Coordinate Mapping Form
Example Data Table
| Source System | Input Values | Transform Applied | Target System | Mapped Result |
|---|---|---|---|---|
| Polar | r = 5, θ = 53.1301° | Scale X = 2, Scale Y = 1, Translate = (1, -2, 0) | Cartesian 2D | X = 7, Y = 2 |
| Cartesian 3D | X = 1, Y = 2, Z = 2 | No transform | Spherical | r = 3, θ = 63.4349°, φ = 48.1897° |
| Cylindrical | ρ = 4, θ = 30°, Z = 6 | Rotate Z = 15°, Translate = (0, 1, -2) | Cartesian 3D | X ≈ 1.0353, Y ≈ 4.8637, Z = 4 |
Formula Used
Polar to Cartesian: x = r cos(θ), y = r sin(θ)
Cylindrical to Cartesian: x = ρ cos(θ), y = ρ sin(θ), z = z
Spherical to Cartesian: x = r sin(φ) cos(θ), y = r sin(φ) sin(θ), z = r cos(φ)
Polar from Cartesian: r = √(x² + y²), θ = atan2(y, x)
Spherical from Cartesian: r = √(x² + y² + z²), θ = atan2(y, x), φ = arccos(z / r)
Scaling and shear: x′ = sx·x + shxy·y, y′ = sy·y + shyx·x, z′ = sz·z
Transformation order: convert to Cartesian, apply scale and shear, rotate around x, y, z, reflect, then translate.
How to Use This Calculator
- Select the source coordinate system and the target coordinate system.
- Enter the point values using the labels shown for the selected source system.
- Choose decimal precision for cleaner or more detailed output.
- Set any optional scaling, shearing, rotation, reflection, or translation values.
- Click Map Coordinates to show results above the form.
- Review the converted output, Cartesian summaries, displacement metrics, and graph.
- Use the CSV or PDF buttons to save the calculation.
FAQs
1. What does this coordinate mapping tool do?
It converts a point from one coordinate system to another, then applies optional transformations such as scaling, rotation, shearing, reflection, and translation before showing the mapped result.
2. Which coordinate systems are supported?
The tool supports Cartesian 2D, Cartesian 3D, Polar, Cylindrical, and Spherical systems. Every input is converted into Cartesian form internally before the final target output is computed.
3. How are spherical angles defined here?
Azimuth θ is measured in the xy-plane from the positive x-axis. Inclination φ is measured from the positive z-axis down toward the point, which matches a common mathematics and physics convention.
4. Why does the graph sometimes appear in 2D and sometimes in 3D?
The chart switches automatically. It uses 3D when the systems or mapped coordinates include meaningful z-values. Otherwise, it shows a 2D view for faster comparison of original and mapped positions.
5. What is the difference between displacement and mapped magnitude?
Mapped magnitude measures the distance of the mapped point from the origin. Displacement measures how far the mapped point moved away from the original point after all transformations were applied.
6. Can I use negative angles?
Yes. Negative angular values are accepted and interpreted using standard trigonometric direction rules. Negative radii are blocked for radial systems because they usually create ambiguity in practical coordinate mapping workflows.
7. What do the CSV and PDF exports include?
The CSV export saves the source, target, Cartesian values, output values, and summary metrics. The PDF export captures the on-screen result section so you can archive or share a clean report.
8. Is this tool suitable for teaching and modelling?
Yes. It is useful for classroom demonstrations, geometry checks, motion studies, graphics preparation, robotics path thinking, and coordinate validation tasks that need both numeric and visual feedback.