Gnomonic projection

Last updated August 21, 2024

A gnomonic projection, also known as a central projection or rectilinear projection, is a perspective projection of a sphere, with center of projection at the sphere's center, onto any plane not passing through the center, most commonly a tangent plane. Under gnomonic projection every great circle on the sphere is projected to a straight line in the plane (a great circle is a geodesic on the sphere, the shortest path between any two points, analogous to a straight line on the plane).^[1] More generally, a gnomonic projection can be taken of any $n$ -dimensional hypersphere onto a hyperplane.

The projection is the $n$ -dimensional generalization of the trigonometric tangent which maps from the circle to a straight line, and as with the tangent, every pair of antipodal points on the sphere projects to a single point in the plane, while the points on the plane through the sphere's center and parallel to the image plane project to points at infinity; often the projection is considered as a one-to-one correspondence between points in the hemisphere and points in the plane, in which case any finite part of the image plane represents a portion of the hemisphere.^[2]

The gnomonic projection is azimuthal (radially symmetric). No shape distortion occurs at the center of the projected image, but distortion increases rapidly away from it.

The gnomonic projection originated in astronomy for constructing sundials and charting the celestial sphere. It is commonly used as a geographic map projection, and can be convenient in navigation because great-circle courses are plotted as straight lines. Rectilinear photographic lenses make a perspective projection of the world onto an image plane; this can be thought of as a gnomonic projection of the image sphere (an abstract sphere indicating the direction of each ray passing through a camera modeled as a pinhole). The gnomonic projection is used in crystallography for analyzing the orientations of lines and planes of crystal structures. It is used in structural geology for analyzing the orientations of fault planes. In computer graphics and computer representation of spherical data, cube mapping is the gnomonic projection of the image sphere onto six faces of a cube.

In mathematics, the space of orientations of undirected lines in 3-dimensional space is called the real projective plane, and is typically pictured either by the "projective sphere" or by its gnomonic projection. When the angle between lines is imposed as a measure of distance, this space is called the elliptic plane. The gnomonic projection of the 3-sphere of unit quaternions, points of which represent 3-dimensional rotations, results in Rodrigues vectors. The gnomonic projection of the hyperboloid of two sheets, treated as a model for the hyperbolic plane, is called the Beltrami–Klein model.

History

The gnomonic projection is said to be the oldest map projection, speculatively attributed to Thales who may have used it for star maps in the 6th century BC.^[2] The path of the shadow-tip or light-spot in a nodus-based sundial traces out the same hyperbolae formed by parallels on a gnomonic map.

Properties

The gnomonic projection is from the centre of a sphere to a plane tangent to the sphere (Fig 1 below). The sphere and the plane touch at the tangent point. Great circles transform to straight lines via the gnomonic projection. Since meridians (lines of longitude) and the equator are great circles, they are always shown as straight lines on a gnomonic map. Since the projection is from the centre of the sphere, a gnomonic map can represent less than half of the area of the sphere. Distortion of the scale of the map increases from the centre (tangent point) to the periphery.^[2]

If the tangent point is one of the poles then the meridians are radial and equally spaced (Fig 2 below). The equator cannot be shown as it is at infinity in all directions. Other parallels (lines of latitude) are depicted as concentric circles.

If the tangent point is on the equator then the meridians are parallel but not equally spaced (Fig 3 below). The equator is a straight line perpendicular to the meridians. Other parallels are depicted as hyperbolae.

If the tangent point is not on a pole or the equator, then the meridians are radially outward straight lines from a pole, but not equally spaced (Fig 4 below). The equator is a straight line that is perpendicular to only one meridian, indicating that the projection is not conformal. Other parallels are depicted as conic sections.

