This simulator shows the unavoidable trade-offs of flattening a round Earth onto a flat map. It renders a coastline outline and graticule under five classic map projections — Mercator, Mollweide, Equirectangular, Azimuthal Equal-Area and Stereographic — each computed from the latitude–longitude grid using its own forward formula. Overlaid Tissot indicatrices turn infinitesimal circles on the sphere into ellipses on the map, making area and angle distortion visible at a glance. Because no flat map can preserve area, shape and distance at once, comparing projections reveals what each one sacrifices.
Each projection maps geographic coordinates (lon, lat) to map x, y with its forward equation — for example Mercator uses y = ln(tan(π/4 + φ/2)), while Mollweide solves an auxiliary angle θ by Newton iteration to stay equal-area. The Tissot indicatrix draws a circle where there is no distortion and an ellipse where shape or area is stretched, colour-coded green, orange or red by the ratio of its axes.
Click a projection button (Mercator, Mollweide, Equirectangular, Azimuthal EA or Stereographic) to redraw the map. Use the three checkboxes to toggle the Graticule (lat/lon grid), the Tissot Indicatrix ellipses and the Coastlines. The side panel reports the projection name, its type (conformal, equal-area or neither) and its area and angle distortion characteristics.
The Mercator projection inflates area without limit toward the poles, which is why Greenland can look as large as Africa even though Africa is about fourteen times bigger. Mercator survives because it is conformal: it preserves local angles, so a constant compass bearing plots as a straight line — invaluable for navigation.
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.
The Earth is not flat, but maps are—map projections are mathematical transformations that convert positions on the three-dimensional ellipsoidal surface of the Earth to a two-dimensional plane. Every projection introduces distortions: it is mathematically impossible to represent a curved surface on a plane without distorting either shape (conformal vs. non-conformal), area (equal-area vs. non-equal-area), distance, or direction. The choice of projection depends on the application—navigators need conformal projections (preserving angles and shapes), while area calculations require equal-area projections.
The Mercator projection, developed in 1569 by Gerardus Mercator for nautical navigation, is conformal (preserves local shapes and angles) but severely distorts area toward the poles—Greenland appears larger than Africa, though Africa is actually 14 times larger. The Web Mercator (EPSG:3857) used by Google Maps and OpenStreetMap is a spherical simplification of Mercator that enables efficient tile caching. The Transverse Mercator, Universal Transverse Mercator (UTM), and various national grid systems are used for accurate local mapping and surveying, trading broad coverage for local accuracy.
This simulator demonstrates multiple map projections (Mercator, Mollweide, Robinson, Sinusoidal, Azimuthal Equidistant), visualizes the Tissot indicatrix (small circles placed on the globe that distort to ellipses on the map, revealing the nature and magnitude of distortion), and converts GPS coordinates (latitude/longitude) to projected plane coordinates. You can observe how different projections preserve different properties and how GPS coordinates relate to the multiple reference frames (datums) used in mapping.
What is the difference between a map datum and a map projection?
A datum is the mathematical model of the Earth's shape and size used as the reference for geographic coordinates (latitude/longitude). WGS84 (World Geodetic System 1984) is the global datum used by GPS—it defines an ellipsoid that best fits the entire Earth's surface. NAD83 is the North American datum; local datums are tailored to regional surveys. A map projection then converts the 3D ellipsoidal coordinates to 2D plane coordinates. Using the wrong datum with a projection can introduce coordinate errors of hundreds of meters—critical in engineering and legal boundary surveys.
Why does the Mercator projection make Greenland look bigger than Africa?
Mercator is conformal—it preserves local angles and shapes, making navigation by compass bearing (rhumb lines appear as straight lines on the map) straightforward. To achieve conformality, Mercator scales distances equally in all local directions at each point. As latitude increases toward the poles, the scaling factor increases dramatically: at 60° latitude, both north-south and east-west scales are doubled compared to the equator. This uniform local scaling preserves shape but doubles area at 60°N/S and inflates areas by a factor of ~576 near the poles, making Greenland (~2.2 million km²) appear similar in size to Africa (~30 million km²).
What is the Universal Transverse Mercator (UTM) system?
UTM divides the Earth into 60 longitude zones of 6° width and applies a Transverse Mercator projection (the cylinder of the Mercator is rotated 90°, wrapping around a meridian rather than the equator) to each zone. Within each zone, UTM provides accurate conformal mapping with scale error less than 1 part in 1000—sufficient for large-scale engineering and topographic mapping. Coordinates are given in meters of Easting (from a false origin 500,000 m west of the central meridian) and Northing (from the equator, with 10,000,000 m added in the southern hemisphere). UTM underlies most national mapping agencies and military grid reference systems.
The Tissot indicatrix, introduced by French cartographer Nicolas Auguste Tissot in 1859, places small circles of equal size on the globe's surface and projects them onto the map. The resulting ellipses at each location reveal the local distortion: if the ellipse is a circle (same size as others), there is neither shape nor area distortion. Elongated ellipses indicate shape distortion; circles of different sizes indicate area distortion; both deformations together indicate combined distortion. Conformal projections (Mercator) show circles of varying sizes (no shape distortion, area distortion); equal-area projections show ellipses of equal area but varying shapes.
GPS receivers calculate position by trilateration: measuring the time-of-flight of radio signals from multiple satellites (each broadcasting its precise orbital position and transmission time) to determine distance (d = c·Δt, where c is the speed of light). Four satellite distances are needed: three to determine 3D position and one to correct the receiver's imprecise clock. Atmospheric delays (ionospheric and tropospheric refraction) and multipath reflections introduce errors. Differential GPS (DGPS), Wide Area Augmentation System (WAAS), and Real-Time Kinematic (RTK) positioning correct these errors by comparing measurements with known fixed receivers, achieving centimeter-level accuracy for surveying.