modelling Level 3

Earth as a Rotating Sphere

"The Earth is a sphere rotating on an axis. Every point on the surface traces a circle once per day. To model Earth observation from space, we need to work in spherical coordinates and transform between latitude/longitude and Cartesian xyz. This model builds that foundation."

Prerequisites: spherical coordinates, angular velocity, coordinate transforms, trigonometry

26 Feb 2026 Updated 12 Mar 2026 12 min read

1. The Question

How do we describe positions on a rotating sphere?

A satellite orbits Earth, passing over different latitudes and longitudes. A ground station needs to know: when will the satellite be overhead? What angle should the antenna point?

To answer these questions, we need:

A coordinate system for locations on Earth’s surface
A way to transform between spherical and Cartesian coordinates
An understanding of Earth’s rotation — how positions change with time

The mathematical question: How do we model the Earth as a geometric object and work with its coordinates?

2. The Conceptual Model

Earth as a Sphere

Simplification: Treat Earth as a perfect sphere with radius $R = 6371$ km.

Reality: Earth is an oblate spheroid — slightly flattened at the poles (equatorial radius 6378 km, polar radius 6357 km). The difference is small (~0.3%) for many applications.

Geographic coordinates:

Latitude (φ or lat): Angle north or south of the equator (−90° to +90°)
Longitude (λ or lon): Angle east or west of the prime meridian (−180° to +180° or 0° to 360°)

Convention:

North latitude is positive, south is negative
East longitude is positive (0° to 180°), west is negative (0° to −180°)
The prime meridian (0° longitude) passes through Greenwich, England

Spherical Coordinates

In spherical coordinates, a point is defined by:

Radial distance $r$ from the origin (Earth’s center)
Polar angle $\theta$ (co-latitude, measured from the north pole: 0° to 180°)
Azimuthal angle $\phi$ (longitude, measured from the prime meridian)

Relationship to latitude:

\[\theta = 90° - \text{latitude}\]

For a point on Earth’s surface, $r = R$ (constant).

3. Building the Mathematical Model

Converting Spherical to Cartesian Coordinates

Given latitude $\text{lat}$, longitude $\text{lon}$, and radius $R$:

Cartesian coordinates $(x, y, z)$:

\[x = R \cos(\text{lat}) \cos(\text{lon})\] \[y = R \cos(\text{lat}) \sin(\text{lon})\] \[z = R \sin(\text{lat})\]

Coordinate system:

x-axis: Points from Earth’s center through (lat=0°, lon=0°) — equator at prime meridian
y-axis: Points from Earth’s center through (lat=0°, lon=90°E) — equator at 90°E
z-axis: Points from Earth’s center through the North Pole (lat=90°)

Units: Use radians for trig functions:

\[\text{lat}_{\text{rad}} = \text{lat}_{\text{deg}} \times \frac{\pi}{180}\]

Converting Cartesian to Spherical Coordinates

Given $(x, y, z)$:

\[r = \sqrt{x^2 + y^2 + z^2}\] \[\text{lat} = \arcsin\left(\frac{z}{r}\right) \times \frac{180}{\pi}\] \[\text{lon} = \arctan2(y, x) \times \frac{180}{\pi}\]

Note: Use atan2(y, x) (two-argument arctangent) to get the correct quadrant for longitude.

4. Earth’s Rotation

Angular Velocity

Earth rotates once every sidereal day = 23 hours, 56 minutes, 4 seconds ≈ 86164 seconds.

Angular velocity:

\[\omega = \frac{2\pi}{T} = \frac{2\pi}{86164} \approx 7.292 \times 10^{-5} \text{ rad/s}\]

In degrees per second:

\[\omega_{\text{deg}} = \frac{360°}{86164} \approx 0.00417°/\text{s} \approx 15.04°/\text{hour}\]

Linear Velocity at the Equator

A point on the equator traces a circle of radius $R = 6371$ km.

