Map math formula reference

Every map mathematics formula from the MapMath guide, free to consult. Each formula links to the chapter that derives it and shows working JavaScript code — bookmark this page if you write geospatial code regularly.

Notation:

φ

= latitude (radians),

λ

= longitude (radians),

R

= Earth's mean radius (6,371 km).

Distance

Haversine formula

a = sin^{2} (\frac{Δ φ}{2}) + cos φ_{1} \cdot cos φ_{2} \cdot sin^{2} (\frac{Δ λ}{2})

c = 2 \cdot arctan 2 (a, 1 - a)

d = R \cdot c

Great-circle distance between two points on a sphere. Numerically stable across all distances; accurate to roughly 0.5% when using WGS84 mean radius. The standard choice for almost all geospatial distance calculations.

→ Distance on a sphere — Haversine and friends

Spherical law of cosines

d = arccos (sin φ_{1} sin φ_{2} + cos φ_{1} cos φ_{2} cos Δ λ) \cdot R

Direct formulation of great-circle distance. Mathematically equivalent to Haversine but loses precision at short distances due to floating-point limits near arccos(1). Use Haversine instead for production code.

→ Distance on a sphere — Haversine and friends

Equirectangular approximation

Δ x = (λ_{2} - λ_{1}) \cdot cos (\frac{φ _{1} + φ _{2}}{2}) \cdot R

Δ y = (φ_{2} - φ_{1}) \cdot R

d \approx Δ x^{2} + Δ y^{2}

Fast flat-Earth approximation. Accurate to ~0.1% over distances under 50 km; error grows quadratically beyond that. Useful for sorting/clustering nearby points where absolute accuracy doesn't matter.

→ Distance on a sphere — Haversine and friends

Bearing & destination

Initial bearing (forward azimuth)

θ = arctan 2 (sin Δ λ \cdot cos φ_{2}, cos φ_{1} \cdot sin φ_{2} - sin φ_{1} \cdot cos φ_{2} \cdot cos Δ λ)

bearing = (θ \cdot \frac{180}{π} + 360) mod 360

Compass bearing from point 1 to point 2 at the moment of departure. Note: the bearing changes continuously along a great-circle path; use the final bearing formula at arrival.

→ Bearings — compass directions on a sphere

Destination point

φ_{2} = arcsin (sin φ_{1} cos δ + cos φ_{1} sin δ cos θ)

λ_{2} = λ_{1} + arctan 2 (sin θ sin δ cos φ_{1}, cos δ - sin φ_{1} sin φ_{2})

Given a start point, a bearing θ, and an angular distance δ = d/R, compute the destination. Used for geofences, range rings, dead reckoning, and any 'move a point N km in direction θ' operation.

→ Destination points — moving on a sphere

Projections

Web Mercator — forward (lat/lon → x, y)

x = R \cdot λ

y = R \cdot ln (tan (\frac{π}{4} + \frac{φ}{2}))

Maps spherical coordinates to a flat plane (EPSG:3857). Used by Google Maps, OpenStreetMap, Mapbox, and virtually every web map. Distorts area severely near the poles; valid latitude range is ±85.05113°.

→ Map projections — flattening the globe

Web Mercator — inverse (x, y → lat/lon)

λ = \frac{x}{R}

φ = 2 \cdot arctan (e^{y / R}) - \frac{π}{2}

The inverse projection — recover lat/lon from Mercator x, y. Essential for click-to-coordinate conversion in interactive maps.

→ Map projections — flattening the globe

UTM zone number

zone = ⌊ \frac{λ + 180}{6} ⌋ + 1

UTM divides the globe into 60 zones, each 6° of longitude wide. Within a zone, UTM gives metre-accurate flat coordinates suitable for surveying and most engineering work.

→ Coordinate transforms and reference systems

Tile math

Lat/lon to tile (z/x/y)

n = 2^{z}

x_{tile} = ⌊ n \cdot \frac{λ + 180}{360} ⌋

y_{tile} = ⌊ n \cdot \frac{1 - ln ( tan φ + sec φ ) / π}{2} ⌋

Given a lat/lon and zoom level z, find which slippy-map tile contains that point. This is the math behind every URL like `tile.openstreetmap.org/{z}/{x}/{y}.png`.

