Projection -Property -Globe-to-map 0%loadingFPS 0

Drag to turn the globe (or tilt the map), wheel or pinch to zoom. Orange discs are Tissot circles: every one is a true circle of equal radius on the sphere, so each ellipse shows how that map stretches shapes and areas. Cyan = great circle (shortest path), yellow = straight ruler line on the map. Pins A and B are the two cities chosen in the panel.

Map Projections: Tissot Circles, Area and Angle Distortion, Globe-to-Map Morph and Real Distances

Peel the globe flat, one projection at a time. Equal circles on the Earth show exactly how each map lies about shape and size, and a ruler on the map is compared with the real great-circle distance.

Projection

Overlays

Distance between two places

Pin APin B

Live readouts

Scale factors along the central meridian (or away from the centre)

Largest angular distortion (degrees)

Experiment: which projection lies least?

Every projection is sampled over the whole sphere (3° grid, weighted by true area). The mean absolute log of the area scale ln s measures size error, the mean ω measures shape error. A good compromise keeps both small; no map can make both zero.

Independent numerical checks

Controls

Choose a projection, then drag the Globe-to-map slider or press Flatten to watch the sphere unroll. Dragging on the picture turns the globe or tilts the map.
Central meridian turns the map to put any longitude in the middle. Centre latitude is available for the azimuthal projections only (they can be centred on any point, e.g. the pole).
Pick two places; the panel compares the true shortest distance with what a ruler measures on the map.

The science

Sphere. Points are (latitude φ, longitude λ) on a sphere of mean radius R = 6371.0088 km. A projection is a function (φ, λ) → (x, y). Everything here is computed on the sphere; the real Earth is an ellipsoid whose flattening changes distances by up to about 0.5%.

Mercator. x = Rλ, y = R ln tan(π/4 + φ/2). Scale factors: meridian h = sec φ, parallel k = sec φ, so area scales as sec² φ (about 10.5 at 72°N: Greenland looks about 10 times too big). It is conformal (angles preserved).

Lambert cylindrical equal-area. x = Rλ, y = R sin φ, so h = cos φ, k = 1/cos φ and hk = 1: areas are exact. Sinusoidal: x = Rλ cos φ, y = Rφ. Mollweide: x = (2√2/π) Rλ cos θ, y = √2 R sin θ with 2θ + sin 2θ = π sin φ. Winkel tripel: average of the equirectangular and Aitoff maps with standard parallel arccos(2/π).

Azimuthal projections (centre at angular distance c = arccos(sin φ₀ sin φ + cos φ₀ cos φ cos Δλ)): equidistant r = Rc; stereographic r = 2R tan(c/2); Lambert equal-area r = 2R sin(c/2).

Tissot indicatrix. The Jacobian J of the map (per unit angle on the sphere, with the parallel direction divided by cos φ) turns an infinitesimal circle into an ellipse whose semi-axes a ≥ b are the singular values of J. Area scale s = ab, maximum angle deviation ω = 2 arcsin((a − b)/(a + b)). Conformal maps have a = b, equal-area maps have ab = 1. The Theorema Egregium (Gauss) says a sphere has curvature, a plane has none, so no map can be both.

Distance. The great-circle angle uses the haversine formula, d = 2R arcsin √(sin²(Δφ/2) + cos φ₁ cos φ₂ sin²(Δλ/2)). The yellow ruler is the straight segment between the two pins on the flat map, converted with the equator scale, which is what a person who ignores distortion would read off.

Honest simplifications. Coastlines come from a 2048 × 1024 raster of the world, so details finer than about 20 km are not resolved. Colours are schematic. Areas quoted for Greenland (2.17 million km²) and Africa (30.4 million km²) are typical rounded values, and the apparent-area comparison uses the local area scale at one representative point of each, not an integral over the outline. The Tissot circles are drawn on the sphere as true small circles and projected exactly, so they are slightly different from the infinitesimal ellipse when the radius is large.

Not a navigation tool. Real-world surveying uses ellipsoid-based projections such as UTM.