Gnomonic Projection¶
A center-of-sphere mapping of an open hemisphere to a tangent plane that turns visible great-circle arcs into straight lines.
Core Idea¶
The gnomonic projection sends points on a sphere to a plane by extending a line from the sphere's center through each point until it meets a plane tangent to the sphere at a chosen chart center. On the plane-facing open hemisphere, each source point has one finite image. The defining geometric reward is that every visible great-circle arc becomes a straight line. The reason is structural: a great circle lies in a plane through the sphere's center, and that plane intersects the tangent map plane in a line.[1][2]
This line property does not mean that the chart preserves length, area, angles or the whole globe. If \(R\) is sphere radius and \(c\) is angular distance from the tangent point, planar radial distance is \(\rho=R\tan c\). As \(c\) approaches \(90^\circ\), the radius and scale diverge; horizon points have no finite image. Only less than a hemisphere fits on any finite sheet, and distortion increases rapidly away from the center.[1][2] Thus the projection makes a particular kind of line easy to reason with in exchange for severe coverage and metric limitations.
For a concise coordinate statement, take a sphere of radius \(R\), a unit vector \(\mathbf n\) toward the chart center, and a point \(\mathbf p\) on the sphere with \(\mathbf n\cdot\mathbf p>0\). The tangent plane is \(\mathbf n\cdot\mathbf q=R\), and central projection gives \(\mathbf q=R\mathbf p/(\mathbf n\cdot\mathbf p)\). Choosing axes within the plane then yields ordinary two-dimensional map coordinates. The denominator encodes the horizon: at \(\mathbf n\cdot\mathbf p=0\) the finite image disappears. This coordinate form is a direct expression of the USGS central-ray construction, not an extra claim about an ellipsoidal Earth.[1]
Structural Signature¶
Sig role-phrases: spherical directional carrier → sphere-center rays → selected tangent plane → front open-hemisphere domain → great-circle line invariant → radial distortion gradient.
- Spherical directional carrier. Earth positions on a spherical reference model or sky directions on a celestial unit sphere supply the source points and their great circles. Replace the sphere by an arbitrary surface and the same straight-line theorem does not follow.[1][3]
- Sphere-center rays. The source point is joined to the sphere center, which is the projection point. Moving it to a point on the surface gives a different perspective mapping; sending it to infinity gives a parallel projection. Centrality is what makes every great-circle plane project to a plane line.[1]
- Selected tangent plane. A chart plane touches the sphere at its chosen center. That choice locates the locally undistorted center and determines which hemisphere faces the plane. Rotating or recentering changes coverage but not the construction.[2]
- Front open-hemisphere domain. Only source points with a ray meeting the plane on the chosen forward side and at finite distance form the one-to-one chart. The horizon great circle has no finite image, and the back hemisphere is not part of the ordinary chart.[1][2]
- Great-circle line invariant. Any great-circle arc that remains inside that domain appears on a straight line. This is the property used for route plotting; it does not straighten arbitrary small circles or make planar line length equal spherical distance.[1][4]
- Radial distortion gradient. \(\rho=R\tan c\) and increasing radial scale make the outer field unsuitable for uncorrected metric reading. Eliminating that cost while retaining the same plane and central rays is not an available adjustment; the geometry couples the two.[1][2]
What It Is Not¶
- It is not a finite map of an entire hemisphere. Points at its horizon are at infinity, so any finite chart contains strictly less than the open hemisphere's full angular range.[2]
- It is not a map of the back hemisphere by identifying antipodes. Such identification belongs to an optional projective extension, not the ordinary one-to-one, front-facing geographic or celestial chart. Treating both sides as one map would conflate distinct places or directions.
- It is not a distance- or area-preserving map. The line property selects one geometric invariant and sacrifices others. Only the center is free of local distortion.[2]
- It is not an exact ellipsoidal-geodesic straightener. The theorem concerns great circles of the modeled sphere; the shortest path on a reference ellipsoid need not be exactly that spherical great circle.[1]
- It is not Mercator. A straight Mercator course is a rhumb line of constant bearing, not in general a great circle; the two charts answer different navigation questions.[4][5]
- It is not every rectilinear photograph. A camera can generate analogous central mapping of viewing directions to an image plane, but the title's cartographic/celestial identity requires an explicit spherical directional carrier and tangent-plane geometry.
