Coordinate Transformation Calculator
Use two inputs for 2D systems. The third field is needed for 3D systems and ignored for 2D conversions.
Visualization
Example Data Table
| Case | Input System | Input Values | Converted Output | Insight |
|---|---|---|---|---|
| 1 | Cartesian (2D) | x = 3, y = 4 | r = 5, θ ≈ 53.13° | Classic right triangle relationship. |
| 2 | Polar | r = 10, θ = 30° | x ≈ 8.66, y = 5.00 | Radius projects into horizontal and vertical components. |
| 3 | Cartesian (3D) | x = 2, y = 2, z = 5 | ρ ≈ 2.83, θ = 45°, r ≈ 5.74 | Cylindrical radius ignores the vertical component. |
| 4 | Cylindrical | ρ = 6, θ = 60°, z = 2 | x = 3.00, y ≈ 5.20, r ≈ 6.32 | Planar radius and height define spherical distance. |
| 5 | Spherical | r = 8, θ = 45°, φ = 60° | x ≈ 4.90, y ≈ 4.90, z = 4.00 | Azimuth controls direction, polar angle controls elevation. |
Formula Used
This tool converts between common two-dimensional and three-dimensional coordinate systems. It applies standard trigonometric and distance relationships. For spherical conversion, the convention is azimuth in the xy-plane and polar angle measured from the positive z-axis.
- 2D Cartesian to Polar: r = √(x² + y²), θ = atan2(y, x)
- Polar to 2D Cartesian: x = r cos θ, y = r sin θ
- 3D Cartesian to Cylindrical: ρ = √(x² + y²), θ = atan2(y, x), z = z
- Cylindrical to 3D Cartesian: x = ρ cos θ, y = ρ sin θ, z = z
- 3D Cartesian to Spherical: r = √(x² + y² + z²), θ = atan2(y, x), φ = arccos(z / r)
- Spherical to 3D Cartesian: x = r sin φ cos θ, y = r sin φ sin θ, z = r cos φ
How to Use This Calculator
- Select the source coordinate system that matches your available data.
- Choose whether your angular values are in degrees or radians.
- Enter the point label if you want named results and chart labeling.
- Fill in the numeric fields. Use two values for 2D systems and three for 3D systems.
- Set decimal precision to control output detail in tables and exports.
- Press Transform Coordinates to view results above the form.
- Review the graph to verify direction, quadrant, and spatial placement.
- Download CSV or PDF if you need to share or archive the result.
Frequently Asked Questions
1. What systems does this tool support?
It supports Cartesian 2D, polar, Cartesian 3D, cylindrical, and spherical coordinates. The tool converts within matching dimensional groups and visualizes the resulting point automatically.
2. Does the angle unit affect the output?
Yes. Input angles are interpreted using the selected unit. Output angles are displayed in the same chosen unit, making review and export more consistent.
3. How is the spherical angle defined here?
This tool uses θ as the azimuth in the xy-plane and φ as the polar angle from the positive z-axis. Always confirm this convention when comparing with textbooks or software.
4. Why use atan2 instead of arctan?
atan2 identifies the correct quadrant by using both horizontal and vertical components. That prevents angle ambiguity, especially when x is zero or negative.
5. Can I use negative radii or distances?
You can enter them, but interpretation depends on your mathematical convention. In most practical coordinate systems, radial distance is treated as nonnegative for clarity and consistency.
6. Why is the third input ignored sometimes?
For two-dimensional systems, only two values define the point. The third input remains available because the same form also handles three-dimensional transformations.
7. What do the CSV and PDF downloads contain?
They export the current transformation table shown in the results section. This makes it easy to document coordinate conversions for reports, classes, and engineering notes.
8. Is the graph useful for accuracy checks?
Yes. The plot helps confirm sign direction, quadrant, and relative distance from the origin. It is especially useful when validating angle conventions and spatial interpretation.