Limacon of Pascal

Mathematica Notebook for This Page.

History

Discovered and named after Etienne Pascal, father of Blaise Pascal. Also discussed by Roberval in 1650.

From Howard Eves's “An Intro to the hist of math”, 6th ed., 1989. Problem Studies 4.7, p.128.:

… the curve had already been given by Albrecht Durer (1471 〜 1528) in the early sixteenth century.

Description

Limacon of Pascal describe a family of curves. It is a special case of epitrochoid. (See: Curve Family Index) It can also be defined as a conchoid of a circle. Cardioid and trisectrix are special cases of Limacon of Pascal.

Limacon of Pascal as a conchoid:

1. Let there be a fixed point O on a circle.
2. Draw a line passing O and P, where P is any point on the circle.
3. On this line, mark points Q1 and Q2 such that distance[P,Q1] == distance[P,Q2] == k, where k is a constant.
4. Repeat step 2, 3 for different choice of P. The locus of Q is the Limacon of Pascal.

Formulas

• Parametric: (k + 2 r Cos[t]) {Cos[t], Sin[t]}.
• Polar: R == (k + 2 r Cos[t]).
• Cartesian: (x^2 + y^2 -2 r x)^2 == k^2 (x^2 + y^2).

Properties

Special Cases

• if 2 r > k, there is a inner loop.
• if 2 r == k, there is a cusp. (it is a cardioid)
• If 2 r ≤ k, it is a dimpled limacon.
• If r == k, it is the trisectrix

Epitrochoid

Limacon of Pascal is a special case of epitrochoid, when the rolling and fixed circles has equal radius. i.e., it is the trace of a point Q fixed to a circle that rolls around another circle of the same size.

Let radius of circle B and A be r, and Let the distance from the tracing point Q to the center of circle B be h. The parametric formula is then {2 r Cos[t] + h Cos[2 t], 2 r Sin[t] + h Sin[2 t]} with a period of 2 π.

Pedal

Limacon of Pascal is the pedal of a circle with respect to any point in the plane. It is also the envelope of circles with centers on a given circle C and each circle passing through a fixed point P in the plane. (See: limacon of Pascal graphics gallery)

Moving Pedal Point Curve Tracing

Moving Point P Curve Tracing

Relation to Conic Sections

Limacon of Pascal is the inversion of conic sections with respect to a focus.

Limacon of Pascal Graphics Gallery.

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.

limacon of Pascal.