Spherical coordinates, angular velocity, and the geometry of our rotating planet
2026-02-26
Every time a GPS receiver displays a latitude and longitude, it has performed a coordinate transformation: satellite signals arrive as timing measurements, which are converted to distances, which are converted to a position in three-dimensional Cartesian space, which is converted to a latitude/longitude/altitude triplet referenced to an ellipsoidal model of Earth’s shape. The chain of transformations is invisible to the user, but understanding it is essential for anyone building geospatial software, correcting satellite imagery, or designing an orbit.
Earth is, to first approximation, a sphere of radius 6,371 km rotating once every 23 hours, 56 minutes, and 4 seconds around an axis tilted 23.5° from the perpendicular to its orbital plane. Every point on the surface traces a circle of latitude once per sidereal day. Describing positions on that sphere — and keeping track of how those positions move as it rotates — requires a coordinate system suited to spherical geometry. This model builds spherical coordinates from first principles, derives the transformation to Cartesian (x, y, z) coordinates, and establishes the rotational kinematics that underpin satellite tracking, Earth observation geometry, and the calculation of ground tracks in subsequent models.
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: 1. A coordinate system for locations on Earth’s surface 2. A way to transform between spherical and Cartesian coordinates 3. 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?
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
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).
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}
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.
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}
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}
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)
Problem: A point on Earth’s surface is at latitude 30°N, longitude 120°E. Earth’s radius is 6371 km.
(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
Below is an interactive 3D visualization of Earth with controllable rotation.
<label>
Latitude:
<input type="range" id="lat-slider" min="-90" max="90" step="5" value="40">
<span id="lat-value">40</span>°
</label>
<label>
Longitude:
<input type="range" id="lon-slider" min="-180" max="180" step="5" value="0">
<span id="lon-value">0</span>°
</label>
<label>
Rotation speed:
<input type="range" id="rot-slider" min="0" max="5" step="0.5" value="1">
<span id="rot-value">1.0</span>× real time
</label>
<div class="button-group">
<button id="toggle-rotation">Pause Rotation</button>
<button id="reset-view">Reset View</button>
</div><p id="earth-coords"></p>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.
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|.
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)
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
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.
Trig functions in most programming languages expect radians, not degrees.
Always convert:
\text{radians} = \text{degrees} \times \frac{\pi}{180}
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 wraps at ±180°. When computing differences:
\Delta lon = lon_2 - lon_1
If |\Delta lon| > 180°, subtract 360° to get the short way around.
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.
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)
Geography: Uses latitude (from equator) and longitude.
Physics: Uses co-latitude (from pole) and azimuth.
Both describe the same geometry — just different angle conventions.
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.