1. What is the Limacon Polar Equation?

The limacon is a polar curve defined by the equation r = a + b cosθ (or r = a + b sinθ). It produces various shapes including cardioids, dimpled limacons, and loops depending on the relationship between parameters a and b.

2. How Does the Calculator Work?

The calculator uses the limacon equation:

Where:

• \( a \) — Constant parameter (unitless)
• \( b \) — Constant parameter (unitless)
• \( \theta \) — Angle in radians
• \( r \) — Radial distance from origin (unitless)

Explanation: The equation calculates the radial distance r for a given angle θ based on the specified parameters a and b.

3. Applications of Limacon Curves

Details: Limacon curves have applications in mathematics, physics, engineering, and computer graphics. They are used in gear design, antenna radiation patterns, and various geometric modeling applications.

4. Using the Calculator

Tips: Enter parameters a and b (unitless values) and angle θ in radians. The calculator will compute the corresponding radial distance r.

5. Frequently Asked Questions (FAQ)

Q1: What shapes can a limacon produce?
A: Depending on the ratio a/b, limacons can produce cardioids (a=b), dimpled limacons (1 < a/b < 2), loops (a/b < 1), or convex shapes (a/b ≥ 2).

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

Q3: What are some special cases of limacons?
A: When a = 0, it becomes a circle. When a = b, it becomes a cardioid. When b = 0, it becomes a circle centered at the origin.

Q4: Can the calculator handle negative values?
A: Yes, the calculator accepts negative values for parameters a, b, and angle θ.

Q5: How accurate are the results?
A: Results are calculated with high precision using PHP's built-in trigonometric functions, typically accurate to about 14 decimal places.