Trisectrix

above: Artistic rendering of trisectrix's conchoids.

Mathematica Notebook for This Page.

History

Description

Trisectrix is a general name for curves that can be used to trisect a angle. The name trisectrix is often reserved for a special case of limacon of Pascal, which we discuss here. Other famous trisectrix include trisectrix of Maclaurin and conchoid of Nicomedes.

Trisectrix can be generated as a conchoid or epitrochoid. (see limacon of Pascal for definition).

Constructing Trisectrix as a Conchoid

Formulas

• Polar: r == (1+2 * Cos[t]) , 0 ≤ t < 2 π.
• Cartesian: (-2 x + x^2 + y^2)^2 == x^2 + y^2 trisectrix.gcf

Properties

Trisecting a Angle

Let P be any point on the outer loop of the curve. The angle formed by P, {1,0}, {3,0} is three times larger than the angle {0,0}, P, {1,0}.

Graphic Gallery

Parallels of trisectrix.

Normals and tangent circles of a trisectrix. Note that the evolute of trisectrix is NOT nephroid.

Conchoids of trisectrix.
Conchoid of trisectrix with a moving point 1; Artistic Redering

string art based on trisectrix.

Trisectrix string art animation

trisectrix.pdf

Related Web Sites

See: Websites on Plane Curves, Printed References On Plane Curves.

The MacTutor History of Mathematics archive↗.

Wikipedia: Trisectrix↗.

© 1995-2008 by Xah Lee.