Scope of Application¶
Gnomonic projection is used where a spherical directional problem benefits from line images of great-circle arcs. In terrestrial cartography, a chart may assist great-circle route inspection, but one uses other chart conventions and calculations for operational bearings and distances. A NOAA-hosted hydrographic manual describes plotting a long great-circle track on a gnomonic chart and transferring selected positions to a Mercator chart, where course work is easier.[4][5]
In positional astronomy, the same geometry is applied to celestial directions. Starlink's SLALIB takes stellar right ascension and declination and a chosen tangent point and returns gnomonic tangent-plane coordinates; its status distinctions flag cases too far from the axis or on the antistar side. Its ASTROM plate model uses a conventional gnomonic transformation around a plate center before a separate optical-distortion adjustment. This is a literal second setting, not simply a metaphorical “sky map.”[3][6]
The exact all-great-circles-straight statement is spherical. Earth is not a perfect sphere, so a real geodesic or high-precision survey may need ellipsoidal methods. It also does not say that a physical aircraft or ship must or will follow the drawn arc; currents, winds, constraints, and course management are separate issues.
Clarity¶
The name is often summarized as “great circles become straight lines,” which is true only after saying where. The complete statement is: a great-circle arc lying within the selected front open hemisphere maps to a finite straight line segment. A great circle as a full closed loop crosses the chart horizon; no single ordinary gnomonic sheet displays it in full.[1][2]
The distinction between line and metric is equally important. A straight segment on this plane identifies a great-circle course in the spherical model, but its planar length is not automatically the spherical arc length. The difference grows away from the tangent point because the radial tangent relation accelerates toward the horizon. A user seeking true area or stable global appearance should choose another projection.[1]
Manages Complexity¶
On a sphere, a shortest route between non-antipodal points is part of a great circle. Gnomonic projection converts the shape of such an in-domain route into a plane-line problem. This can simplify drawing, intersection and comparison of great-circle arcs without repeatedly visualizing them on a curved globe. In astronomy, a local celestial neighborhood similarly becomes a two-coordinate tangent-plane working space for star positions.[1][4][3]
The simplification does not erase its own bookkeeping. A useful chart must retain its tangent point, hemisphere, spherical reference model and distortion field. If those are suppressed, the deceptively simple straight line invites misuse as a constant-bearing or constant-scale line. The abstraction manages complexity by converting one geometric invariant into planar form while making the lost invariants explicit.
Abstract Reasoning¶
Given two source directions \(\mathbf p_1,\mathbf p_2\) within the same chart hemisphere, project them from the sphere center to the tangent plane. The plane through the sphere center and both directions defines their great circle; its intersection with the tangent plane is the straight line through their images. This proves why the plotted straight segment represents the visible spherical great-circle arc, subject to the usual non-antipodal endpoint assumption.[1]
Before drawing a conclusion from the chart, test whether the relevant source points and arc stay in the front domain, whether the spherical approximation is adequate, and whether the desired output is great-circle geometry rather than true planar distance or constant bearing. For celestial astrometry, also distinguish the ideal tangent-plane coordinates from instrumental distortion; Starlink models that distortion separately.[1][6]
Knowledge Transfer¶
The construction transfers literally from an Earth-centered spherical map to a celestial sphere of viewing directions: the carrier is a sphere, rays emerge from its center, a tangent plane is chosen, and great circles map to lines. A navigator uses the line property to inspect a modeled route; an astronomer uses the same mapping to express nearby right-ascension/declination positions in a plate plane. The purposes differ, but the geometric roles are unchanged.[4][3]
That literal transfer does not extend automatically to an arbitrary camera image, a small circle or an ellipsoidal geodesic. A broader “central projection preserves projective lines” skeleton may be investigated as a future-prime question. The named gnomonic construction remains domain-specific to spherical/tangent-plane geometry.
