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WorldBiG and PolBiG Grid Systems

1. Projection Characteristics

The WorldBiG grid uses the Lambert Azimuthal Equal-Area (LAEA) spherical projection. This choice is based on two key features:

2. Mathematical Foundations

Forward Transformation (Geographic → Grid): $$p = \sqrt{\frac{2}{1 + \sin\alpha \sin\phi + \cos\alpha \cos\phi \cos(\lambda - \beta)}}$$ $$x = R \cdot p \cdot \cos\phi \sin(\lambda - \beta)$$ $$y = R \cdot p \cdot (\cos\alpha \sin\phi - \sin\alpha \cos\phi \cos(\lambda - \beta))$$
Inverse Transformation (Grid → Geographic): $$\rho = \sqrt{x^2 + y^2}, \quad c = 2 \arcsin\left(\frac{\rho}{2R}\right)$$ $$\phi = \arcsin\left(\cos c \sin\alpha + \frac{y \sin c \cos\alpha}{\rho}\right)$$ $$\lambda = \beta + \operatorname{arctg}\left(\frac{x \sin c}{\rho \cos\alpha \cos c - y \sin\alpha \sin c}\right)$$

Technical note: In programming implementations, use the \(\operatorname{arctg}\) instead of a standard atan2(y, x) to ensure correctness across all quadrants.
Here’s a simple CONVERTER (x, y)<->(φ, λ)

3. Earth Radius (R) and WorldBiG

PolBiG: A specific implementation of WorldBiG for Poland (\(\alpha=52, \beta=19\)) using a constant radius of \(R = 6371\) km (consistent with Google Earth).

WorldBiG also supports the Gaussian radius (the mean radius of curvature for the WGS84 ellipsoid), providing enhanced precision for professional GIS applications.

4. Binary Structure

Recursive division ensures seamless densification. Each square side is halved at every level:

1(NW)2(NE)
3(SW)4(SE)

5. Labeling

Letter labels (e.g., hi) define 64x64 km squares. Sub-units append a binary suffix (e.g., hi 1342), creating a hierarchical and human-readable notation.

Calculator WorldBiG.org