Fig 1. A great circle projects to a straight line in the gnomonic projection
Fig 2. Gnomonic projection centered on the north pole
Fig 3. Gnomonic projection centered on the equator
Fig 4. Gnomonic projection centered on latitude 40° north

As with all azimuthal projections, angles from the tangent point are preserved. The map distance from that point is a function r(d) of the true distance d, given by

r(d)=R\,\tan {\frac {d}{R}}

where R is the radius of the Earth. The radial scale is

r'(d)={\frac {1}{\cos ^{2}{\frac {d}{R}}}}

and the transverse scale

{\frac {1}{\cos {\frac {d}{R}}}}

so the transverse scale increases outwardly, and the radial scale even more.

Use

Gnomonic projections are used in seismic work because seismic waves tend to travel along great circles. They are also used by navies in plotting direction finding bearings, since radio signals travel along great circles. Meteors also travel along great circles, with the Gnomonic Atlas Brno 2000.0 being the IMO's recommended set of star charts for visual meteor observations. Aircraft and ship navigators use the projection to find the shortest route between start and destination. The track is first drawn on the gnomonic chart, then transferred to a Mercator chart for navigation.

The gnomonic projection is used extensively in photography, where it is called rectilinear projection, as it naturally arises from the pinhole camera model where the screen is a plane.^[3] Because they are equivalent, the same viewer used for photographic panoramas can be used to render gnomonic maps ( view as a 360° interactive panorama ).

The gnomonic projection is used in astronomy where the tangent point is centered on the object of interest. The sphere being projected in this case is the celestial sphere, R = 1, and not the surface of the Earth.

In astronomy, gnomic projection star charts of the celestial sphere can be used by observers to accurately plot the straight line path of a meteor trail.^[4]

Comparison of the Gnomonic projection and some azimuthal projections centred on 90° N at the same scale, ordered by projection altitude in Earth radii. (click for detail)

Related Research Articles

In geography, latitude is a coordinate that specifies the north–south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from −90° at the south pole to 90° at the north pole, with 0° at the Equator. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude and longitude are used together as a coordinate pair to specify a location on the surface of the Earth.

The Mercator projection is a conformal cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation due to its ability to represent north as "up" and south as "down" everywhere while preserving local directions and shapes. However, as a result, the Mercator projection inflates the size of objects the further they are from the equator. In a Mercator projection, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator. Despite these drawbacks, the Mercator projection is well-suited to marine navigation and internet web maps and continues to be widely used today.

A sphere is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. Formally, a sphere is the set of points that are all at the same distance $r$ from a given point in three-dimensional space. That given point is the center of the sphere, and $r$ is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians.

In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.

The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. It may be implemented in spherical or rectangular coordinates, both defined by an origin at the centre of Earth, a fundamental plane consisting of the projection of Earth's equator onto the celestial sphere, a primary direction towards the March equinox, and a right-handed convention.

In cartography, a map projection is any of a broad set of transformations employed to represent the curved two-dimensional surface of a globe on a plane. In a map projection, coordinates, often expressed as latitude and longitude, of locations from the surface of the globe are transformed to coordinates on a plane. Projection is a necessary step in creating a two-dimensional map and is one of the essential elements of cartography.

A circle of latitude or line of latitude on Earth is an abstract east–west small circle connecting all locations around Earth at a given latitude coordinate line.

In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere, onto a plane perpendicular to the diameter through the point. It is a smooth, bijective function from the entire sphere except the center of projection to the entire plane. It maps circles on the sphere to circles or lines on the plane, and is conformal, meaning that it preserves angles at which curves meet and thus locally approximately preserves shapes. It is neither isometric nor equiareal.

<span class="mw-page-title-main">Rhumb line</span> Arc crossing all meridians of longitude at the same angle

In navigation, a rhumb line, rhumb, or loxodrome is an arc crossing all meridians of longitude at the same angle, that is, a path with constant bearing as measured relative to true north.

In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H $, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.$

The transverse Mercator map projection is an adaptation of the standard Mercator projection. The transverse version is widely used in national and international mapping systems around the world, including the Universal Transverse Mercator. When paired with a suitable geodetic datum, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.