Circumference:

\[C = 2\pi R = 2\pi \times 6371 \approx 40030 \text{ km}\]

Linear velocity:

\[v_{\text{eq}} = \frac{C}{T} = \frac{40030 \text{ km}}{86164 \text{ s}} \approx 0.465 \text{ km/s} = 465 \text{ m/s}\]

Linear Velocity at Latitude φ

At latitude φ, the radius of the circular path is:

\[r_{\phi} = R \cos(\phi)\]

Linear velocity:

\[v(\phi) = \omega R \cos(\phi)\]

Examples:

Equator (φ = 0°): $v = 465$ m/s
45° latitude: $v = 465 \times \cos(45°) = 329$ m/s
North Pole (φ = 90°): $v = 0$ (just spins in place)

5. Worked Example by Hand

Problem: A point on Earth’s surface is at latitude 30°N, longitude 120°E. Earth’s radius is 6371 km.

(a) Convert to Cartesian coordinates (x, y, z).
(b) What is the linear velocity of this point due to Earth’s rotation?
(c) After 6 hours, what is the new longitude? What are the new Cartesian coordinates?

Solution

(a) Cartesian coordinates

Convert to radians:

\[\text{lat} = 30° \times \frac{\pi}{180} = 0.5236 \text{ rad}\] \[\text{lon} = 120° \times \frac{\pi}{180} = 2.0944 \text{ rad}\]

Compute:

\[x = 6371 \times \cos(0.5236) \times \cos(2.0944)\] \[= 6371 \times 0.866 \times (-0.5) = -2759 \text{ km}\] \[y = 6371 \times \cos(0.5236) \times \sin(2.0944)\] \[= 6371 \times 0.866 \times 0.866 = 4777 \text{ km}\] \[z = 6371 \times \sin(0.5236) = 6371 \times 0.5 = 3186 \text{ km}\]

Cartesian coordinates: (−2759, 4777, 3186) km

(b) Linear velocity

\[v = \omega R \cos(\text{lat}) = 7.292 \times 10^{-5} \times 6371 \times \cos(30°)\] \[= 7.292 \times 10^{-5} \times 6371 \times 0.866 \approx 0.402 \text{ km/s} = 402 \text{ m/s}\]

(c) New longitude after 6 hours

Earth rotates:

\[\Delta \text{lon} = 15.04° \times 6 = 90.24°\]

New longitude:

\[\text{lon}_{\text{new}} = 120° + 90.24° = 210.24°\]

(Or equivalently: −149.76°)

New Cartesian coordinates:

\[\text{lon}_{\text{rad}} = 210.24° \times \frac{\pi}{180} = 3.669 \text{ rad}\] \[x = 6371 \times 0.866 \times \cos(3.669) = 6371 \times 0.866 \times (-0.866) = -4777 \text{ km}\] \[y = 6371 \times 0.866 \times \sin(3.669) = 6371 \times 0.866 \times (-0.5) = -2759 \text{ km}\]

$z = 3186 \text{ km}$ (unchanged — latitude doesn’t change due to rotation)

New coordinates: (−4777, −2759, 3186) km

6. Computational Implementation

Below is an interactive 3D visualization of Earth with controllable rotation.

Latitude: 40° Longitude: 0° Rotation speed: 1.0× real time

Try this:

Watch the pink marker move as Earth rotates
Change latitude: See how the z-coordinate changes
Change longitude: Initial position in the x-y plane
Set rotation speed to 0: Freeze the Earth
The graticule (grid lines) shows meridians and parallels

Key insight: As Earth rotates, longitude changes but latitude remains constant. The Cartesian coordinates rotate around the z-axis.

7. Interpretation

Great Circles and Geodesics

The shortest path between two points on a sphere is along a great circle — a circle whose center coincides with Earth’s center.

Examples:

All meridians (lines of longitude) are great circles
The equator is a great circle
Parallels (lines of latitude) are NOT great circles (except the equator)

Flight paths: Aircraft follow great circle routes to minimize distance.

Calculation: The great circle distance between two points $(lat_1, lon_1)$ and $(lat_2, lon_2)$:

\[d = R \cdot \arccos\left(\sin(lat_1) \sin(lat_2) + \cos(lat_1) \cos(lat_2) \cos(\Delta lon)\right)\]

Where $\Delta lon =

lon_2 - lon_1

Time Zones

Earth rotates 360° in 24 hours → 15° per hour.

Time zones are (roughly) 15° wide. When you cross 15° of longitude, local solar time changes by 1 hour.

