Limaçon

Definition

A limaçon is a type of polar curve defined by the equation $r = a + b\cos(\theta)$ or $r = a + b\sin(\theta)$. Its shape varies based on the values of $a$ and $b$, resulting in different forms including loops, dimpled shapes, or cardioid-like figures.

5 Must Know Facts For Your Next Test

1. The general form of the limaçon equation is $r = a + b\cos(\theta)$ or $r = a + b\sin(\theta)$.
2. If \(|a| = |b|\), the limaçon produces a cardioid shape.
3. If \(|a| > |b|\), the limaçon will have an outer loop but no inner loop (dimpled).
4. If \(|a| < |b|\), the limaçon contains an inner loop.
5. Limaçons with $b = 0$ reduce to circles with radius $a$.

Review Questions

• What are the standard equations for a limaçon in polar coordinates?
• How does varying the relationship between $a$ and $b$ affect the shape of a limaçon?
• What specific condition results in a cardioid-shaped limaçon?

Related terms

Polar Coordinates: A coordinate system where each point on a plane is determined by an angle and distance from a reference point.

Cardioid: A special type of limaçon where $|a| = |b|$, resulting in a heart-shaped curve.

$r=acos(kθ)$: $A type of polar equation that produces spiral shapes known as rose curves when k is an integer.$