🗺️ GPS Map Projection

A map projection is a mathematical rule that converts positions on the curved Earth, given as longitude and latitude, into flat x and y coordinates on a map. Because a sphere cannot be unrolled onto a plane without stretching, every projection introduces some distortion of area, shape, distance or direction.

Each projection has its own forward formula applied to the grid of coordinates. Mercator uses y = ln(tan(π/4 + φ/2)); Mollweide solves an auxiliary angle by Newton iteration to keep areas correct; Equirectangular maps longitude and latitude linearly; and the azimuthal and stereographic projections project the sphere onto a plane tangent at the pole.

They are Tissot indicatrices: a tiny circle on the globe becomes an ellipse on the map. A round green ellipse means little distortion, while orange and red show increasing area or shape error, judged here by the ratio of the ellipse axes. Their orientation and size reveal exactly where and how the projection distorts the surface.

A conformal projection such as Mercator or Stereographic preserves local angles and shapes but distorts area. An equal-area projection such as Mollweide or Azimuthal Equal-Area keeps relative sizes correct but distorts shapes. No projection can be both at once, so cartographers choose based on whether accurate shapes or accurate areas matter more.

The projection formulae are mathematically faithful, so the distortion patterns shown are realistic. The coastlines, however, are a deliberately simplified outline of the continents drawn as coarse polylines, intended to illustrate how each projection bends the world rather than to serve as a precise atlas.