Examples¶
Geographic great-circle route chart¶
Imagine two distant geographic points represented on a spherical reference globe, both within a gnomonic chart centered near their route. Their great-circle plane passes through the globe center. On the chart, the visible route appears as a straight segment. The NOAA-hosted hydrographic account describes the historical/technical use of such a chart to inspect a long great-circle path and then transfer route positions to a Mercator chart for subsequent course handling. The straight gnomonic line is a route-shape aid, not a promise that planar inches equal nautical distance.[1][4]
Mapped back: spherical directional carrier = modeled Earth sphere and route endpoints; sphere-center rays = lines from the globe center through geographic positions; selected tangent plane = regional chart surface; front open-hemisphere domain = the route segment inside the chosen chart's view; great-circle line invariant = straight plotted route segment; radial distortion gradient = distances and bearings require other measurement conventions, especially far from the chart center.
Celestial tangent-plane astrometry¶
Take nearby stars specified by right ascension and declination around a chosen plate center. Starlink's SLALIB maps those spherical sky directions into gnomonic tangent-plane standard coordinates, and ASTROM uses the same ideal projection before modeling optical departures from it. Here the task is to relate angular sky coordinates to a flat detector or plate model, not to plan a journey. The great-circle theorem remains a property of the mapping, even when no observer is drawing a route.[3][6]
Mapped back: spherical directional carrier = celestial sphere of star directions; sphere-center rays = radial directions in the angular model; selected tangent plane = plate-centered coordinate plane; front open-hemisphere domain = stars in the finite field around that center; great-circle line invariant = any chart-contained celestial great-circle arc is straight; radial distortion gradient = tangent-plane scale varies with angular offset, while optical distortion is a further, separately modeled effect.
Boundary: a Mercator rhumb line¶
A straight line on a Mercator navigation chart can indicate a constant-bearing rhumb line rather than the shortest great-circle path. Its straightness does not establish the sphere-center rays or great-circle line invariant of the gnomonic construction. This is a negative comparison, not a third positive gnomonic case.[5]
Structural Tensions¶
T1 — Straight great circles versus coverage. Central projection makes visible great-circle arcs straight, but the horizon flees to infinity and one finite sheet cannot display half the sphere, much less all of it. Moving outward from the center gains angular reach only at rapidly increasing scale cost. Diagnostic: Does the desired region fit comfortably inside one chart, or would recentering and multiple charts be clearer?[2]
T2 — Line geometry versus metric fidelity. Plane-line simplicity aids route shape and sky-coordinate work, but area, length, angle and local shape increasingly distort away from the center. Trusting the line invariant while measuring uncorrected planar lengths or areas trades a genuine benefit for a false metric. Diagnostic: Is the intended inference about the locus of a great circle or about quantities the projection does not preserve?[1]
T3 — Spherical exactness versus physical-Earth accuracy. The all-great-circles-straight theorem is exact on a sphere, whereas higher-accuracy terrestrial work may use an ellipsoid whose geodesics differ. Keeping the spherical model simplifies chart reasoning; upgrading the Earth model changes what counts as a shortest route. Diagnostic: Is a spherical great-circle approximation sufficient for the decision at hand?[1]
Structural–Framed Character¶
The mapping is strongly structural within geometry. Evaluative weight: none in the definition; “good for navigation” is a task judgment. Human-practice dependence: low for the theorem, moderate for choosing a chart center and tolerable distortion. Institutional origin: none constitutes the mapping; USGS and observatory conventions document it, not create it. Vocabulary travel: “gnomonic” transfers literally between geographic and celestial spheres, but camera-graphics uses require an explicit direction-sphere construction rather than a name match. Import versus recognition: applying the term recognizes center-of-sphere/tangent-plane geometry; it does not import a cartographic preference into unrelated mappings.[1][3]
Its character: a domain-specific formal projection of spherical geometry. Its possible broader projective skeleton remains a future-prime question; observed use in two fields does not remove the spherical carrier from this identity.
Structural Core vs. Domain Accent¶
The core relation is central rays from a sphere to a plane, with the incident-plane argument turning great circles into straight image lines. The cost follows from the same construction: \(\rho=R\tan c\) and a nonfinite horizon. These together, not merely the word “projection,” identify the entry.
