Gnomonic ProjectionEdit

The gnomonic projection is a mathematical tool used in cartography and astronomy that maps points on the surface of a sphere onto a plane. It achieves this by projecting from the center of the sphere onto a plane that is tangent to the sphere at a chosen point. A defining feature is that every great circle on the sphere—those paths representing the shortest distance between two points—appears as a straight line on the plane. This property makes the projection particularly useful for planning routes and for analyzing spherical geometry, even though it comes with significant distortions away from the projection point.

As an azimuthal projection, the gnomonic map is part of a family of projections that emphasize directions from the center point. It has found applications beyond geographic maps, including astronomy, where star charts can be drawn by projecting the celestial sphere onto a tangent plane. The projection is named after the gnomon, an ancient instrument for measuring the sun’s position, reflecting the geometric idea of projecting from a central point onto a plane. For more on related ideas, see Azimuthal projection and Gnomon.

History

The concept of projecting from a sphere’s center onto a plane has roots dating back to early geographers who sought practical representations of a spherical world. The gnomonic projection was developed as a mathematical abstraction that could render great-circle routes as straight lines, a feature that proved valuable for navigators who needed the shortest paths between two points. Over time, the projection was analyzed in greater mathematical detail, leading to explicit formulas and a clearer understanding of distortion patterns. See also Marinus of Tyre for early discussions of spherical projection concepts, and for the general family of projections, Azimuthal projection.

Mathematical formulation

The gnomonic projection maps points on a sphere of radius R onto a plane tangent to the sphere at a chosen projection center P0, defined by latitude φ1 and longitude λ1. For any point with latitude φ and longitude λ, let Δλ = λ − λ1. The angular distance c from the projection center to the point satisfies cos c = sin φ1 sin φ + cos φ1 cos φ cos Δλ. The plane coordinates (x, y) of the projection are then given by:

x = cos φ sin Δλ / cos c
y = (cos φ1 sin φ − sin φ1 cos φ cos Δλ) / cos c

Points with cos c > 0 lie in the hemisphere visible from the projection center; points with cos c = 0 lie on the horizon and map to infinity. The projection therefore has a natural, finite extent only for the hemisphere centered on the projection point. Because the scale factor grows with c, shapes and areas become increasingly distorted toward the edge of the map. Great circles on the sphere map to straight lines on the plane, a property that underpins many navigational and analytical uses. For a broader discussion of projection geometries, see Map projection and Tissot's indicatrix.

Properties

Type and construction: gnomonic is an azimuthal projection produced by central projection from the sphere’s center onto a tangent plane at the center point.
Geometric property: all great circles map to straight lines, which simplifies plotting great-circle routes.
Distortion pattern: neither area-preserving nor angle-preserving; distortion increases with angular distance from the projection center.
Limitations: a single gnomonic map cannot cover the entire sphere without departing from a single plane; the plane representation becomes infinite at the horizon of the projection.

Applications

Navigation and flight planning: the straight-line rendition of great-circle routes makes it a convenient tool for determining shortest paths between distant points, especially in early navigation and for plotting initial route concepts. See Navigation and Great-circle.
Astronomy and star charts: for celestial mapping, the gnomonic projection projects the sky onto a tangent plane at a chosen celestial pole, with great circles on the celestial sphere (such as meridians and hour circles) appearing as straight lines. See Star chart and Celestial sphere.
Local mapping and surveying: when the area of interest is relatively small and centered near a chosen point, the projection provides minimal distortion, which can be advantageous for engineering and land use planning. See Cartography and Geodesy.

Limitations and alternatives

Because distortion grows with distance from the projection point, gnomonic maps are most reliable only for relatively small regions near the center. For global maps, other projections—such as conformal, equal-area, or compromise projections—are typically preferred, depending on whether the goal is to preserve shapes, preserve areas, or achieve a balanced representation. See Conformal map, Equal-area projection, and Compass rose for related concepts.

Controversies and debates

Projections are tools with specific goals, and choosing one reflects priorities rather than moral values. From a practical, results-oriented perspective, the gnomonic projection is valued for its exact rendering of great circles as straight lines, which makes it a natural choice for navigation calculations and for certain analytical tasks in spherical geometry. Critics who argue that map representations should be oriented toward broader social considerations sometimes claim that all map choices encode implicit biases. In response, proponents of the gnomonic approach argue that geometry itself is neutral and that distortions are mathematical realities of projecting a curved surface onto a plane. They contend that debates about “equity” in map design should not override the need for precise tools when the objective is shortest-path navigation, initial route planning, or the straightforward depiction of spherical relationships. In practice, the selection of a projection is best guided by purpose: what must be preserved, what must be measured, and how the map will be used in real-world tasks. See also Cartography and Geodesy for broader discussions of projection choices and their implications.