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