Spherical Coordinate System

The spherical coordinate system represents points in three-dimensional space using three values: radius, elevation (latitude), and azimuth (longitude). This system is especially useful for describing positions on a sphere.

Defining the Axes

  • z-axis: The vertical axis passing through the center of the sphere.
  • x-axis: Passes through the equator of the sphere.
  • y-axis: Perpendicular to both the x and z axes.

With these axes, we can describe both translation and rotation in 3D space, giving us 6 degrees of freedom (DOF).

Latitude and Longitude

To specify a point on a perfect unit sphere, we use:

  • Latitude (\(\lambda\)): The angle of elevation from the equator, measured by rotating around the y-axis (in the x-z plane).
  • Longitude (\(\theta\)): The angle of azimuth, measured by rotating around the z-axis (in the x-y plane).

These two angles uniquely determine the position of a point on the sphere.

Step-by-Step Conversion

  1. Elevation (Latitude):
    Elevate by an angle \(\lambda\) from the equator. This sets the vertical position (\(z\)) and the radius of the horizontal circle at that elevation.

    \[z = r \sin \lambda\]

    For a unit sphere (\(r = 1\)):

    \[z = \sin \lambda\]

  2. Azimuth (Longitude):
    Rotate by an angle \(\theta\) around the z-axis to determine the position in the x-y plane. The radius of the circle at elevation \(\lambda\) is \(r \cos \lambda\).

    \[x = \cos \lambda \cdot \cos \theta\] \[y = \cos \lambda \cdot \sin \theta\]

Spherical to Cartesian Conversion

For a unit sphere, the Cartesian coordinates \((x, y, z)\) are:

\[\begin{align*} x &= \cos \lambda \cdot \cos \theta \\ y &= \cos \lambda \cdot \sin \theta \\ z &= \sin \lambda \end{align*}\]

Where:

  • \(\lambda\) is the latitude (elevation from the equator)
  • \(\theta\) is the longitude (azimuthal angle)

This convention allows us to map any point on the surface of a sphere using latitude and longitude.

Note: At the poles (\(\lambda = \pm \frac{\pi}{2}\)), the azimuthal angle \(\theta\) becomes irrelevant because all longitudes converge at the poles. Any value of \(\theta\) represents the same point at the poles, resulting in a singularity at these locations.