1. What is a Limacon?

A limacon is a polar curve defined by the equation r = a + b*cos(θ) or r = a + b*sin(θ). It's a member of the family of curves called conchoids and can produce various shapes including cardioids, dimpled limacons, and convex limacons depending on the ratio of a to b.

2. How Does the Calculator Work?

The calculator uses the limacon polar equation:

Where:

• \( r \) — Radius (distance from origin)
• \( a \) — Constant term
• \( b \) — Coefficient of cosine term
• \( \theta \) — Angle in radians

Explanation: The equation describes how the radius varies with angle in polar coordinates. The shape of the curve depends on the relationship between a and b.

3. Importance of Limacon Calculation

Details: Limacons are important in mathematics for studying polar curves and have applications in physics, engineering, and computer graphics for modeling periodic phenomena.

4. Using the Calculator

Tips: Enter constants a and b, and the angle θ in radians. The calculator will compute the radius r at that angle.

5. Frequently Asked Questions (FAQ)

Q1: What shapes can a limacon produce?
A: When a/b < 1, it has an inner loop; when a/b = 1, it's a cardioid; when 1 < a/b < 2, it's dimpled; when a/b ≥ 2, it's convex.

Q2: Can I use degrees instead of radians?
A: The equation requires radians. Convert degrees to radians by multiplying by π/180.

Q3: What's the difference between cos(θ) and sin(θ) versions?
A: They're rotated versions of each other. The cos version is symmetric about the x-axis, while sin is symmetric about the y-axis.

Q4: Are there real-world applications of limacons?
A: Yes, they appear in gear design, antenna radiation patterns, and the study of planetary orbits.

Q5: What happens when b = 0?
A: The equation reduces to r = a, which is simply a circle with radius a.