Enter a Three-Dimensional Point
Use matching length units for all three coordinates.
Formula Used
The calculator treats x and y as the horizontal plane, then uses z as the vertical coordinate. It reports both the two-dimensional projection and the full three-dimensional polar description.
Spatial radius: ρ = √(x² + y² + z²)
Azimuth: θ = atan2(y, x)
Elevation: α = atan2(z, r)
Inclination: φ = arccos(z / ρ)
Why atan2 is used: atan2 reads both x and y. It preserves the correct quadrant, unlike a basic arctangent calculation. The normalized azimuth converts a negative direction into the usual positive rotation range.
How to Use This Calculator
- Enter the point's x, y, and z rectangular coordinates.
- Choose degrees or radians for every reported angle.
- Select elevation or inclination for the main vertical angle.
- Set the decimal precision needed for your measurements.
- Select Convert Coordinates to view results above the form.
- Check the location label and calculation steps before using values elsewhere.
Example Data
These examples use degrees and show common coordinate situations.
| Point (x, y, z) | Planar Radius r | Spatial Radius ρ | Azimuth θ | Elevation α |
|---|---|---|---|---|
| (3, 4, 0) | 5.000 | 5.000 | 53.130° | 0.000° |
| (3, 4, 5) | 5.000 | 7.071 | 53.130° | 45.000° |
| (-3, 4, 5) | 5.000 | 7.071 | 126.870° | 45.000° |
| (0, 0, 8) | 0.000 | 8.000 | Undefined | 90.000° |
Rectangular Coordinates and Polar Position
Rectangular coordinates describe location with horizontal, vertical, and depth values. A three-point entry uses x, y, and z. Polar conversion describes that same position with distance and angles. The calculator first reads the x and y plane. It then includes z to measure full three-dimensional distance. This makes the result useful for drafting, engineering, robotics, surveying, and science exercises.
What the Results Mean
The planar radius measures distance from the origin to the point projection on the xy plane. The spatial radius measures direct distance from the origin to the entered point. Azimuth identifies the rotation around the z axis. Elevation shows the upward or downward tilt from the xy plane. Inclination gives the alternative angle measured downward from the positive z axis. Both vertical angles describe the same point differently.
Quadrants Matter
An inverse tangent can lose quadrant information. For example, points with positive x and negative y need a fourth-quadrant azimuth. This tool uses atan2 logic. It examines both horizontal coordinates before selecting the angle. Therefore, the displayed azimuth remains correct for every quadrant and for points placed directly on an axis. The normalized angle is also provided for conventions that require positive rotations only.
Choosing Degrees or Radians
Degrees are common in navigation, drawing, and classroom geometry. Radians are common in programming, higher mathematics, and trigonometric modeling. Select the unit that matches your work. The calculator converts all reported angles consistently. Radius values do not change when you switch units. Only the angle labels and numerical angle values change. Keep the selected unit consistent when copying results into another calculation.
Special Cases
The origin has zero spatial radius. Direction is undefined there because no line extends away from the origin. A point on the z axis has no unique azimuth. The calculator identifies those cases instead of presenting a misleading direction. Points on the xy plane have zero elevation. Points below that plane have negative elevation. These checks help you interpret results accurately before using them in designs or models.
Practical Accuracy Tips
Use the same length unit for x, y, and z. You may enter millimeters, meters, feet, or another consistent unit. The output radius uses that same unit. Increase decimal places when measurements contain small differences. Reduce decimal places for simple estimates. Review the calculation steps whenever values appear unexpected. A negative coordinate is valid and often changes only the direction. It does not automatically make distance negative.
Where This Conversion Helps
Three-dimensional polar values simplify many position problems. A robot arm can use distance and direction to approach a target. A surveyor can describe an elevated point from a reference station. A developer can place objects around a camera. A student can check spherical-coordinate homework. The calculator offers both projection information and full spatial information, making it useful when a task requires more detail than a two-dimensional polar conversion. Compare measurements across different coordinate systems. Communicate locations clearly across technical teams. Conversion prevents avoidable angle mistakes during modeling. It supports repeatable technical work.
Frequently Asked Questions
1. What coordinates does this calculator accept?
It accepts one three-dimensional rectangular point: x, y, and z. Each value may be positive, negative, or zero. Use one consistent length unit for all entries.
2. What is the difference between planar and spatial radius?
Planar radius measures the point's distance from the origin on the xy plane. Spatial radius measures direct distance from the origin through three-dimensional space.
3. Why does the calculator use atan2?
atan2 uses both x and y values. This identifies the correct quadrant and avoids incorrect azimuths that can occur with a basic inverse tangent formula.
4. What does a negative elevation mean?
A negative elevation means the point lies below the xy plane. The point still has a positive spatial radius because distance cannot be negative.
5. Why is azimuth undefined on the z axis?
Every horizontal direction reaches the z axis. Because the point has no x or y projection, there is no single azimuth to report.
6. Should I choose degrees or radians?
Choose the unit required by your task. Degrees work well for geometry and navigation. Radians are often preferred in programming, calculus, and scientific formulas.
7. What is inclination?
Inclination is the angle measured from the positive z axis toward the point. Elevation instead starts at the xy plane. Both describe the same direction.
8. Can I enter decimal values?
Yes. Enter decimal, whole-number, positive, or negative coordinate values. Then select a suitable number of decimal places for the displayed results.
9. What happens when all coordinates are zero?
The point is the origin. Both radii are zero, but azimuth and vertical direction are undefined because the point has no outward direction.
10. Does changing angle units alter the radius?
No. Switching between degrees and radians changes only angle values. The planar and spatial radii stay unchanged because they are distances.
11. Is this the same as spherical coordinates?
It provides the core spherical-coordinate values: spatial radius, azimuth, and a vertical angle. It also gives the xy projection radius for extra clarity.