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Witch of Agnesi

Parallels of Witch of Agnesi. witch_parall.gcf

Mathematica Notebook for This Page.

History

Studied by Maria Gaetana Agnesi (1718-1799) 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.

Description

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.

Tracing Witch of Agnesi Construction of the Witch

Formulas

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

• Parametric: {2*t, 2/(1+t^2)}, 0 < t < ∞.
• Parametric: 2 {Tan[t], Cos[t]^2}, -π/2 < t < π/2.
• Cartesian: y*(x^2+4)==8 witch.gcf

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.

Properties

Graphics Gallery

Normals, and Evolute of witch of Agnesi.

Left: the Witch and its normals. Right: The Witch and its normals up to center of osculating circle.

Left: Tangent circles of the Witch. Right: The Witch (blue) and its evolute (red).

Artistic work based on the Conchoids of witch of Agnesi. Right: various conchoids of the Witch. Conhoids wrt a moving point
Conchoid of the Witch

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.

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, Printed References On Plane Curves.

Robert Yates: Curves and Their Properties.

The MacTutor History of Mathematics archive.

Wikipedia: Witch of Agnesi.