→ Tile math — z, x, y and the slippy map convention

Tile (z/x/y) to lat/lon (NW corner)

λ = \frac{x}{2 ^{z}} \cdot 360 - 180

φ = arctan (sinh (π \cdot (1 - \frac{2 y}{2 ^{z}}))) \cdot \frac{180}{π}

Recover the lat/lon of a tile's northwest corner. Combined with the tile to its southeast, this gives the full geographic bounding box of a tile.

→ Tile math — z, x, y and the slippy map convention

Pixel resolution at zoom z

resolution = \frac{2 π R}{256 \cdot 2 ^{z}} \approx \frac{156 , 543.03}{2 ^{z}} m/px

How many real-world metres each pixel represents at zoom z, at the equator. Halves with every zoom level. Useful for sizing markers, buffers, and accuracy thresholds in screen space.

→ Pixel and world coordinates

Polygons & geometry

Shoelace formula — planar polygon area

A = \frac{1}{2} i \sum (x_{i} \cdot y_{i + 1} - x_{i + 1} \cdot y_{i})

Computes the area of a polygon from its vertex coordinates, on a flat plane. Works for any simple polygon (no self-intersections). The signed version (without absolute value) tells you winding direction.

→ Polygons — area, centroid, and containment

Spherical polygon area

A = \frac{R ^{2}}{2} i \sum (λ_{i + 1} - λ_{i}) \cdot (2 + sin φ_{i} + sin φ_{i + 1})

Area of a polygon on a sphere, accounting for Earth's curvature. Required for any polygon spanning more than ~100 km — planar shoelace gives wrong answers at country/continent scale.

→ Polygons — area, centroid, and containment

Polygon centroid (planar)

C_{x} = \frac{1}{6 A} i \sum (x_{i} + x_{i + 1}) \cdot (x_{i} y_{i + 1} - x_{i + 1} y_{i})

C_{y} = \frac{1}{6 A} i \sum (y_{i} + y_{i + 1}) \cdot (x_{i} y_{i + 1} - x_{i + 1} y_{i})

The geometric center of mass of a polygon (not the average of vertices, which is wrong for irregular shapes). Useful for label placement, cluster centers, and pivot points.

→ Polygons — area, centroid, and containment

Point in polygon — ray casting

inside ⟺ # {ray crossings} mod 2 = 1

Cast a ray from the test point to infinity in any direction; count how many polygon edges it crosses. Odd = inside, even = outside. Simple, robust, and the standard implementation in turf.js and most GIS libraries.

→ Polygons — area, centroid, and containment

Spatial indexing

Geohash precision

Geohash encodes lat/lon as a short base-32 string. Each character adds 5 bits of precision, alternating between latitude and longitude. Cell size shrinks rapidly with length:

Length	Cell size (mid-latitudes)	Typical use
1	~5,000 km × 5,000 km	continents
4	~40 km × 20 km	cities
6	~1.2 km × 600 m	neighbourhoods
8	~38 m × 19 m	buildings
10	~1.2 m × 0.6 m	parking spaces

→ Spatial indexing — Geohash, Quadtrees, S2, H3

3D & ECEF

Geodetic → ECEF (lat/lon/alt → X, Y, Z)

N = \frac{a}{1 - e ^{2} sin ^{2} φ}

X = (N + h) cos φ cos λ

Y = (N + h) cos φ sin λ

Z = (N (1 - e^{2}) + h) sin φ

Convert lat/lon/altitude to Earth-Centered Earth-Fixed Cartesian coordinates. WGS84 parameters: a = 6,378,137 m, e² ≈ 0.006694. Used in GPS computations, satellite tracking, and any 3D Earth modeling where distance must be measured through the planet.

→ ECEF — 3D Earth-centered coordinates

ECEF Euclidean distance

d = (X_{2} - X_{1})^{2} + (Y_{2} - Y_{1})^{2} + (Z_{2} - Z_{1})^{2}

Straight-line distance through the Earth between two ECEF points. For airline distance use Haversine (great-circle along the surface); ECEF distance is shorter because it cuts through the planet.

→ ECEF — 3D Earth-centered coordinates