Spherical Coordinates Calculator

The Spherical Coordinates Calculator converts between Cartesian coordinates $$(x, y, z)$$ and spherical coordinates $$(\rho, \theta, \varphi)$$. Spherical coordinates describe a point in 3D space by its distance from the origin, an azimuthal angle in the $$xy$$-plane, and a polar angle from the positive $$z$$-axis. They are the natural coordinate system for problems with radial symmetry about a point, such as gravitational fields, electromagnetic radiation, and quantum mechanics.

The conversion from Cartesian to spherical (physics convention) is:

$$\rho = \sqrt{x^2 + y^2 + z^2}$$

$$\theta = \text{atan2}(y, x) \quad \text{(azimuthal angle, } 0° \text{ to } 360°\text{)}$$

$$\varphi = \arccos\left(\frac{z}{\rho}\right) \quad \text{(polar angle, } 0° \text{ to } 180°\text{)}$$

The conversion from spherical to Cartesian is:

$$x = \rho \sin\varphi \cos\theta$$

$$y = \rho \sin\varphi \sin\theta$$

$$z = \rho \cos\varphi$$

This calculator uses the physics convention (ISO standard) where $$\theta$$ is the azimuthal angle and $$\varphi$$ is the polar (zenith) angle. The mathematics convention swaps these two symbols.

The volume element in spherical coordinates is:

$$dV = \rho^2 \sin\varphi \, d\rho \, d\theta \, d\varphi$$

The Jacobian factor $$\rho^2 \sin\varphi$$ reflects how coordinate cells grow with distance from the origin and thin out near the poles.

Spherical coordinates are indispensable in physics. The gravitational potential of a point mass $$M$$ is $$V = -GM/\rho$$, depending only on the radial distance. The Coulomb potential of a point charge follows the same form. The Schrodinger equation for the hydrogen atom, when separated in spherical coordinates, yields the familiar quantum numbers $$n$$, $$l$$, and $$m$$ corresponding to radial, polar, and azimuthal modes. The angular solutions are spherical harmonics $$Y_l^m(\theta, \varphi)$$, which also appear in multipole expansions, computer graphics (environment mapping), and geophysics (Earth's gravitational field).

In astronomy, celestial coordinates (right ascension and declination) are spherical coordinates centered on the observer. In geophysics, latitude and longitude are spherical coordinates on Earth's surface. In acoustics, sound radiation from a point source spreads spherically, with intensity falling as $$1/\rho^2$$. In computer graphics, environment maps and HDRI lighting use spherical parameterization.

The Laplacian in spherical coordinates is:

$$\nabla^2 f = \frac{1}{\rho^2}\frac{\partial}{\partial\rho}\left(\rho^2\frac{\partial f}{\partial\rho}\right) + \frac{1}{\rho^2\sin\varphi}\frac{\partial}{\partial\varphi}\left(\sin\varphi\frac{\partial f}{\partial\varphi}\right) + \frac{1}{\rho^2\sin^2\varphi}\frac{\partial^2 f}{\partial\theta^2}$$