limacon - calculus
TRANSCRIPT
![Page 1: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/1.jpg)
CALCULuS
LIMACON
![Page 2: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/2.jpg)
FANTASTIC FOUR
PRESENTS
1.MOHIT BALHARA2.UTKARSH ANAND
3.ABHISHEK CHIIKARA4.HARIKESH KUMAR
![Page 3: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/3.jpg)
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
![Page 5: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/5.jpg)
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
![Page 6: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/6.jpg)
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
![Page 7: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/7.jpg)
The first one is the concaved. Second one is dimpled.The third one is cardioid.
CONSTRUCTION OF LIMACON
![Page 8: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/8.jpg)
Limacon — Pedal Curve of a Circle
![Page 9: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/9.jpg)
For , the inner loop has area.
=
=
= where .
AREA OCCUPIED BY LIMACON
![Page 10: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/10.jpg)
For , the outer loop has area
=
=
=
Area Contd.
![Page 11: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/11.jpg)
Thus, the area between the loops is
But there is a special case for b = a/2
Area Contd.
![Page 12: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/12.jpg)
In the special case of b=a/2 , these simplify to
Area Contd.
¿
¿
¿
![Page 13: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/13.jpg)
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
![Page 14: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/14.jpg)
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.
![Page 15: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/15.jpg)
Limacon Evolute means a limacon which is the locus of the centre of curvature of another limacon. Some examples are,
LIMACON EVOLUTE
![Page 16: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/16.jpg)
The catacaustic of a circle for a radiant point is the limacon evolute.
It has parametric equations.
=
=
Contd.
![Page 17: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/17.jpg)
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
![Page 18: Limacon - Calculus](https://reader035.vdocuments.mx/reader035/viewer/2022070602/587b14591a28abb15c8b6fbd/html5/thumbnails/18.jpg)
THANK YOU