Spherical to Cartesian Coordinates Conversion

In spherical coordinates, a point in space is described by three values: the radial distance of that point from the origin, the inclination angle from the positive z-axis, and the azimuth angle from the positive x-axis and the intersection between the plane passing through the point and the origin perpendicular to the z-axis.

The spherical coordinates are usually represented as (r, theta, phi), where:

• r is the radial distance,

• theta is the inclination angle (ranging from 0 to pi), and

• phi is the azimuth angle (ranging from 0 to 2pi).

The conversion from spherical to Cartesian coordinates is given by the following equations:

• x = r * cos(theta)

• y = r * sin(theta) * cos(phi)

• z = r * sin(theta) * sin(phi)

For example, if we have a point in spherical coordinates as (1, pi/2, 0), the conversion to Cartesian coordinates would be:

• x = r * cos(pi/2) = 0

• y = r * sin(pi/2) * cos(0) = 1

• z = r * sin(pi/2) * sin(0) = 0

So the Cartesian coordinates for this point would be (0, 1, 0).

It’s important to note that in a 2D space (or when r and theta are defined, but phi is not), the conversion simplifies to:

• x = r * cos(theta)

• y = r * sin(theta)

The spherical_to_cartesian function implements these conversions. For the multi-dimension case see the article here N-sphere.