terican

BIPM-ratified constants · v1.0

Converter

Spherical, coordinates calculator.

Convert between Cartesian (x, y, z) and spherical (ρ, θ, φ) coordinates using standard math/ISO formulas. Supports degrees and radians.

From

cartesian

to_spherical

1 to_spherical =1.7321Selected Coordinate Component

Equivalents

Precision: 6 dp · Notation: Decimal · 3 units

ρ or x

Firstfirst1.7321

θ or y

Secondsecond45

φ or z

Thirdthird54.7356

Common pairings

1 to_sphericalequals1.7321 first
1 to_sphericalequals45 second
1 to_sphericalequals54.7356 third
1 to_cartesianequals1 first
1 to_cartesianequals0 second
1 to_cartesianequals6.12e-17 third

The conversion

How the value
is computed.

What Are Spherical Coordinates?

Spherical coordinates describe any point in three-dimensional space using three values: the radial distance ρ (rho), the azimuthal angle θ (theta), and the polar angle φ (phi). This system excels at problems with spherical symmetry, including gravitational fields, electromagnetic wave propagation, and fluid flow around spherical objects. A spherical coordinates converter translates between this system and the familiar Cartesian (x, y, z) representation.

The Conversion Formulas

Cartesian to Spherical

Given a point (x, y, z) in Cartesian space, spherical coordinates are computed in three steps:

  • ρ = √(x² + y² + z²) — the Euclidean distance from the origin to the point; always non-negative
  • θ = atan2(y, x) — the azimuthal angle in the xy-plane measured from the positive x-axis, ranging from −π to π radians (−180° to 180°)
  • φ = arccos(z / ρ) — the polar angle measured from the positive z-axis downward, ranging from 0 to π radians (0° to 180°)

Always compute ρ first, since both θ and φ depend on its value.

Spherical to Cartesian

To recover Cartesian coordinates from (ρ, θ, φ), apply the inverse transformation:

  • x = ρ sin(φ) cos(θ)
  • y = ρ sin(φ) sin(θ)
  • z = ρ cos(φ)

Variable Definitions

ρ (rho) is the radial distance — the straight-line length from the origin to the point. It equals zero only at the origin and is always non-negative: ρ = √(x² + y² + z²).

θ (theta) is the azimuthal angle, sweeping counterclockwise around the z-axis in the xy-plane, analogous to geographic longitude. The two-argument function atan2(y, x) is used instead of arctan(y/x) because it correctly resolves all four quadrants, returning values across the full range −π to π.

φ (phi) is the polar angle from the positive z-axis. At φ = 0° the point lies on the positive z-axis (north pole); at φ = 90° it lies in the equatorial xy-plane; at φ = 180° it lies on the negative z-axis (south pole). This follows the math/ISO 80000-2 convention used by this calculator.

Math/ISO vs. Physics Convention

This calculator follows the math/ISO 80000-2 convention, where θ denotes the azimuthal angle and φ denotes the polar angle. Many physics texts and engineering references reverse these symbols. According to Wolfram MathWorld, both conventions appear widely in the literature — always verify the convention before applying a formula from an external source to avoid systematic errors in computed angles.

Worked Example: Cartesian to Spherical

Convert the Cartesian point (3, 4, 5) to spherical coordinates:

  • ρ = √(3² + 4² + 5²) = √50 ≈ 7.071
  • θ = atan2(4, 3) ≈ 0.9273 rad ≈ 53.13°
  • φ = arccos(5 / 7.071) = arccos(0.7071) ≈ 0.7854 rad = 45.00°

Verification via inverse conversion: x = 7.071 × sin(45°) × cos(53.13°) ≈ 3.00, y = 7.071 × sin(45°) × sin(53.13°) ≈ 4.00, z = 7.071 × cos(45°) ≈ 5.00 — confirming the result.

The Volume Element and Jacobian

When evaluating triple integrals in spherical coordinates, the volume element is dV = ρ² sin(φ) dρ dθ dφ. The factor ρ² sin(φ) is the Jacobian determinant of the transformation. As explained by Paul's Online Math Notes on Spherical Coordinates, this Jacobian ensures that equal increments in ρ, θ, and φ map to correct physical volumes — essential for computing masses, charge densities, and electromagnetic potentials over spherical domains.

