Distance & Geofencing Math

Technical Deep Dive & Specifications

The Haversine Formula Derivation

The Haversine formula calculates the great-circle distance between two points on a sphere given their longitudes and latitudes:

$$\Delta\varphi = (\text{lat}_2 - \text{lat}_1) \times \frac{\pi}{180}$$

$$\Delta\lambda = (\text{lon}_2 - \text{lon}_1) \times \frac{\pi}{180}$$

$$a = \sin^2\left(\frac{\Delta\varphi}{2}\right) + \cos\left(\text{lat}_1 \times \frac{\pi}{180}\right) \cdot \cos\left(\text{lat}_2 \times \frac{\pi}{180}\right) \cdot \sin^2\left(\frac{\Delta\lambda}{2}\right)$$

$$c = 2 \cdot \text{atan2}\left(\sqrt{a}, \sqrt{1-a}\right)$$

$$d = R \cdot c$$

Where:

• $R$ is the mean volumetric radius of Earth:
  • $R = 6,371\text{ km}$ ($6,371,000\text{ meters}$)
  • $R = 3,958.8\text{ statute miles}$
  • $R = 3,440.0\text{ nautical miles}$
• $\varphi$ is latitude in radians.
• $\lambda$ is longitude in radians.
• $d$ is the shortest distance over the Earth's surface.

Pure JavaScript Haversine Implementation

/**
 * Calculates great-circle distance between two points using the Haversine formula.
 * @param {number} lat1 Latitude of point 1 (decimal degrees)
 * @param {number} lon1 Longitude of point 1 (decimal degrees)
 * @param {number} lat2 Latitude of point 2 (decimal degrees)
 * @param {number} lon2 Longitude of point 2 (decimal degrees)
 * @param {'km' | 'm' | 'miles'} unit Desired output unit (default: 'm')
 * @returns {number} Distance in chosen units
 */
function calculateDistance(lat1, lon1, lat2, lon2, unit = 'm') {
  const EARTH_RADII = {
    m: 6371000,
    km: 6371,
    miles: 3958.8
  };

  const R = EARTH_RADII[unit] || EARTH_RADII.m;
  const toRad = (deg) => (deg * Math.PI) / 180;

  const dLat = toRad(lat2 - lat1);
  const dLon = toRad(lon2 - lon1);
  const rLat1 = toRad(lat1);
  const rLat2 = toRad(lat2);

  const a =
    Math.sin(dLat / 2) * Math.sin(dLat / 2) +
    Math.cos(rLat1) * Math.cos(rLat2) * Math.sin(dLon / 2) * Math.sin(dLon / 2);

  const c = 2 * Math.atan2(Math.sqrt(a), Math.sqrt(1 - a));

  return R * c;
}

Calculating Initial Compass Bearing (Forward Azimuth)

To determine which direction (in degrees from True North) a user must face to travel toward a destination:

$$\theta = \text{atan2}\left(\sin(\Delta\lambda)\cdot\cos(\varphi_2), \cos(\varphi_1)\cdot\sin(\varphi_2) - \sin(\varphi_1)\cdot\cos(\varphi_2)\cdot\cos(\Delta\lambda)\right)$$

function calculateBearing(lat1, lon1, lat2, lon2) {
  const toRad = (deg) => (deg * Math.PI) / 180;
  const toDeg = (rad) => (rad * 180) / Math.PI;

  const y = Math.sin(toRad(lon2 - lon1)) * Math.cos(toRad(lat2));
  const x =
    Math.cos(toRad(lat1)) * Math.sin(toRad(lat2)) -
    Math.sin(toRad(lat1)) * Math.cos(toRad(lat2)) * Math.cos(toRad(lon2 - lon1));

  const bearingRad = Math.atan2(y, x);
  const bearingDeg = (toDeg(bearingRad) + 360) % 360; // Normalize 0 - 360°

  return bearingDeg;
}

The Circular Geofencing State Machine

A Geofence is a virtual geographic boundary defined by a center coordinate $(lat_C, lon_C)$ and a radius $r$ in meters:

[ User Moves ] ──► Compute distance(userPos, fenceCenter)
                         │
         ┌───────────────┴───────────────┐
         ▼                               ▼
  distance <= radius              distance > radius
         │                               │
         ▼                               ▼
 [ INSIDE ZONE ]                  [ OUTSIDE ZONE ]
 (Trigger 'ENTER' if               (Trigger 'EXIT' if
  previous was OUTSIDE)             previous was INSIDE)

💻 Interactive Code Playground

Starter Code

Line-by-Line Code Breakdown

• Lines 114–128 (haversineDistance): Implements standard Haversine equations, explicitly converting inputs from decimal degrees into radians using deg * Math.PI / 180.
• Lines 130–140 (calculateBearing): Computes trigonometric arctangent of spherical delta vectors to return a normalized azimuth between $0^\circ$ and $360^\circ$.
• Lines 155–163: Evaluates whether distMeters <= radius to dynamically trigger inside/outside geofence status.

Expected Browser Render Output

📐 Haversine Distance & Geofencing Math
Point A: 37.7749, -122.4194  |  Point B: 37.7792, -122.4140

[ ⚡ Compute Spatial Metrics ]

DISTANCE (METERS)        DISTANCE (MILES)         COMPASS BEARING
671.4 m                  0.42 mi                  43.5°

[ 🚫 OUTSIDE GEOFENCE (671.4m > 500m) ]