# Limacon Of Pascal

## 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.

## Formula

• 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.

### 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)

### Relation to Conic Sections

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

## Related Web Sites

Robert Yates: Curves and Their Properties.

The MacTutor History of Mathematics archive

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