Real-World Applications

  • Astronomy and navigation: Celestial right ascension and declination, satellite orbit propagation, and GPS Earth-centered coordinate frames
  • Physics: Gravitational and electric potential of spherical charge distributions, hydrogen atom wave functions in quantum mechanics, solving Laplace's equation with spherical boundary conditions
  • Engineering: Antenna gain pattern specification, radar cross-section measurement, and sonar beam steering
  • Computer graphics: Environment map sampling, skybox rendering, and physically based rendering light probes

Special Cases and Singularities

At the origin (0, 0, 0), ρ = 0 and both angles θ and φ are mathematically undefined. On the z-axis (x = y = 0, z ≠ 0), θ is indeterminate and is conventionally set to 0. These are coordinate singularities inherent to the spherical parameterization — they reflect non-uniqueness of the coordinate representation at special points, not any physical infinity. Numerical implementations must guard against division by zero when ρ = 0.

Reference

Frequently asked questions

What is the formula for converting Cartesian coordinates to spherical coordinates?
The three formulas are ρ = √(x² + y² + z²), θ = atan2(y, x), and φ = arccos(z / ρ). Here ρ is the radial distance from the origin, θ is the azimuthal angle in the xy-plane measured from the positive x-axis with range −180° to 180°, and φ is the polar angle from the positive z-axis with range 0° to 180°. Always compute ρ first, since both θ and φ depend on it.
What is the difference between the math convention and physics convention for spherical coordinates?
In the math/ISO 80000-2 convention, θ is the azimuthal angle in the xy-plane and φ is the polar angle from the z-axis. In the physics convention these symbols are swapped: θ becomes the polar angle and φ becomes the azimuthal angle. Mixing the two conventions produces incorrect numerical values for both angles. Always check which convention a textbook or software package uses before applying its formulas, especially when comparing results across different sources.
How do you convert spherical coordinates back to Cartesian (x, y, z)?
Given spherical coordinates (ρ, θ, φ), apply x = ρ sin(φ) cos(θ), y = ρ sin(φ) sin(θ), and z = ρ cos(φ). For example, the spherical point (5, 30°, 60°) gives x = 5 × sin(60°) × cos(30°) ≈ 3.75, y = 5 × sin(60°) × sin(30°) ≈ 2.165, and z = 5 × cos(60°) = 2.50. This inverse transformation is unique for all points except those on the z-axis and at the origin.
Why does the spherical coordinates formula use atan2 instead of arctan for the azimuthal angle?
The standard arctan function returns values only in the range −90° to 90° and cannot distinguish between diagonally opposite quadrants. For instance, arctan(1/1) and arctan(−1/−1) both return 45°, yet points (1, 1) and (−1, −1) have azimuthal angles of 45° and −135° respectively. The two-argument atan2(y, x) uses the individual signs of both x and y to determine the correct quadrant, returning an unambiguous result across the full −180° to 180° range.
What are the most common real-world applications of spherical coordinates?
Spherical coordinates appear in astronomy (right ascension and declination for star positions, satellite orbit propagation), physics (gravitational and electric potential of point masses, hydrogen atom wave functions in quantum mechanics), engineering (antenna radiation gain patterns, radar cross-section analysis, sonar beam steering), computer graphics (environment map and skybox sampling in physically based rendering), and mathematics (evaluating triple integrals over spherical volumes using the volume element ρ² sin(φ) dρ dθ dφ).
What happens at the poles and the origin in spherical coordinates?
At the origin (0, 0, 0), ρ = 0 and both θ and φ are undefined because infinitely many angle pairs satisfy the inverse transformation equations there. At the north pole (x = 0, y = 0, z > 0), φ = 0 and θ is indeterminate — conventionally assigned the value 0. At the south pole (z < 0), φ = π with θ similarly undefined. These are coordinate singularities arising from the mathematical parameterization and must be handled explicitly in numerical implementations to prevent division-by-zero errors.