ECEF and 3D coordinate systems

Most of the formulas in this guide project the Earth onto a flat surface. ECEF doesn't. It's a 3D Cartesian system with the origin at Earth's centre — the coordinate system GPS hardware and 3D rendering engines actually use.

ECEF — Earth-Centered, Earth-Fixed

ECEF places the origin at Earth's centre of mass. The axes are fixed to the Earth and rotate with it:

• X-axis: points to the intersection of the equator and prime meridian (0°N, 0°E)
• Y-axis: points to 0°N, 90°E
• Z-axis: points to the North Pole

Units are metres. At Lahore (31.52°N, 74.36°E, ≈220 m):

$X \approx 1{,}186{,}000 \text{ m}, \quad Y \approx 4{,}249{,}000 \text{ m}, \quad Z \approx 3{,}309{,}000 \text{ m}$

Geodetic → ECEF

Given latitude $\varphi$, longitude $\lambda$, and altitude $h$ (metres above the ellipsoid):

$$N(\varphi) = \frac{a}{\sqrt{1 - e^2 \sin^2 \varphi}}$$ $$X = (N + h) \cos\varphi \cos\lambda$$ $$Y = (N + h) \cos\varphi \sin\lambda$$ $$Z = \bigl(N(1 - e^2) + h\bigr) \sin\varphi$$

where $a = 6{,}378{,}137$ m, $e^2 = 0.00669437999014$ (WGS84).

const A  = 6_378_137.0;
const E2 = 0.00669437999014;

function geodeticToECEF(latDeg, lonDeg, altM = 0) {
  const lat = latDeg * Math.PI / 180;
  const lon = lonDeg * Math.PI / 180;
  const N = A / Math.sqrt(1 - E2 * Math.sin(lat) ** 2);
  return {
    x: (N + altM) * Math.cos(lat) * Math.cos(lon),
    y: (N + altM) * Math.cos(lat) * Math.sin(lon),
    z: (N * (1 - E2) + altM) * Math.sin(lat),
  };
}

ECEF → Geodetic

The inverse has no closed form — use Bowring's iterative method or Zhu's formula:

function ecefToGeodetic(x, y, z) {
  const p = Math.sqrt(x * x + y * y);
  const lon = Math.atan2(y, x);

  // Bowring iteration
  let lat = Math.atan2(z, p * (1 - E2));
  for (let i = 0; i < 5; i++) {
    const N  = A / Math.sqrt(1 - E2 * Math.sin(lat) ** 2);
    lat = Math.atan2(z + E2 * N * Math.sin(lat), p);
  }
  const N = A / Math.sqrt(1 - E2 * Math.sin(lat) ** 2);
  const alt = p / Math.cos(lat) - N;

  return { lat: lat * 180 / Math.PI, lon: lon * 180 / Math.PI, alt };
}

ENU — the local frame

ECEF is a global system, but for local navigation you want a frame with East, North, Up axes anchored to a specific point. ENU gives you that.