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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 an ideal sphere, stereographic projection sends each point except one chosen projection point to the plane tangent at its antipode: draw a line from the excluded point through the surface point and mark its intersection with that plane. A north-polar map commonly uses the South Pole as projection point and a plane at the North Pole. The excluded pole has no finite image; locations approaching it move ever farther out. This geometry makes the map conformal—infinitesimal angles are preserved—while scale and area are not preserved across the map.[1]

The projection's practical attraction is selective fidelity. Small local shapes and crossing angles remain faithful, but their sizes vary with position. On the sphere, circles become circles or straight lines, the latter when they pass through the omitted projection point. Operational cartographic variants for an ellipsoidal Earth use conformal formulas and may select a standard latitude away from the pole; they are not literally the same ray drawing through an ellipsoid.[1][2]

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.

Structural Signature

Sig role-phrases:

  • Surface model: the sphere in the geometric definition or an ellipsoid in an operational grid.
  • Projection center and plane: the excluded spherical point and target plane, with an aspect chosen for the region of interest.
  • Point correspondence: one planar location for each admitted surface location; the excluded point tends to infinity.
  • Local angular fidelity: a conformal map preserves the angle between tiny crossing features.
  • Scale placement: a design choice about where scale is exact or least distorted, with larger error elsewhere.[1][2]

Conformality is local, not a promise that a continent keeps its global outline or area. A tiny circle near any ordinary point is represented as a tiny circle up to local scale; far-separated features can be enlarged by very different factors. The standard parallel in a practical polar grid deliberately relocates the zone of near-true scale.[1][2]

What It Is Not

It is not a globally equal-area map, nor a map in which every great-circle route is straight. The latter is the characteristic of the gnomonic projection from the sphere's center, which is a different construction. It is also not the same as merely drawing a round polar map: polar equal-area and equidistant projections have different distortion contracts.[1]

The ideal ray construction is specifically spherical. Snyder's USGS manual explains that ellipsoidal stereographic formulas are nonperspective adaptations chosen to preserve conformality. A catalog or data product that says “polar stereographic” may therefore specify an ellipsoid, central meridian and true-scale parallel. Treating its grid coordinates as the simple tangent-sphere formula can introduce avoidable geolocation errors.[1][2]

Scope of Application

The USGS manual sets out polar, equatorial and oblique aspects, forward and inverse equations, and scale analysis. A mapmaker chooses the aspect and standard scale to place the area of interest near a low-distortion zone. The projection is valuable where local orientation or shapes matter more than comparing area, but no single aspect is a good global equal-area display.[1]

The National Snow and Ice Data Center (NSIDC) uses polar stereographic grids for sea-ice products. Its WGS84 northern grid EPSG:3413, which it recommends for new products, has a central longitude of −45° and true scale at 70°N, a choice aimed at low distortion near marginal ice zones. NSIDC reports roughly 6% grid distortion at the pole and larger distortion toward the grid edge. The selected true-scale latitude is not the same as a sphere-tangent map's true scale at 90°N.[2]

Clarity

For a spherical north-polar tangent aspect of radius R, at latitude φ the radial map distance from the North Pole is ρ = 2R tan((90°−φ)/2). At 60°N this becomes 2R tan15° ≈ 0.536R. The formula makes the center/pole relation concrete: 90°N maps to radius zero; as latitude approaches the excluded South Pole, the tangent grows without bound. This is an author calculation from Snyder's spherical equations, not a number quoted from an NSIDC grid.[1]

The same spherical aspect has local length scale k = 2/(1 + sin φ). At 60°N, k ≈ 1.072: a very small feature is enlarged by about 7.2% in every local direction, so its infinitesimal area factor is k² ≈ 1.149. Equal local directional scaling preserves angles, but the area enlargement explains why conformal does not mean equal-area. The factors would differ in NSIDC's ellipsoidal 70° true-scale setup.[1][2]

Manages Complexity

The projection separates three questions often conflated in map reading: where a location lands, which local relations are preserved, and how scale varies. It maps almost all of a sphere to a plane in a reversible way, turning local angular geometry into something usable on a grid. Yet the omitted pole and unbounded outer extent mean that no finite flat sheet depicts the whole sphere without a cut or limit.[1]

NSIDC's grid illustrates applied compromise. By making scale true at 70° rather than at the pole, it accepts about 6% distortion at the pole to reduce error in a zone important for sea-ice data. The user guide explicitly contrasts its stereographic grids with equal-area EASE grids, which answer a different measurement need. Choice of projection is therefore a choice of what to preserve, not a contest with one universally best map.[2]

Abstract Reasoning

Changing the construction point changes the invariants. Project from the sphere center and great circles become straight on a gnomonic plane, but conformality is lost. Keep stereographic conformality and local angles survive, while scale must vary with position. A cartographic assertion should thus name the projection, aspect, datum and scale setting before applying a property.[1]

