Vector's direction cosines and spherical earth model (Part 2)

In a spherical Earth model, the direction cosines of a position vector are essentially the components of the unit vector that points to a specific latitude (φ) and longitude (λ). [1]

The Coordinate Mapping

To relate these, we use an Earth-Centered, Earth-Fixed (ECEF) Cartesian system where:

• x-axis: Points toward the Prime Meridian (0° longitude) at the Equator.
• y-axis: Points toward 90° East longitude at the Equator.
• z-axis: Points toward the North Pole (90° North latitude). [2, 3]

Conversion Formulas

If a point on Earth has latitude φ and longitude λ, its direction cosines (l, m, n) are calculated as:

1. Component along the x-axis

   

2. Component along the y-axis

   

3. Component along the z-axis

   

   [1, 4, 5]

Key Differences from Standard Physics

It is important to note that "standard" spherical coordinates (ρ, θ, φ) used in math/physics differ from geographic coordinates: [1]

• Latitude (φ): Measured from the Equator (up/down). In standard spherical math, the "polar angle" is measured from the North Pole (downward).
• Relationship: The polar angle θ in math is equal to (90° - φ) in geography. This is why the z-component uses sin φ for latitude instead of the usual cos θ. [1, 5, 6]

Inverse Relationship

If you already have the direction cosines

, you can find the geographic coordinates using:

• Latitude:

  

• Longitude:

  

  [4, 7]

This vector-based approach is often called the n-vector representation and is used in navigation software because it avoids "singularities" (mathematical glitches) at the North and South Poles. [8]

[1] https://moodle2.units.it [2] https://vulms.vu.edu.pk [3] https://math.univ-lyon1.fr [4] https://via-technology.aero [5] https://www.youtube.com [6] https://en.wikipedia.org [7] https://www.movable-type.co.uk [8] https://en.wikipedia.org