The domain accent is the explicit sphere, tangent plane, chart hemisphere and great-circle geometry. Remove them and the line theorem no longer identifies gnomonic projection. No live prime parent is asserted here: broad Projection is related, but its current V2 treats idempotence as constitutive, whereas a sphere-to-plane map cannot be applied twice with matching input/output types. A catalog-level clarification might someday establish a clean parent. Until then the portable central-projection motif is a future-prime question, and the draft stays an explicitly unparented DAG proposal rather than inventing a strict edge.
Instantiates / Related Primes¶
- No asserted parent (proposed root). None of the checked live nodes is a necessary genus with a compatible full definition.
- Related, declined as strict parent — Projection. It names a much broader mapping family and includes cartographic cases, but its live idempotence requirement blocks strict type-correct inheritance here. A future catalog repair may change that decision.
- Related, not parent — Perspective. Its live identity concerns representation of depth to a viewer, not the spherical great-circle theorem.
- Sibling contrasts — Hammer and Wiechel Projections. They also map a sphere to a plane but prioritize different invariants; neither is a genus of this central construction.
- Sibling contrast — Isometric Projection. It uses parallel orthographic rays from a 3D object and equal axis scale, not sphere-center rays.
Neighborhood in Abstraction Space¶
Gnomonic Projection sits in a moderately populated region (55th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Named Physical Phenomena & Theoretical Constructs (16 abstractions)
Nearest neighbors
- Equirectangular Projection — 0.89
- Albers Equal-Area Conic Projection — 0.88
- Vincenty's formulae — 0.86
- Supplementary Angles — 0.84
- Stationary synchronous orbit — 0.84
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
- Mercator projection. Tell: straight rhumb lines express constant bearing; general great-circle routes curve there.[5]
- Orthographic or isometric projection. Tell: parallel rays replace the sphere-center point, so the universal visible-great-circle straight-line property is absent.
- Stereographic projection. Tell: its projection point lies on the sphere; it has different circle and angle properties.
- Hammer or Wiechel projection. Tell: their area-preserving constructions do not use the gnomonic center-ray/tangent-plane relation.
- Rectilinear camera view. Tell: an image plane and central ray alone are not enough unless source directions are explicitly treated as points of a sphere and the gnomonic domain/invariant applies.
- An ellipsoidal geodesic map. Tell: the exact gnomonic theorem applies to spherical great circles, not automatically to the shortest paths on a reference ellipsoid.
References¶
[1] John P. Snyder, Map Projections—A Working Manual, U.S. Geological Survey Professional Paper 1395 (1987), §22 “Gnomonic Projection,” pp. 164–168. https://pubs.usgs.gov/pp/1395/report.pdf . registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r
[2] John P. Snyder and Philip M. Voxland, An Album of Map Projections, U.S. Geological Survey Professional Paper 1453 (1989), “Gnomonic Projection,” printed p. 116. https://pubs.usgs.gov/pp/1453/report.pdf . registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i
[3] Starlink Project, SLALIB—Positional Astronomy Library, “SLA_S2TP: Spherical to Tangent Plane,” input/output and notes. https://starlink.eao.hawaii.edu/star/OLD_20240118/docs/sun67.htx/sun67ss166.html . registry ↩a ↩b ↩c ↩d ↩e ↩f
[4] Shalowitz, Interpretation and Use of Nautical Charts, chapter 6, printed p. 349 and note 150, NOAA-hosted original hydrographic text on gnomonic route plotting and transfer to Mercator chart. https://nauticalcharts.noaa.gov/publications/docs/shore-and-sea-boundaries/volume-2/cse-library-shalowitz-v2-p2-ch6.pdf . registry ↩a ↩b ↩c ↩d ↩e ↩f
[5] U.S. Office of Coast Survey, “Nautical Cartography,” contrasting straight constant-course rhumb lines on Mercator charts. https://nauticalcharts.noaa.gov/learn/nautical-cartography.html . registry ↩a ↩b ↩c ↩d
[6] Starlink Project, ASTROM Basic Astrometry Program, User Note 5.18, “Method,” p. 8; conventional gnomonic projection of celestial coordinates to ideal plate coordinates before separate distortion modeling. https://starlink.eao.hawaii.edu/docs/sun5.pdf . registry ↩a ↩b ↩c