Limacon Of Pascal

limacon of Pascal limacon of Pascal
Left: Several Limacon of Pascal, having different parameters. Right: A artistic rendering.

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History

Discovered and named after Étienne Pascal (1588 – 1651), father of Blaise Pascal (1623 – 1662). Also discussed by Gilles de Roberval (1602 – 1675) in 1650.

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

An Introduction to the History of Math Buy at amazon By Howard Eves, 6th edition, Problem Studies 4.7, p.128.

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.
limacon of Pascal
Limacon as a Conchoid Limacon Family
conOfCircle r1k1
r = 1, k = 1
conOfCircle r1k2
r = 1, k = 2
conOfCircle r1k3
r = 1, k = 3

Formula

Properties

Special Cases

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.

limacon of Pascal
Limacon as Epitrochoid
epitrochoid 1 1 2
Epitrochoid {1,1,2}

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.

LimaconGG2 2 conOfCircle1
Moving Pedal Point. Curve Tracing

Envelope of Circles

Limacon of Pascal is 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)

LimaconGG4 2 limaconOfPEnvOfCirc
Limacon of Pascal is the envelope of circles, with 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 limaconOfPInvCon
Inversion of Limacons

Limacon 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.