Compare five map projections: Equirectangular, Mercator, Mollweide, Lambert Azimuthal Equal-Area and Sinusoidal. Tissot's indicatrices reveal how each projection distorts area, shape and angles.
No flat map can perfectly represent a sphere — every projection makes trade-offs. Mercator preserves angles (conformal) but wildly distorts area near the poles. Mollweide preserves area but distorts shapes. Tissot's indicatrices show these distortions as ellipses.
Switch between projections to see the graticule reshape. Tissot's indicatrices are coloured by area-ratio distortion. Toggle layers to compare distortion patterns.
Greenland appears the same size as Africa on a Mercator map, but Africa is actually 14× larger. The Peters projection (equal-area) sparked a 'map war' in the 1970s over which projection schools should teach.
This simulation maps the curved surface of a sphere onto a flat plane using five classic projections: Equirectangular, Mercator, Mollweide, Lambert Azimuthal Equal-Area and Sinusoidal. Each is defined by explicit forward equations — for example, Mercator uses y = ln(tan(π/4 + φ/2)), while Mollweide solves 2θ + sin2θ = πsinφ by Newton iteration. A 30° graticule is drawn by projecting longitude and latitude lines point by point.
Buttons switch between the five projections and toggle the graticule and Tissot layers. Tissot's indicatrices are small geodesic circles (12° radius) projected onto the map; the code measures each projected ellipse's area by the shoelace formula and colours it by its area ratio relative to the equator. Projections underpin all of cartography, GPS and web maps, where no single map can keep both shape and area true.
What does this simulation show?
It shows how five different map projections flatten the globe, drawing a latitude/longitude graticule for each. Overlaid Tissot indicatrices reveal how shape and area are distorted at points across the map, so you can compare the trade-offs each projection makes.
What is a Tissot's indicatrix?
It is a tiny circle drawn on the sphere that becomes an ellipse once projected. Its size shows local area distortion and its elongation shows shape distortion. Here each indicatrix is a 12-degree geodesic circle sampled with 24 points and coloured by its area ratio, cyan for near-true and orange for inflated.
Why can't a flat map be perfect?
A sphere has intrinsic curvature that no flat sheet can match without stretching or tearing, a result formalised by Gauss's Theorema Egregium. Every projection must therefore sacrifice something: area, shape, distance or direction. This simulation lets you see exactly which property each projection chooses to preserve.
Each button selects one of the five projections and redraws the graticule using that projection's forward equations. The information box updates to list the projection's type, and whether area, angles and shape are preserved or distorted.
The Graticule toggle shows or hides the 30-degree latitude and longitude grid, with the equator and prime meridian drawn slightly brighter. The Tissot toggle shows or hides the indicatrix ellipses. You can turn either off to study the other more clearly.
Mercator is conformal, meaning it preserves angles and local shapes, which made it ideal for navigation because a constant compass bearing is a straight line. The cost is severe area distortion: its formula y = ln(tan(π/4 + φ/2)) sends area to infinity at the poles, so the map is clamped near 84 degrees.
These three preserve area: a region's relative size on the map matches its true size on the globe, so their Tissot ellipses all enclose roughly the same area. To achieve this they distort shape instead, which is why landmasses appear sheared or squashed near the edges of these projections.
For each indicatrix the simulation projects the small circle, then uses the shoelace formula to find the polygon's area. It compares this to the same circle drawn on the equirectangular reference at the equator. The average of these ratios is shown in the legend as the Tissot area factor.
Yes, they use the standard mathematical formulae for each projection, such as Lambert azimuthal equal-area centred on 0°N, 0°E. The Tissot ellipses are a genuine numerical estimate rather than an analytic one, so they are approximate but faithfully reflect each projection's real distortion pattern.
Web maps like Google Maps use a Mercator variant for its straight-line bearings and tiling convenience. Mollweide and equal-area projections are favoured for thematic world maps of population or climate, where comparing region sizes fairly matters more than preserving shape.