Limaçons as Loci and Other Polar Curves

polar curve

limaçon: r = a + b cos(θ)

a

1

b

1

θ

0.78

n

1

k

1

m

1

d

1

α

0.78

A limaçon is the locus of a point

P

that lies on a variable line (obtained by varying the angle

θ

) passing through a fixed point (the pole, taken to be the origin) on a circle with radius

b/2

(shown dashed);

P

is a fixed distance

a

(shown with a purple line) from the other point of intersection of the line with the circle. By varying

a

and

b

, various types of limaçons are obtained, namely the circle, trisectrix, cardioid, limaçon with inner loop, dimpled limaçon, and oval (convex) limaçon. In addition, other popular plane curves (roses, spirals, lines, and lemniscates) with their polar equations are shown.

References

[1] G. B. Thomas, Jr., Thomas' Calculus, 11th ed., Upper Saddle River, NJ: Pearson, 2005 pp. 714–725.

Permanent Citation

Roberta Grech

"Limaçons as Loci and Other Polar Curves"
http://demonstrations.wolfram.com/LimaconsAsLociAndOtherPolarCurves/
Wolfram Demonstrations Project
Published: June 3, 2012