Calculator Input
Example Data Table
| Radius ρ | Azimuth θ | Polar φ | Unit | Convention | x | y | z |
|---|---|---|---|---|---|---|---|
| 10 | 45 | 60 | Degrees | Inclination | 6.1237 | 6.1237 | 5.0000 |
| 8 | 30 | 20 | Degrees | Elevation | 6.5104 | 3.7588 | 2.7362 |
| 5 | 1.5708 | 1.0472 | Radians | Inclination | 0.0000 | 4.3301 | 2.5000 |
Formula Used
Inclination convention: x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ), z = ρ cos(φ).
This mode treats φ as the polar angle measured downward from the positive z-axis.
Elevation convention: x = ρ cos(φ) cos(θ), y = ρ cos(φ) sin(θ), z = ρ sin(φ).
This mode treats φ as the elevation angle measured upward from the xy-plane.
The calculator converts degrees and gradians into radians before applying trigonometric functions.
How to Use This Calculator
- Enter a non-negative radius value.
- Enter the azimuth angle around the z-axis.
- Enter the polar or elevation angle.
- Select degrees, radians, or gradians.
- Choose the convention used by your textbook or project.
- Set the decimal precision for the final coordinates.
- Press the calculate button.
- Download the result as CSV or PDF when needed.
Advanced Guide to Spherical Coordinate Conversion
What This Calculator Does
Spherical coordinates describe a point by distance and two angles. They are useful when a point sits naturally around a center. Many geometry, physics, robotics, graphics, and engineering problems use them. This calculator converts those values into Cartesian coordinates. Cartesian coordinates give x, y, and z distances along three perpendicular axes.
Why Convention Matters
The main challenge is convention. Some courses use phi as an inclination angle. It starts at the positive z-axis. Other fields use phi as an elevation angle. It starts at the xy-plane. The two systems look similar, but they produce different z values. This tool includes both options, so the result can match your source.
Input Meaning
The radius shows distance from the origin. It cannot be negative. The azimuth angle rotates around the z-axis. It usually starts from the positive x-axis. The second angle controls vertical position. In inclination mode, zero points upward. In elevation mode, zero remains on the xy-plane.
Calculation Process
The calculator first reads the selected angle unit. Degrees and gradians are converted into radians. Radian values are then used by the sine and cosine functions. The x value depends on horizontal projection and cosine of azimuth. The y value depends on horizontal projection and sine of azimuth. The z value depends on the chosen vertical formula.
Result Checks
The cylindrical radius is also shown. It equals the distance from the z-axis. The distance check rebuilds the original radius from x, y, and z. This helps catch typing errors. It also confirms that rounding has not changed the point much.
Practical Uses
Use this calculator for vector problems, three-dimensional plotting, camera direction, antenna direction, orbital examples, and coordinate geometry. It is also helpful for checking homework steps. The export buttons make it easier to save one result for reports. The example table gives tested values for quick comparison.
Best Practice
Always confirm the angle convention before solving. A correct formula with the wrong convention gives a wrong point.
FAQs
What are spherical coordinates?
Spherical coordinates describe a point using a radius, an azimuth angle, and one vertical angle. They are common in three-dimensional maths, physics, graphics, and engineering problems.
What is the Cartesian result?
The Cartesian result is an ordered point written as x, y, and z. These values show distance along three perpendicular axes from the origin.
Which convention should I choose?
Choose inclination if your second angle is measured from the positive z-axis. Choose elevation if it is measured from the xy-plane.
Can the radius be negative?
This calculator requires a zero or positive radius. A negative radius can be handled in some theoretical systems, but it often causes confusion in standard coordinate conversion.
Are degrees and radians supported?
Yes. The calculator supports degrees, radians, and gradians. It converts degrees and gradians into radians before applying trigonometric formulas.
Why is the distance check included?
The distance check recomputes the radius from x, y, and z. It helps verify that the converted point still matches the original spherical distance.
What does cylindrical radius mean?
Cylindrical radius is the distance from the z-axis. It is calculated from x and y only, so it ignores vertical height.
Can I export the result?
Yes. After calculation, use the CSV or PDF buttons. CSV is useful for spreadsheets. PDF is useful for reports, notes, or printed records.