Polar Coordinate Converter

What are Polar Coordinates?

Polar coordinates represent points using a distance from the origin (radius r) and an angle (θ) from the positive x-axis. This system is particularly useful for circular and rotational motion.

Coordinate Systems

• Cartesian: (x, y) - rectangular coordinates
• Polar: (r, θ) - distance and angle

Conversion Formulas

Cartesian to Polar:

• r = √(x² + y²)
• θ = arctan(y/x) (with quadrant adjustment)

Polar to Cartesian:

• x = r × cos(θ)
• y = r × sin(θ)

Key Concepts

Radius (r)

The distance from the origin to the point. Always non-negative in standard notation.

Angle (θ)

Measured counterclockwise from the positive x-axis. Can be in degrees or radians.

Quadrant Considerations

When converting from Cartesian to polar, the angle must be adjusted based on the quadrant:

• Quadrant I: θ = arctan(y/x)
• Quadrant II: θ = arctan(y/x) + 180°
• Quadrant III: θ = arctan(y/x) + 180°
• Quadrant IV: θ = arctan(y/x) + 360°

Special Cases

• Origin: (0, 0) = (0, any angle)
• Positive x-axis: (a, 0) = (a, 0°)
• Positive y-axis: (0, b) = (b, 90°)
• Negative x-axis: (-a, 0) = (a, 180°)

Physics & Engineering

• Circular motion analysis
• Wave propagation
• Antenna radiation patterns
• Rotational mechanics
• Electromagnetic fields

Mathematics

• Complex number representation
• Trigonometric calculations
• Calculus in polar form
• Graphing polar equations
• Vector analysis

Computer Graphics

• 2D rotation algorithms
• Circular interpolation
• Radar displays
• Game development
• Animation systems