane Curves, Plane Curves Books.

Robert Yates: Curves and Their Properties.

The MacTutor History of Mathematics archive

Witch of Agnesi.

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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