Limacon of Pascal

above: Limacon of Pascal(s) and a artistic reflection.

Mathematica Notebook for This Page.

History

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

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.

Epitrochoid {1,1,2}
Limacon as Epitrochoid

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 Pi.

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.

Inversion of Limacons

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↗.

Wikipedia: limacon of Pascal↗.