1. What is a Polar Graph?

A polar graph represents mathematical functions in polar coordinates where each point is determined by a distance from a reference point (r) and an angle from a reference direction (θ). This system is useful for graphing equations that have circular or rotational symmetry.

2. How Does the Polar Grapher Work?

The grapher converts polar equations to Cartesian coordinates using:

Where:

• \( r(\theta) \) — The polar equation (function of θ)
• \( \theta \) — Angle in radians
• \( x, y \) — Cartesian coordinates

3. Common Polar Equations

Examples:

• Circle: r = a
• Cardioid: r = a(1 ± cosθ)
• Rose curves: r = a cos(bθ) or r = a sin(bθ)
• Archimedean spiral: r = aθ
• Lemniscate: r² = a² cos(2θ)

4. Using the Calculator

Tips: Enter your polar equation using θ as the variable. Use parameters a, b, etc. for constants that you want to adjust. Set the θ range (typically 0 to 2π for complete graphs).

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between polar and Cartesian graphs?
A: Polar graphs use angle and distance from origin, while Cartesian graphs use x,y coordinates. Polar is better for circular/rotational patterns.

Q2: How do I graph r = sin(3θ)?
A: This creates a 3-petal rose curve. Enter "sin(3*theta)" in the equation field.

Q3: Why is my graph incomplete?
A: You may need to increase θ Max to 2π (6.28) or more for complete patterns.

Q4: Can I graph multiple equations?
A: This calculator graphs one equation at a time. For multiple graphs, plot them separately.

Q5: How do I make a spiral?
A: Try equations like "a*theta" (Archimedean spiral) or "a^theta" (logarithmic spiral).