Convert between Cartesian and Spherical systems
Step 1: Identify Current Coordinate System
Determine if you have Cartesian (x, y, z) or Spherical (r, θ, φ) coordinates.
Why: The direction of conversion determines which formulas to apply.
Step 2: Ensure All Three Values Are Known
Verify you have x, y, z (Cartesian) or r, θ, φ (Spherical) values.
Why: 3D space requires three independent values; missing one makes conversion impossible.
Step 3: Check Angle Units
Confirm if angles are in degrees or radians; conversions usually require radians.
Why: Trigonometric functions operate on radians; degree input produces incorrect results.
Step 4: Apply Conversion Formulas
Cartesian-to-Spherical: r = √(x² + y² + z²), θ = atan2(y, x), φ = acos(z/r).
Why: These formulas define the geometric relationships between the two coordinate systems.
Step 5: Verify Results Make Sense Geometrically
Check that r is non-negative and angles fall within expected ranges (0 to 2π for θ, 0 to π for φ).
Why: Out-of-range results indicate calculation errors or unconventional coordinate conventions.
r: Radial distance
θ (theta): Azimuthal angle in the xy-plane
φ (phi): Inclination angle from the z-axis
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