Limaçons as Loci and Other Polar Curves

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

A limaçon is the locus of a point that lies on a variable line (obtained by varying the angle ) passing through a fixed point (the pole, taken to be the origin) on a circle with radius (shown dashed); is a fixed distance (shown with a purple line) from the other point of intersection of the line with the circle. By varying and , 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.

Contributed by: Roberta Grech (June 2012)
Open content licensed under CC BY-NC-SA




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

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.