Every flat map distorts our spherical world. Select any latitude on Earth to calculate real-time area enlargement ratios (sec²φ), scale factors, and Tissot Indicatrix ellipse deformations across projections.
Choose a preset region or adjust the latitude slider from Equator (0°) to the North/South Pole (90°).
Preserves local angles and shape exactly, but inflates area exponentially (sec²φ) as you move toward the poles.
Projected onto a polyhedron unfolding. Keeps relative sizes and shapes of landmasses remarkably consistent worldwide.
Guarantees exact relative land areas everywhere (1.00×), but stretches continents vertically near Equator and compresses near poles.
Neither conformal nor equal-area. Striking a visual balance between shape preservation and area inflation across mid-latitudes.
One of the fundamental mathematical theorems of differential geometry is Carl Friedrich Gauss's Theorema Egregium ("Remarkable Theorem"). Gauss proved that the Gaussian curvature of a surface is invariant under local isometry. Because a sphere has constant positive curvature (+1) while a flat sheet of paper or computer screen has zero curvature (0), it is mathematically impossible to flatten the surface of the Earth onto a 2D plane without distorting area, shape, distance, or direction.
The Mercator projection is a cylindrical conformal projection created by Gerardus Mercator in 1569 for marine navigation. Being conformal means that intersecting rhumb lines (lines of constant compass bearing) cross at true right angles everywhere on the map.
To maintain these local angles, the vertical scale stretching factor (k) must equal the horizontal scale stretching factor (h) at every latitude (φ):
As latitude φ approaches 90° (the poles), cos(φ) approaches zero, making sec(φ) approach infinity!
Introduced by French mathematician Nicolas Auguste Tissot in 1859, Tissot's Indicatrix characterizes local distortion by placing infinitesimal circles on the globe and observing what happens when they are projected onto a 2D map: