Formula and Method for Spherical Coordinate Conversion
Spherical coordinates describe a point in 3D space using a radial distance and two angles instead of three perpendicular distances. This calculator uses the physics (ISO 80000-2) convention: r is the radial distance from the origin, θ (theta) is the polar angle measured from the positive z-axis (0° to 180°), and φ (phi) is the azimuthal angle measured in the xy-plane from the positive x-axis (0° to 360°). Note that many math textbooks swap θ and φ, so always confirm which convention a formula uses before mixing sources.
Converting Cartesian to spherical coordinates
Given (x, y, z), the radial distance is r = √(x² + y² + z²). The polar angle is θ = arccos(z / r), which is always between 0° and 180°. The azimuthal angle is φ = atan2(y, x); using atan2 instead of a plain arctangent keeps the correct quadrant automatically, and this calculator adds 360° when the result is negative so φ falls in the standard 0°-360° range. At the origin (x = y = z = 0), r = 0 and θ, φ are undefined — the calculator reports them as 0 by convention.
Converting spherical to Cartesian coordinates
Given (r, θ, φ) with θ and φ converted to radians, the Cartesian coordinates are x = r·sin(θ)·cos(φ), y = r·sin(θ)·sin(φ), and z = r·cos(θ). These formulas fall directly out of right-triangle trigonometry: r·sin(θ) is the point's distance from the z-axis (its cylindrical radius), which is then split into x and y components using φ, while r·cos(θ) gives the height along the z-axis.
Common mistakes and real-world applications
- Mixing conventions: some sources (especially many calculus textbooks) swap θ and φ, using θ for the azimuthal angle and φ for the polar angle — check this before comparing results across sources.
- Degrees vs. radians: trigonometric functions in most programming languages expect radians; forgetting to convert degrees to radians (or back) is the most common source of wrong answers.
- Out-of-range θ: the polar angle must stay within 0°-180° (0 to π radians); values outside that range do not correspond to a valid point in this convention.
- Applications: spherical coordinates are used in physics for gravitational and electric fields, in navigation and astronomy for specifying direction (right ascension/declination, azimuth/elevation), in 3D graphics for camera orbit controls, and in engineering for antenna radiation patterns.