Limacon - Calculus
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The document discusses the limacon, a polar curve defined by the equation r=b+acos, detailing its historical origins, properties, and different forms based on parameters a and b. It explains conditions for its shapes, such as being convex, dimpled, or having an inner loop, and provides equations related to its area and arc length. Additionally, references and examples of the limacon are included, highlighting its significance in mathematics.
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- 2. FANTASTIC FOUR PRESENTS 1.MOHIT BALHARA 2.UTKARSH ANAND 3.ABHISHEK CHIIKARA 4.HARIKESH KUMAR
- 3. ? The Limacon is a polar curve of the form having Equation r=b+acos ? Cartesian Equation (x2 + y2 - 2ax)2 = b2(x2 + y2) ? Some basic curves of limacon. INTRODUCTION
- 4. Some other examples of Limacon
- 5. ? Limacon is also called the Limacon of Pascal. It was first investigated by Drer, who gave a method for drawing it in book Underweysung der Messung (1525). It was rediscovered by ?tienne 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
- 6. ? As we know the equation of Limacon is r=b+acos? ? 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
- 7. The first one is the concaved. Second one is dimpled. The third one is cardioid. CONSTRUCTION OF LIMACON
- 8. Limacon Pedal Curve of a Circle
- 9. ? For , the inner loop has area. = = = where . AREA OCCUPIED BY LIMACON
- 10. ? For , the outer loop has area = = = Area Contd.
- 11. ? Thus, the area between the loops is ? But there is a special case for b = a/2 Area Contd.
- 12. ? In the special case of b=a/2 , these simplify to Area Contd. = = =
- 13. ? 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
- 14. ? 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.
- 15. ? Limacon Evolute means a limacon which is the locus of the centre of curvature of another limacon. Some examples are, LIMACON EVOLUTE
- 16. ? The catacaustic of a circle for a radiant point is the limacon evolute. It has parametric equations. = = Contd.
- 18. THANK YOU