3D coordinate converter

How to convert 3D coordinates

1. Choose Cartesian, cylindrical or spherical as the source, then enter only the three visible values.
2. Angles use degrees. Azimuth is counterclockwise from +x in the xy-plane; polar angle is from +z and must be 0°–180°.
3. Select Calculate to show all three equivalent representations. Editing a value or source mode invalidates the old result.
4. Copy records the three forms, including any undefined-angle notice. Clear removes numeric inputs but keeps the selected source.

Formulas and conventions

Cartesian uses signed x, y and z. Cylindrical radius r = √(x²+y²), while spherical radius ρ = √(x²+y²+z²).

Azimuth θ = atan2(y,x). Polar angle φ = atan2(r,z), which is equivalent to arccos(z/ρ) away from the origin.

For cylindrical input, x=r cosθ and y=r sinθ. For spherical input, x=ρ sinφ cosθ, y=ρ sinφ sinθ and z=ρ cosφ.

r = √(x²+y²) · ρ = √(x²+y²+z²) · θ = atan2(y,x) · φ = atan2(r,z)

Worked examples

Limits and undefined angles

• This is mathematical 3D coordinate conversion, not GPS, latitude/longitude, a map, camera pose or 3D rendering.
• Cylindrical and spherical radii may be zero but not negative. Spherical polar angle is limited to 0°–180°; azimuth input may be any degree value and is normalized.
• At the origin, radius is zero and both angles are undefined. The output says so instead of inventing a unique direction.
• On the positive or negative z-axis away from the origin, polar angle is defined but azimuth is not unique; the output marks azimuth undefined.

Frequently asked questions