Coordinated Universal Time (UTC): Reference time at the prime meridian (0° longitude).

Local time:

\[\text{Local time} = \text{UTC} + \frac{\text{longitude}}{15°}\]

(East is +, west is −, adjusted for daylight saving and political boundaries)

Centrifugal and Coriolis Effects

Earth’s rotation creates fictitious forces in the rotating reference frame:

Centrifugal force: Pulls objects away from the rotation axis

Strongest at the equator
Contributes to Earth’s equatorial bulge

Coriolis force: Deflects moving objects

Rightward in the Northern Hemisphere, leftward in the Southern Hemisphere
Causes large-scale wind patterns (trade winds, westerlies)
Essential for understanding cyclones, ocean currents, and projectile trajectories

8. What Could Go Wrong?

Mixing Up Latitude Conventions

Geographic latitude (used here): Angle from the equator (−90° to +90°).

Co-latitude (polar angle θ in physics): Angle from the North Pole (0° to 180°).

\[\theta = 90° - \text{lat}\]

Always check which convention your formulas use.

Degrees vs. Radians

Trig functions in most programming languages expect radians, not degrees.

Always convert:

\[\text{radians} = \text{degrees} \times \frac{\pi}{180}\]

Assuming a Perfect Sphere

Earth is an oblate spheroid. For precision geodesy (GPS, surveying), use the WGS84 ellipsoid:

Equatorial radius: 6378.137 km
Polar radius: 6356.752 km
Flattening: $f = 1/298.257$

For satellite orbits and large-scale modelling, the difference matters.

Longitude Wraparound

Longitude wraps at ±180°. When computing differences:

\[\Delta lon = lon_2 - lon_1\]

If $

\Delta lon

> 180°$, subtract 360° to get the short way around.

9. Extension: Satellite Ground Tracks

A satellite in orbit traces a ground track — the path of the point directly below it on Earth’s surface.

For a satellite at altitude $h$ above the equator:

Orbital period: $T_{\text{orbit}}$
Earth rotation period: $T_{\text{Earth}} = 86164$ s

Ground track shift per orbit:

\[\Delta lon = 360° \times \frac{T_{\text{orbit}}}{T_{\text{Earth}}}\]

Example: Low Earth orbit satellite with $T_{\text{orbit}} = 90$ minutes:

\[\Delta lon = 360° \times \frac{5400}{86164} \approx 22.5°\]

After one orbit, the ground track is 22.5° west of the starting point (Earth rotated eastward underneath).

Phase II models will derive orbital mechanics and predict when satellites pass overhead.

10. Math Refresher: Spherical Coordinates

Definition

In 3D, a point can be described by:

Cartesian $(x, y, z)$: Rectangular coordinates
Spherical $(r, \theta, \phi)$: Radial distance, polar angle, azimuthal angle

Conversion spherical → Cartesian:

\[x = r \sin\theta \cos\phi\] \[y = r \sin\theta \sin\phi\] \[z = r \cos\theta\]

(This is the physics convention with θ from the z-axis.)

Conversion Cartesian → spherical:

\[r = \sqrt{x^2 + y^2 + z^2}\] \[\theta = \arccos\left(\frac{z}{r}\right)\] \[\phi = \arctan2(y, x)\]

Geographic vs. Physics Convention

Geography: Uses latitude (from equator) and longitude.

Physics: Uses co-latitude (from pole) and azimuth.

Both describe the same geometry — just different angle conventions.

Summary

Earth is modelled as a sphere with radius $R = 6371$ km
Latitude φ and longitude λ define positions on the surface
Spherical to Cartesian: $x = R \cos(lat) \cos(lon)$, etc.
Earth rotates at $\omega = 7.292 \times 10^{-5}$ rad/s
Linear velocity at the equator: 465 m/s, decreases with latitude as $v = \omega R \cos(lat)$
Great circles are the shortest paths on a sphere
Coordinate transforms are essential for satellite tracking and Earth observation

Phase II Preview: The next phase will introduce orbital mechanics — how satellites move, how to predict their positions, and how to calculate when they pass overhead. We’ll derive Kepler’s laws, compute ground tracks, and model satellite swaths.