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Limacon - Calculus
- 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+acosΘ 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 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
- 6. 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
- 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=2𝜋 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