CALCULuS

LIMACON

FANTASTIC FOUR

PRESENTS

1.MOHIT BALHARA2.UTKARSH ANAND

3.ABHISHEK CHIIKARA4.HARIKESH KUMAR

The Limacon is a polar curve of the form having Equation r=b+acosΘ

Cartesian Equation (x2 + y2 - 2ax)2 = b2(x2 + y2)

Some basic curves of limacon.

INTRODUCTION

Some other examples of Limacon

Limacon is also called the Limacon of Pascal. It was first investigated by Dürer, who gave a method for drawing it in book Underweysung der Messung (1525). It was rediscovered by Étienne Pascal, father of Blaise Pascal, and named by Gilles-Personne Roberval in 1650 (MacTutor Archive).

The word “Limacon" comes from the Latin limax, meaning "snail."

HISTORY

As we know the equation of Limacon is r=b+acosѲ

In it, If,b>=2a the Limacon is convex. If, 2a>b>a the Limacon is dimpled. If, b=a the Limacon degenerates to

a cardioid. If, b<a the Limacon has an inner loop. If, b=a/2 it is a trisectrix.

SOME BASIC CONDITIONS

The first one is the concaved. Second one is dimpled.The third one is cardioid.

CONSTRUCTION OF LIMACON

Limacon — Pedal Curve of a Circle

For  , the inner loop has area.

=

=

= where  .

AREA OCCUPIED BY LIMACON

For  , the outer loop has area

=

=

=

Area Contd.

Thus, the area between the loops is

But there is a special case for b = a/2

Area Contd.

In the special case of  b=a/2 , these simplify to

Area Contd.

¿

¿

¿

The parametric form of Limacon is: x= (b + a cos t) cos t y = (b + a cos t) sin t gives the arc length  s(t) as a function of  t as

where E(z,k) is an elliptic integral of the second kind.

PARAMETRIC FORM

Let t=2  gives the arc length of the entire curve as

where E(k)  is a complete elliptic integral of the second kind.

Contd.

Limacon Evolute means a limacon which is the locus of the centre of curvature of another limacon. Some examples are,

LIMACON EVOLUTE

The catacaustic of a circle for a radiant point is the limacon evolute.

It has parametric equations.

=

=

Contd.

en.wikipedia.org/wiki/Limaçon

http://mathworld.wolfram.com/Limacon.html

http://www-history.mcs.st-and.ac.uk/Curves/Limacon.html

www.mathwords.com/l/limacon.htm

BIBLIOGRAPHY

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