Swatchai's Web Adventure: เมษายน 2026

วันศุกร์ที่ 17 เมษายน พ.ศ. 2569

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:

Component along the x-axis
$l = \cos \alpha = \cos \phi \cos \lambda$
Component along the y-axis
$m = \cos \beta = \cos \phi \sin \lambda$
Component along the z-axis
$n = \cos \gamma = \sin \phi$
[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

(l, m, n)

, you can find the geographic coordinates using:

Latitude:
$\phi = \arcsin(n)$
Longitude:
$\lambda = \operatorname{atan2}(m, l)$
[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

Vector's direction cosines and spherical earth model (Part 1)

Direction cosines are the cosines of the angles that a vector makes with the positive x, y, and z axes. For a vector

$\vec{v} = \langle v_x, v_y, v_z \rangle$

in three-dimensional space, the direction cosines (often denoted as l, m, and n) are calculated by dividing each component of the vector by its magnitude. [1, 2, 3, 4]

Finding the Direction Cosines of a Vector | Formula ... Understand Unit Vectors and Direction Cosines of Vectors ...

Formulas and Calculations

The direction cosines relate to the direction angles α (with the x-axis), β (with the y-axis), and γ (with the z-axis) as follows: [5, 6, 7]

Calculate Magnitude: First, find the length of the vector:
$|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}$
Determine Cosines:

$l = \cos \alpha = \frac{v_x}{|\vec{v}|}$

$m = \cos \beta = \frac{v_y}{|\vec{v}|}$

$n = \cos \gamma = \frac{v_z}{|\vec{v}|}$

[8, 9, 10]

Essential Properties

Unit Vector: The direction cosines of a vector are exactly the components of its corresponding unit vector.
Fundamental Identity: The sum of the squares of the direction cosines always equals one:
$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = l^2 + m^2 + n^2 = 1$
Direction Ratios: Any set of numbers proportional to the direction cosines are called direction ratios (often a, b, c). [2, 11, 12, 13, 14, 15]