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MATHEMATICAL ATLAS · 14 PROJECTIONS INDEXED

The Projection Lab

Every map projection is a mathematical compromise between area, conformal angle, distance, and direction. Explore 14 landmark projections with real-time Tissot indicatrix deformation metrics.

THE PROJECTION DISTORTION CRUCIBLE

Mercator Projection (1569)

Inventor: Gerardus Mercator · Family: Cylindrical

01 / THE REFERENCE

One round Earth.

A curved surface. Every flat map makes a different compromise.
02 / PRESERVE LOCAL ANGLES

Mercator

Area factor at 60°N: 4.00×. Circles preserve their shape.
03 / PRESERVE RELATIVE AREA

Gall–Peters

Area factor at 60°N: 1.00×. Shapes change while areas stay equal.
TEST LATITUDE:60° North
0° (Equator)30° (Subtropics)60° (Greenland)85° (Arctic)
Conformal (Angles)
Equal-Area (Sizes)
Compromise Map
Uninterrupted
CONTINENTAL SCALE DISTORTION RATIO:
True geographic area of Greenland (2.16M km²) is 1/14th (0.07×) of Africa (30.37M km²).
Projected Visual Ratio:1:1 (Equal Size!)
TISSOT INDICATRIX (Lat: 60°N)Area: 4.00×
h=2.00k=2.00
MERIDIAN (h)
2.00
PARALLEL (k)
2.00
ANGULAR (2θ')
0.0°
Cartographic & Political Critique:

"When co-opted into classrooms and wall atlases, Mercator visually inflates imperial powers in North America and Europe while drastically shrinking the African continent, South America, and the equatorial belt."

Formula: h = k = \sec \phiFull Mathematical Dossier

Complete Mathematical Projection Directory

Mercator Projection
Cylindrical
1569
Inventor: Gerardus MercatorEPSG Datum: Spherical

Mercator Projection

Engineered strictly as a marine navigation tool, Mercator stretches latitude intervals toward the poles at the exact rate needed to preserve loxodromes (rhumb lines) as straight lines. This produces monumental area inflation in polar latitudes.

Conformal
Tissot IndicatricesInspect Math
Gall-Peters Projection
Cylindrical
1973
Inventor: James Gall (1855) / Arno Peters (1973)EPSG Datum: Spherical

Gall-Peters Projection

Championed by Arno Peters in the 1970s as a politically just corrective to Mercator, Gall-Peters preserves absolute surface area proportions across the globe at the cost of extreme shape shearing at the poles and vertical elongation at the equator.

Equal-Area
Tissot IndicatricesInspect Math
Robinson Projection
Pseudocylindrical
1963
Inventor: Arthur H. RobinsonEPSG Datum: Spherical

Robinson Projection

Commissioned by the Rand McNally publishing company, Robinson created an empirical lookup table of coordinates designed to minimize both shape and area distortion across the global oceans and continents simultaneously.

Compromise
Tissot IndicatricesInspect Math
Winkel Tripel Projection
Pseudocylindrical
1921
Inventor: Oswald WinkelEPSG Datum: Spherical

Winkel Tripel Projection

Tripel ('triple') refers to Winkel's objective to simultaneously minimize three distortions: area, direction, and distance. Adopted by National Geographic in 1998 to replace Robinson.

Compromise
Tissot IndicatricesInspect Math
Dymaxion Projection (Fuller)
Polyhedral
1954
Inventor: R. Buckminster FullerEPSG Datum: Spherical

Dymaxion Projection (Fuller)

Projected onto the 20 triangular facets of an icosahedron and unfolded flat with no standard 'North-Up' orientation, Fuller's Dymaxion map presents the Earth's continents as an almost contiguous landmass with minimal local shape or area distortion.

CompromiseInterrupted
Tissot IndicatricesInspect Math
Peirce Quincuncial Projection
Quincuncial
1879
Inventor: Charles Sanders PeirceEPSG Datum: Spherical

Peirce Quincuncial Projection

Invented by the American pragmatist philosopher and mathematician C.S. Peirce, this projection maps the globe onto a square where the North Pole is at the center and the South Pole is divided among the four corners, forming a tessellatable quincunx.

Conformal
Tissot IndicatricesInspect Math
Goode Homolosine Projection
Pseudocylindrical
1923
Inventor: John Paul GoodeEPSG Datum: Spherical

Goode Homolosine Projection

Goode combined the Sinusoidal projection (between 40°44'N and 40°44'S) with the Mollweide projection for higher latitudes, interrupting the ocean basins to allow each major continental landmass to be centered on its own local meridian.

Equal-AreaInterrupted
Tissot IndicatricesInspect Math
Mollweide Projection
Pseudocylindrical
1805
Inventor: Karl Brandan MollweideEPSG Datum: Spherical

Mollweide Projection

Published in Leipzig as an improvement on Mercator for astronomical and global geographic distributions, Mollweide represents the entire spherical world within an exact 2:1 ellipse.

Equal-Area
Tissot IndicatricesInspect Math
Lambert Conformal Conic (LCC)
Conic
1772
Inventor: Johann Heinrich LambertEPSG Datum: Spherical

Lambert Conformal Conic (LCC)

Lambert’s mathematical tour de force projects the sphere onto a cone intersecting the globe at two standard parallels. Meridians are straight lines converging at the pole; parallels are concentric circular arcs.

Conformal
Tissot IndicatricesInspect Math
Albers Equal-Area Conic
Conic
1805
Inventor: Heinrich C. AlbersEPSG Datum: Spherical

Albers Equal-Area Conic

Using two standard parallels where scale is exact, Albers Equal-Area Conic preserves true proportional surface area across large east-west landmasses like the contiguous United States, Europe, and China.

Equal-Area
Tissot IndicatricesInspect Math
Azimuthal Equidistant Projection
Azimuthal
1581
Inventor: Al-Biruni (c. 1000 CE) / Guillaume Postel (1581)EPSG Datum: Spherical

Azimuthal Equidistant Projection

Centered on any chosen point (such as the North Pole in the United Nations emblem or a radio broadcasting antenna), every straight line radiating from the center is a true great circle with exact linear scale.

Equidistant
Tissot IndicatricesInspect Math
Orthographic Projection
Azimuthal
-150
Inventor: Hipparchus (c. 150 BCE) / Classical AntiquityEPSG Datum: Spherical

Orthographic Projection

A perspective azimuthal projection where light rays originate from infinity. It portrays the Earth exactly as it appears from deep space, displaying exactly one hemisphere at a time with foreshortening increasing toward the horizon limb.

Compromise
Tissot IndicatricesInspect Math
Waterman Butterfly Projection
Polyhedral
1996
Inventor: Steve WatermanEPSG Datum: Spherical

Waterman Butterfly Projection

Waterman developed this projection based on packing spheres and unfolding a truncated octahedron (WKS polyhedra). Continents remain remarkably compact and undistorted while displaying ocean connectivity.

CompromiseInterrupted
Tissot IndicatricesInspect Math
AuthaGraph Projection
Polyhedral
1999
Inventor: Hajime NarukawaEPSG Datum: Spherical

AuthaGraph Projection

Created by Japanese architect Hajime Narukawa, AuthaGraph divides the spherical earth into 96 regions, projects them onto an inflated tetrahedron, and unfolds into an approximate equal-area rectangle that can tile infinitely.

Equal-Area
Tissot IndicatricesInspect Math