The circle property has a similarly bounded logic. On the ideal sphere, a circle through the omitted point reaches the planar point at infinity and appears as a straight line; another spherical circle appears as a plane circle. That is geometric circle preservation on the extended plane, not a claim that all mapped geographic routes are straight or that ellipsoidal circles inherit the exact spherical theorem without qualification.[1]

Knowledge Transfer

One can transfer the spherical stereographic argument to a different pole or an oblique center by rotating the surface and plane; the omitted point and local-angle property remain, while coordinates change. Transferring it to Earth-data grids requires more care: NSIDC's WGS84 configuration uses ellipsoidal formulas, EPSG code and a true-scale parallel. The function of conformality transfers, but the simple spherical radial number does not.[1][2]

This projection has a particular conformal distortion pattern; a general map projection need not preserve the same properties.

Examples

  1. Snyder's spherical north-polar tangent calculation. Choose a sphere of radius R, project from South Pole onto a plane at North Pole, and take a surface point at 60°N. Snyder's radial and scale formulas give ρ ≈ 0.536R and k ≈ 1.072, hence local area factor about 1.149. Mapped back: surface = sphere; center/plane = South Pole/North Pole tangent plane; correspondence = finite radial point for 60°N, infinity only at excluded South Pole; angular fidelity = same small length scale in all directions; scale placement = exact at North Pole, already enlarged at 60°N. The numerical substitutions are derived here from the manual's equations.[1]

  2. NSIDC WGS84 northern sea-ice grid. Its EPSG:3413 specification uses a north-polar stereographic aspect, WGS84 ellipsoid, central longitude −45°, and true-scale latitude 70°N. NSIDC reports about 6% grid distortion at the pole while keeping lower error near marginal ice zones. Mapped back: surface = WGS84 ellipsoid; center/plane = polar stereographic orientation implemented by conformal ellipsoidal formulas; correspondence = specified geographic-to-grid transform; angular fidelity = conformal projection; scale placement = deliberately true at 70°N rather than at the pole. This operational variant is not the preceding literal sphere-ray construction.[2][1]

Structural Tensions

Local angular fidelity versus area fidelity. Stereographic conformality keeps infinitesimal angles and local shapes faithful, which can be valuable for orientation and feature geometry. Its varying scale distorts area away from the chosen true-scale region, making broad areal comparisons unreliable. An equal-area alternative reverses that priority but cannot preserve every local angle as this projection does. Diagnostic: does the task require local shapes/directions or areal totals, and where must scale be best? NSIDC's 70° choice also shows a within-projection trade: reducing distortion in an application zone shifts nonzero distortion to the pole.[1][2]

Structural–Framed Character

The sphere-to-plane correspondence and angle/scale relation are structural mathematical properties, not social conventions. Their use is evaluative and institutionally framed: USGS documentation and NSIDC grids select aspects and standard latitudes for mapping purposes. Human practice does not cause conformality, but it determines which distortion is acceptable. Vocabulary can travel from ideal geometry to ellipsoidal sea-ice grids only with the changed surface and formulas acknowledged. Recognizing a circular polar map is not enough to import the stereographic name or its scale properties. Its character: a mathematically defined projection family with application-dependent distortion placement and a strict spherical-versus-ellipsoidal boundary.[1][2]

Structural Core vs. Domain Accent

The shared family property is a locally conformal map with varying scale; the one-point-omitting ray construction describes the ideal sphere, while operational ellipsoidal forms use adapted formulas. The domain mechanism is cartographic coordinate transformation with a selected aspect, datum and standard scale; the ideal tangent sphere and NSIDC's WGS84 polar grid are accents. The named entry fails the prime bar because its defining conformal and circle properties require this particular geometry, not a portable social or computational relation. A broader prime about selective fidelity would need independent unlike-domain instances and cannot inherit these cartographic formulas. No strict parent is asserted.[1]

Gnomonic Projection uses a different spherical projection center and makes great-circle arcs straight. Projection, as a general abstraction, does not by itself specify this conformal map.

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

Not to Be Confused With

  • Gnomonic projection from the sphere center, which makes great circles straight but has a different distortion pattern.
  • An equal-area polar grid.
  • A claim that conformal means constant distance or area scale across a whole map.
  • Applying the simple spherical tangent formula unchanged to an ellipsoidal, standard-parallel operational grid.[1][2]

References

[1] John P. Snyder, Map Projections: A Working Manual, USGS Professional Paper 1395 (1987), chapter 21 “Stereographic,” with original report PDF. The manual's original metadata and indexed chapter excerpts were checked; direct web text extraction of the full PDF failed. Numerical values here are author substitutions into the spherical formulas. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s

[2] National Snow and Ice Data Center, “A Guide to NSIDC's Polar Stereographic Projection”, first-party operational specification, tables 3–4 and “Grid distortion” section; WGS84 EPSG:3413/3976 and 70° true-scale setting. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l