Gnomonic projection

# Gnomonic projection

Overview
A gnomonic map projection
Map projection
A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...

displays all great circle
Great circle
A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as opposed to a general circle of a sphere where the plane is not required to pass through the center...

s as straight lines. Thus the shortest route between two locations in reality corresponds to that on the map
Map
A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, regions, and themes....

. This is achieved by projecting, with respect to the center of the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

(hence perpendicular to the surface), the Earth's surface onto a tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

plane. The least distortion occurs at the tangent point. Less than half of the sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

can be projected onto a finite map.
Discussion

Encyclopedia
A gnomonic map projection
Map projection
A map projection is any method of representing the surface of a sphere or other three-dimensional body on a plane. Map projections are necessary for creating maps. All map projections distort the surface in some fashion...

displays all great circle
Great circle
A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as opposed to a general circle of a sphere where the plane is not required to pass through the center...

s as straight lines. Thus the shortest route between two locations in reality corresponds to that on the map
Map
A map is a visual representation of an area—a symbolic depiction highlighting relationships between elements of that space such as objects, regions, and themes....

. This is achieved by projecting, with respect to the center of the Earth
Earth
Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets...

(hence perpendicular to the surface), the Earth's surface onto a tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

plane. The least distortion occurs at the tangent point. Less than half of the sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

can be projected onto a finite map. As a corollary, a rectilinear photographic lens cannot encompass more than 180 degrees for the same reason.

Since meridians and the equator
Equator
An equator is the intersection of a sphere's surface with the plane perpendicular to the sphere's axis of rotation and containing the sphere's center of mass....

are great circles, they are always shown as straight lines.
• If the tangent point is one of the Poles
Poles
thumb|right|180px|The state flag of [[Poland]] as used by Polish government and diplomatic authoritiesThe Polish people, or Poles , are a nation indigenous to Poland. They are united by the Polish language, which belongs to the historical Lechitic subgroup of West Slavic languages of Central Europe...

then the meridians are radial and equally spaced. The equator is at infinity
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

in all directions. Other parallels
Parallels
Parallels may refer to:* Circle of latitude , imaginary east-west circles connecting all locations that share a given latitude* "Parallels", the third track from the 1977 Yes album Going for the One...

are depicted as concentric circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

s.

• If the tangent point is on the equator then the meridians are parallel but not equally spaced. The equator is a straight line perpendicular to the meridians. Other parallels are depicted as hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

e.

• In other cases the meridians are radially outward straight lines from a Pole, but not equally spaced. The equator is a straight line that is perpendicular to only one meridian (which again demonstrates that the projection is not conformal
Conformal map
In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...

).

As for 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

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

and the transverse
Transversality
In mathematics, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. It formalizes the idea of a generic intersection in differential topology...

scale

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

The gnomonic projection is said to be the oldest map projection, developed by Thales
Thales
Thales of Miletus was a pre-Socratic Greek philosopher from Miletus in Asia Minor, and one of the Seven Sages of Greece. Many, most notably Aristotle, regard him as the first philosopher in the Greek tradition...

in the 6th century BC.
The path of the shadow-tip or light-spot in a nodus-based sundial traces out the same hyperbola
Hyperbola
In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

e formed by parallels on a gnomonic map.

Gnomonic projections are used in seismic work because seismic waves tend to travel along great circles. They are also used by navies
Navy
A navy is the branch of a nation's armed forces principally designated for naval and amphibious warfare; namely, lake- or ocean-borne combat operations and related functions...

in plotting direction finding
Direction finding
Direction finding refers to the establishment of the direction from which a received signal was transmitted. This can refer to radio or other forms of wireless communication...

Radio is the transmission of signals through free space by modulation of electromagnetic waves with frequencies below those of visible light. Electromagnetic radiation travels by means of oscillating electromagnetic fields that pass through the air and the vacuum of space...

signals travel along great circles. Meteor
METEOR
METEOR is a metric for the evaluation of machine translation output. The metric is based on the harmonic mean of unigram precision and recall, with recall weighted higher than precision...

s also travel along great circles, and the Gnomonic Atlas Brno 2000.0 is the IMO
International Meteor Organization
The International Meteor Organization was founded in 1988 and has several hundred members. IMO was created in response to an ever-growing need for international cooperation on amateur meteor work...

recommended set of star charts for visual meteor observations.

## History

In 1946 Buckminster Fuller
Buckminster Fuller
Richard Buckminster “Bucky” Fuller was an American systems theorist, author, designer, inventor, futurist and second president of Mensa International, the high IQ society....

patented a projection method similar to the Gnomonic Projection in his cuboctahedral
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...

version of the Dymaxion Map
Dymaxion map
The Dymaxion map or Fuller map is a projection of a world map onto the surface of a polyhedron, which can be unfolded and flattened to two dimensions. The projection depicts the earth's continents as "one island," or nearly contiguous land masses. The arrangement heavily interrupts the map in order...

. The 1954 icosahedral
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....

version he published under the title of AirOcean World Map, and this is the version most commonly referred to today.

• Beltrami–Klein model, the analogous mapping of the hyperbolic plane
Hyperbolic geometry
In mathematics, hyperbolic geometry is a non-Euclidean geometry, meaning that the parallel postulate of Euclidean geometry is replaced...