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Stereographic map projection

Stereographic map projections use a sphere or ellipsoid to produce conformal plane maps that preserve local angles while scale and area vary.

Version
v1 · 2026-10-31 · History
Domain-specific #
14583
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomain
Map Projections → Mathematics
Aliases
Stereographic Projection

Core Idea

For the ideal spherical construction, stereographic projection draws a line from an excluded surface point through another point to the plane tangent at the antipode. The excluded point goes to infinity. The resulting map is conformal: it preserves infinitesimal angles, not global areas or distances. Ellipsoidal cartographic variants retain conformality by formulas that are not literal point projections.[1]

How would you explain it like I'm…

Globe Shadow Map

Put a tiny light at the bottom of a globe and a sheet of paper touching the top. Every spot on the globe casts a shadow onto the paper, which makes a map. Shapes of small things look right, but things near the bottom get stretched huge, and the very bottom spot can't be shown at all.

Line-Through-the-Globe Map

Stereographic map projection is a way to flatten a round globe onto paper. Pick one point on the globe, say the South Pole, and lay a flat sheet touching the opposite point, the North Pole. For any spot on the globe, draw a straight line from the South Pole through that spot and mark where it hits the sheet. Small shapes and the angles where lines cross come out correct, but sizes change across the map, and places near the South Pole spread out farther and farther. The South Pole itself never lands on the map.

Conformal Sphere-to-Plane Projection

Stereographic map projection maps a sphere onto a plane by choosing one projection point and a plane touching the sphere at the opposite point. For any other point on the sphere, draw the line from the projection point through it, and its image is where that line meets the plane. A polar map commonly projects from the South Pole onto a plane at the North Pole; the South Pole itself has no image, and nearby points run off to infinity. The map is conformal, meaning very small angles and shapes are preserved, but scale and area change with position. Circles on the sphere map to circles, or to straight lines if they pass through the projection point. For the real, slightly flattened Earth, cartographers use conformal formulas based on this idea rather than literally drawing rays.

 

Stereographic map projection, for an ideal sphere, sends every point except a chosen projection point to the plane tangent at that point's antipode: the image is where the line from the projection point through the surface point meets the plane. A north-polar version commonly projects from the South Pole onto the plane tangent at the North Pole. The projection point has no finite image, and points approaching it are sent arbitrarily far out. The construction is conformal, preserving infinitesimal angles and local shapes, while scale and area vary across the map. Spherical circles map to circles, or to straight lines when the circle passes through the omitted projection point. This gives selective fidelity: local shapes and crossing angles are faithful, but sizes depend on position. Operational cartographic variants for the ellipsoidal Earth use conformal formulas and may adopt a standard latitude away from the pole, so they are not a literal ray construction through an ellipsoid.

Scope of Application

Snyder's USGS manual describes polar, equatorial and oblique forms. NSIDC uses WGS84 polar stereographic grids for sea-ice data; northern EPSG:3413 has central longitude −45° and true scale at 70°N to limit distortion near marginal ice zones. NSIDC reports about 6% pole grid distortion. That is a configured ellipsoidal grid, not the ideal spherical tangent case.[1][2]

Clarity

For a sphere of radius R in the north-polar tangent aspect, the 60°N parallel maps to radial distance 2R tan15° ≈ 0.536R. Its local scale is 2/(1+sin60°) ≈ 1.072, so tiny areas there are magnified by roughly 1.149. Angles remain faithful while area does not; the South Pole is omitted and approaches infinite map distance.[1]

Manages Complexity

The projection separates location, local angular fidelity and variable scale. NSIDC's 70° true-scale choice shifts low distortion to the sea-ice zone rather than the pole. An equal-area grid serves areal comparison instead, at the cost of different local shape behavior. Neither is universally best.[2]

Abstract Reasoning

Projecting from the sphere center would instead produce a gnomonic map, changing the invariant. Conformality is local: it does not make all great circles straight or preserve global area. Circle-to-circle-or-line behavior belongs to the ideal spherical projection and must not be casually claimed for every ellipsoidal implementation.[1]

Knowledge Transfer

The spherical geometry transfers among polar and oblique aspects with changed coordinates. An operational Earth grid additionally needs its ellipsoid, central longitude and scale setting. A broad analogy to selective fidelity is not a verified prime parent or a license to reuse the spherical equations unchanged.[2]

Neighborhood in Abstraction Space

Stereographic map projection sits in a sparse region of the domain-specific corpus (70th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Coordinate Systems & Spatial Measures (30 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-31

References

[1] John P. Snyder, Map Projections: A Working Manual, USGS Professional Paper 1395 (1987), chapter 21 and original report PDF; indexed excerpts checked, full PDF text extraction unavailable. registry ↩a ↩b ↩c ↩d

[2] NSIDC, “A Guide to NSIDC's Polar Stereographic Projection”, tables 3–4 and grid-distortion section. registry ↩a ↩b ↩c