Convert your Cartesian equation point
Use the origin-based point values from a Cartesian coordinate system.
Example Data Table
These examples use the standard convention. θ is azimuth. φ is the polar angle from the positive z-axis.
| Cartesian point | ρ | θ | φ |
|---|---|---|---|
| (1, 1, 1) | 1.7321 | 45° | 54.7356° |
| (3, 4, 0) | 5 | 53.1301° | 90° |
| (0, 0, 5) | 5 | Undefined | 0° |
| (-2, 2, -1) | 3 | 135° | 109.4712° |
Formula Used
The calculator converts a Cartesian point (x, y, z) into spherical coordinates measured from the origin.
ρ = √(x² + y² + z²)θ = atan2(y, x) for the azimuth angle.φ = arccos(z / ρ) for the standard polar angle.α = atan2(z, √(x² + y²)) for elevation from the xy-plane.The atan2 function preserves the correct quadrant. The calculator marks directional angles as undefined at the origin. It also marks azimuth as undefined anywhere on the z-axis.
How to Use This Calculator
Type the x, y, and z values from your Cartesian equation or coordinate point.
Choose standard polar notation or elevation notation. Match the convention your work requires.
Use degrees for visual interpretation. Use radians for formulas, coding, and many scientific tasks.
Pick two, four, six, or eight decimal places before converting.
Read rho, planar radius, and both angles. Check the note for axis-related cases.
Use CSV for tables. Use PDF for a compact calculation record.
Cartesian to Spherical Coordinates Explained
Cartesian coordinates describe a location with three straight measurements. The x value moves left or right. The y value moves forward or backward. The z value moves up or down. This approach is easy for grids, boxes, and engineering drawings. Spherical coordinates describe the same location through distance and direction. They use a radius plus two angles. The radius reaches outward from the origin. One angle turns around the vertical axis. The other angle measures direction from the positive z-axis.
The conversion begins by finding the radial distance. Square each coordinate value. Add the squares. Then take the square root. This produces rho, the total distance from the origin. A point close to the origin has a small rho. A point farther away has a larger rho. The azimuth angle comes from the x and y values. It shows the point's rotation around the z-axis. The polar angle uses z and rho. It shows how far the point sits from the positive z-axis.
Angle conventions matter in three-dimensional work. This calculator uses the common physics convention by default. Theta is the azimuth. Phi is the polar angle. Theta begins on the positive x-axis and moves toward the positive y-axis. Phi begins on the positive z-axis and moves downward. Geographic and graphics work sometimes use elevation instead. Elevation starts at the xy-plane. It becomes positive above that plane. The calculator can display that alternative form too.
The atan2 function is important for correct azimuth results. It considers both x and y signs. Therefore, it identifies the proper quadrant. A basic arctangent of y divided by x cannot always do this. It may place points on the wrong side of the coordinate system. The calculator also normalizes negative azimuth values. This keeps standard angles between zero and one full turn. Use degrees when reviewing familiar diagrams. Use radians for programming, science, and symbolic work.
The origin requires special treatment. When x, y, and z are all zero, the radius is zero. Direction has no unique meaning there. The calculator clearly labels those angles as undefined. Points on the z-axis also have an undefined azimuth. Every azimuth points to the same axis line. This is expected mathematical behavior, not an error.
Spherical conversion helps with spheres, rotations, radar ranges, navigation models, and field equations. It can simplify problems with radial symmetry. Always check the selected angle convention before sharing an answer. Record the unit for every angle. Keep enough decimal places for your task. Then substitute the result back into the reverse equations when verification matters. Careful notation prevents misunderstandings between textbooks, software, and technical teams.
Before finalizing a result, inspect units, rounding, and the displayed coordinate order. Small input errors can change an angle greatly near an axis. Save exported values for reports or homework. Recalculate whenever the reference convention, measurement source, or required precision changes during future reviews and calculation comparisons.
Frequently Asked Questions
1. What does rho represent?
Rho is the straight-line distance from the origin to the Cartesian point. It is always zero or positive. It is found by taking the square root of x squared, y squared, and z squared added together.
2. What is the difference between polar angle and elevation?
The polar angle starts at the positive z-axis. Elevation starts at the xy-plane. They describe the same vertical direction from different reference lines. Select the convention required by your equation, textbook, or application.
3. Why does the calculator use atan2?
Atan2 uses both x and y values. It identifies the correct azimuth quadrant. A basic arctangent can lose this information and return an angle on the wrong side of the axis.
4. Why is azimuth undefined on the z-axis?
On the z-axis, x and y are both zero. There is no planar direction to measure. Any azimuth describes the same vertical axis, so one unique azimuth cannot be assigned.
5. What happens at the origin?
The radius is zero at the origin. No direction exists because the point does not extend away from the origin. Both directional angles are therefore undefined.
6. Should I use degrees or radians?
Use degrees for diagrams, everyday interpretation, and many classroom exercises. Use radians for calculus, programming, and technical formulas. Match the requested unit before submitting an answer.
7. Can Cartesian coordinates be negative?
Yes. Any of x, y, and z may be negative. Negative values change the point's position and can change the azimuth or vertical angle. The radial distance remains nonnegative.
8. Does coordinate order matter?
Yes. Standard spherical notation is commonly written as rho, azimuth, polar angle. Elevation notation is written as rho, azimuth, elevation. Always state the convention to avoid ambiguity.
9. How can I verify my output?
Convert back to Cartesian form. Use x equals rho times sine of phi times cosine of theta, y equals rho times sine of phi times sine of theta, and z equals rho times cosine of phi.
10. Are decimals accepted?
Yes. The calculator accepts positive and negative decimal values. Use enough input precision for your source data, then choose a suitable output precision from the available options.
11. What can I download after conversion?
You can download the displayed values as CSV for spreadsheets or PDF for sharing and records. Exports include the input point, radial distance, angles, selected unit, and convention.