Witch of Agnesi

Parallels of Witch of Agnesi

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

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 (20 k)

Formulas

• Parametric: 2 a {Tan[t], Cos[t]^2}, -Pi/2 < t < Pi/2.
• Cartesian: y (x^2 + 4 a^2) == 8 a^3

(a is the scaling factor. It is the radius of the circle the Witch is constructed on)

Properties

Graphics Gallery

Normals, and Evolute of witch of Agnesi.

Osculating circles of the Witch.

Conchoids of witch of Agnesi.

Conhoids wrt a moving point (57 k)

Parallels 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.

© copyright 1995-97 by Xah Lee.