Pick a mode, enter the required values, and calculate. Angle inputs and outputs can use different units.
Sample conversions to validate your own inputs and expected outputs.
| Mode | Input | Output (approx.) |
|---|---|---|
| Cartesian → Polar | x=3, y=4 | r=5, θ≈53.130° |
| Polar → Cartesian | r=10, θ=30° | x≈8.660, y=5 |
| Cartesian → Cylindrical | x=1, y=1, z=2 | ρ≈1.414, φ=45°, z=2 |
| Cartesian → Spherical | x=0, y=0, z=5 | r=5, θ=0°, φ=0° |
- Cartesian → Polar (2D): r = √(x²+y²), θ = atan2(y, x)
- Polar → Cartesian (2D): x = r·cos(θ), y = r·sin(θ)
- Cartesian → Cylindrical (3D): ρ = √(x²+y²), φ = atan2(y, x), z = z
- Cylindrical → Cartesian (3D): x = ρ·cos(φ), y = ρ·sin(φ), z = z
- Cartesian → Spherical (3D): r = √(x²+y²+z²), θ = acos(z/r), φ = atan2(y, x)
- Spherical → Cartesian (3D): x = r·sin(θ)·cos(φ), y = r·sin(θ)·sin(φ), z = r·cos(θ)
- Affine (2D): Scale → rotate → translate using a 3×3 homogeneous matrix.
- Affine (3D): Scale → rotate (Z·Y·X) → translate, with a 3×3 rotation matrix.
- Select a transformation mode from the dropdown.
- Enter the required inputs for that mode (shown automatically).
- Choose input and output angle units, then set decimals and normalization.
- Click Calculate. Results appear above the form under the header.
- Use Download CSV or Download PDF to export the latest result.
Why coordinate systems matter
Engineers and analysts often record the same point in different coordinate frames. Polar and cylindrical systems simplify rotation-symmetric problems, while Cartesian coordinates suit linear algebra and plotting. Converting consistently reduces downstream error, especially when multiple tools exchange data. For example, a sensor at x=3, y=4 converts to r=5 and θ≈53.13°, which is easier to compare against a bearing. This calculator standardizes that handoff.
Angle units and normalization
Angles are frequently mixed across datasets: CAD exports may store radians, while reports use degrees. The unit controls here keep inputs and outputs explicit, and the optional normalization wraps azimuths to a standard range. That is helpful for dashboards where 359° and −1° should be treated as the same direction. Internally, stable atan2-based angles avoid quadrant mistakes and behave well near x≈0.
Choosing 2D or 3D modes
Pick 2D modes when z is irrelevant, such as map coordinates or planar mechanisms. Use cylindrical coordinates for pipes, turbines, and circular motion where ρ and φ capture radial behavior and z captures height. Use spherical coordinates for fields and directions where r, θ (inclination), and φ (azimuth) describe a full 3D orientation. The calculator follows the common convention θ from +z, improving comparability with physics texts.
Affine transforms for modeling
Beyond system conversion, many workflows require repositioning points. The affine options apply scaling, rotation, and translation in a controlled order, returning transformed coordinates plus a matrix for auditing. In 2D the 3×3 homogeneous matrix makes chaining transforms straightforward. In 3D the rotation matrix is built in Z·Y·X order, which aligns with many simulation pipelines. Use this for camera rigs, robotics frames, or CAD assembly offsets.
Interpreting exported results
Exports are designed for repeatability. The CSV includes inputs, outputs, and any matrix so you can paste results into spreadsheets or attach them to change logs. The PDF summary is useful for reviews, calculations files, and client communication. If you need consistent rounding across documents, set decimals before calculating; the same formatting is used in tables and exports. Store the settings alongside values to preserve context. For QA, compare sample rows against known trigonometric identities and tolerances daily.
What definition of spherical angles is used?
θ is the inclination measured from the positive z-axis, and φ is the azimuth in the x–y plane from the positive x-axis. This matches many physics and engineering references.
Why does the calculator use atan2?
atan2(y, x) selects the correct quadrant automatically and stays stable when x is near zero. It prevents sign and quadrant mistakes that happen with arctan(y/x).
Should I enable angle normalization?
Enable normalization when you need a consistent azimuth range for comparisons or charts, such as 0–360° or 0–2π. Disable it when you want the raw signed output for analysis.
What order is used for 3D rotations in affine mode?
The rotation matrix is built as Z·Y·X, meaning the point is rotated about X, then Y, then Z in sequence. This is common in simulation pipelines and helps keep results predictable.
How accurate are the results?
Accuracy is driven by floating-point math and your chosen decimals setting. For most practical work, 4–8 decimals is sufficient. Use consistent units and avoid mixing degrees and radians unintentionally.
Can I export without recalculating?
Yes. After you run a calculation, the latest result is stored in the session. Use the export buttons to download CSV or PDF again, even after refreshing the page.
Tip: For spherical coordinates, θ is inclination from +z, φ is azimuth in the x–y plane.