Limaçon
The family r = b + a·cos θ that morphs from an inner-loop snail through the cardioid to a convex oval as b grows.
Adjust parameters
Export
Formulas and key values
- Polar equation
- Area · b ≥ a
a = 1 in the graph; b moves it from an inner loop (b < a) through the cardioid (b = a) to a convex curve (b ≥ 2a).
Explainer video
Make a video of the Limaçon
Opens a video template with the parameters on screen. Every number in the video is computed before it renders; choose landscape or vertical.
Start the video01 · Definition
What is the Limaçon?
The limaçon — French for “snail” — is the polar family r = b + a·cos θ, studied by Étienne Pascal (father of Blaise) in the early 1600s. Geometrically it is the path of a point rigidly attached to a circle rolling around another circle of the same radius, where the point need not lie on the rim.
One slider tells the whole story. With a = 1: for b < 1 the radius goes negative and the curve crosses itself in an inner loop; at b = 1 the loop collapses to the cardioid’s cusp; for 1 < b < 2 the curve is dimpled; and from b ≥ 2 on it is a convex oval. It is the standard example of how a single constant reshapes a polar graph.
02 · Properties
Key properties
- b < a: inner loop (the radius changes sign); b = a: cardioid.
- a < b < 2a: dimpled limaçon; b ≥ 2a: convex.
- Enclosed area (b ≥ a): π(b² + a²/2).
- It is the conchoid of a circle with respect to a point on the circle.
- The inner-loop case is a classic area-between-loops integration exercise.
03 · Questions
Limaçon: common questions
- What is the equation of the Limaçon?
- Polar equation: r = b + a cos θ. a = 1 in the graph; b moves it from an inner loop (b < a) through the cardioid (b = a) to a convex curve (b ≥ 2a).
- What is the area of the Limaçon?
- Area (b ≥ a): A = π(b^2 + (a^2)/2). a = 1 in the graph; b moves it from an inner loop (b < a) through the cardioid (b = a) to a convex curve (b ≥ 2a).
04 · Keep exploring