One-loop limaçon

Definition

A one-loop limaçon is a type of polar curve described by the equation $r = a + b \cos(\theta)$ or $r = a + b \sin(\theta)$, where $|a| > |b|$. It forms a single loop and resembles a distorted circle.

5 Must Know Facts For Your Next Test

1. The general form of the equation of a one-loop limaçon is $r = a + b \cos(\theta)$ or $r = a + b \sin(\theta)$.
2. For a one-loop limaçon, the absolute value of $a$ must be greater than the absolute value of $b$ ($|a| > |b|$).
3. When graphing, if $a > 0$ and $b < 0$, the loop will appear on the opposite side compared to when both are positive.
4. One-loop limaçons have no inner loops because the condition $|a| > |b|$ prevents any loops within the curve.
5. The graph's appearance can change significantly based on whether $\cos(\theta)$ or $\sin(\theta)$ is used in the equation.

Review Questions

• What are the conditions for an equation to represent a one-loop limaçon?
• How does changing from $\cos(\theta)$ to $\sin(\theta)$ in the equation affect the orientation of a one-loop limaçon?
• Explain why a one-loop limaçon cannot have an inner loop.

Related terms

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

Cardioid: A special type of limaçon with exactly one cusp, represented by the equation $r = a + b \cos(\theta)$ or $r = a + b \sin(\theta)$, where $|a| = |b|$.

$n$-leaf Rose: $n$-leaf rose curves are described by equations like $r = a \cos(n \theta)$ or $r = a \sin(n \theta)$. They produce petal-like graphs with 'n' leaves if 'n' is odd, and '2n' leaves if 'n' is even.