Orthographic projection in cartography has been used since antiquity. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective projection in which the sphere is projected onto a tangent plane or secant plane. The point of perspective for the orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, where the horizon is a great circle. The shapes and areas are distorted, particularly near the edges.

The Mollweide projection is an equal-area, pseudocylindrical map projection generally used for maps of the world or celestial sphere. It is also known as the Babinet projection, homalographic projection, homolographic projection, and elliptical projection. The projection trades accuracy of angle and shape for accuracy of proportions in area, and as such is used where that property is needed, such as maps depicting global distributions.

<span class="mw-page-title-main">Scale (map)</span> Ratio of distance on a map to the corresponding distance on the ground

The scale of a map is the ratio of a distance on the map to the corresponding distance on the ground. This simple concept is complicated by the curvature of the Earth's surface, which forces scale to vary across a map. Because of this variation, the concept of scale becomes meaningful in two distinct ways.

The equirectangular projection, and which includes the special case of the plate carrée projection, is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100.

The Lambert azimuthal equal-area projection is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772. "Zenithal" being synonymous with "azimuthal", the projection is also known as the Lambert zenithal equal-area projection.

The General Perspective projection is a map projection. When the Earth is photographed from space, the camera records the view as a perspective projection. When the camera is aimed toward the center of the Earth, the resulting projection is called Vertical Perspective. When aimed in other directions, the resulting projection is called a Tilted Perspective.

<span class="mw-page-title-main">Geodesics on an ellipsoid</span> Shortest paths on a bounded deformed sphere-like quadric surface

The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an oblate ellipsoid, a slightly flattened sphere. A geodesic is the shortest path between two points on a curved surface, analogous to a straight line on a plane surface. The solution of a triangulation network on an ellipsoid is therefore a set of exercises in spheroidal trigonometry.

The Nicolosi globular projection is a polyconic map projection invented about the year 1000 by the Iranian polymath al-Biruni. As a circular representation of a hemisphere, it is called globular because it evokes a globe. It can only display one hemisphere at a time and so normally appears as a "double hemispheric" presentation in world maps. The projection came into use in the Western world starting in 1660, reaching its most common use in the 19th century. As a "compromise" projection, it preserves no particular properties, instead giving a balance of distortions.

References

↑ Williams, C.E.; Ridd, M.K. (1960). "Great Circles and the Gnomonic Projection". The Professional Geographer. 12 (5): 14–16. doi:10.1111/j.0033-0124.1960.125_14.x.
1 2 3 4 Snyder, John P. (1987). Map Projections – A Working Manual. U.S. Geological Survey Professional Paper. Vol. 1395. Washington, D.C.: United States Government Printing Office. pp. 164–168. doi:10.3133/pp1395.
↑ Pegoraro, Vincent (12 December 2016). Handbook of Digital Image Synthesis: Scientific Foundations of Rendering. CRC Press. ISBN 978-1-315-39521-0.
↑ Taibi, Richard (November 25, 2016), Charles Olivier and the Rise of Meteor Science, Springer International Publishing, p. 67, ISBN 9783319445182.

External links

Gnomonic Projection

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[1] Williams, C.E.; Ridd, M.K. (1960). "Great Circles and the Gnomonic Projection". The Professional Geographer. 12 (5): 14–16. doi:10.1111/j.0033-0124.1960.125_14.x.

[Snyder-2] 1 2 3 4 Snyder, John P. (1987). Map Projections – A Working Manual. U.S. Geological Survey Professional Paper. Vol. 1395. Washington, D.C.: United States Government Printing Office. pp. 164–168. doi:10.3133/pp1395.

[3] Pegoraro, Vincent (12 December 2016). Handbook of Digital Image Synthesis: Scientific Foundations of Rendering. CRC Press. ISBN 978-1-315-39521-0.

[Taibi_2016-4] Taibi, Richard (November 25, 2016), Charles Olivier and the Rise of Meteor Science, Springer International Publishing, p. 67, ISBN 9783319445182.

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Gnomonic projection

Contents

History

Properties

Use

See also

Related Research Articles

References

Further reading

External links