In
cartography
Cartography (; from grc, χάρτης , "papyrus, sheet of paper, map"; and , "write") is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an i ...
, a conformal map projection is one in which every angle between two curves that cross each other on Earth (a
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
or an
ellipsoid) is preserved in the image of the projection, i.e. the projection is a
conformal map in the mathematical sense. For example, if two roads cross each other at a 39° angle, then their images on a map with a conformal projection cross at a 39° angle.
Properties
A conformal projection can be defined as one that is locally conformal at every point on the Earth. Thus, every small figure on the earth is nearly similar to its image on the map. The projection preserves the ratio of two lengths in the small domain. All
Tissot's indicatrices of the projections are circles.
Conformal projections preserve only small figures. Large figures are distorted by even conformal projections.
In a conformal projection, any small figure is similar to the image, but the ratio of similarity (
scale) varies by location, which explains the distortion of the conformal projection.
In a conformal projection,
parallels and
meridians cross rectangularly on the map. The converse is not necessarily true. The counterexamples are equirectangular and equal-area cylindrical projections (of normal aspects). These projections expand meridian-wise and parallel-wise by different ratios respectively. Thus, parallels and meridians cross rectangularly on the map, but these projections do not preserve other angles; i.e. these projections are not conformal.
As proven by
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
in 1775, a conformal map projection cannot be equal-area, nor can an
equal-area map
In map projection, equal-area maps preserve area measure, generally distorting shapes in order to do that. Equal-area maps are also called equivalent or authalic. An equal-area map projection cannot be conformal, nor can a conformal map projectio ...
projection be conformal. This is also a consequence of
Carl Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
's 1827 ''
Theorema Egregium
Gauss's ''Theorema Egregium'' (Latin for "Remarkable Theorem") is a major result of differential geometry, proved by Carl Friedrich Gauss in 1827, that concerns the curvature of surfaces. The theorem says that Gaussian curvature can be determi ...
''
emarkable Theorem
List of conformal projections
*
Mercator projection (conformal cylindrical projection)
**Mercator projection of normal aspect (Every
rhumb line is drawn as a straight line on the map.)
**
Transverse Mercator projection
The transverse Mercator map projection (TM, TMP) 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 Mercat ...
***
Gauss–Krüger coordinate system (This projection preserves lengths on the central meridian on an ellipsoid)
**Oblique Mercator projection
***
Space-oblique Mercator projection
Space-oblique Mercator projection is a map projection devised in the 1970s for preparing maps from Earth-survey satellite data. It is a generalization of the oblique Mercator projection that incorporates the time evolution of a given satellite gro ...
(a modified projection from Oblique Mercator projection for satellite orbits with the earth rotation within near conformality)
*
Lambert conformal conic projection
**Oblique conformal conic projection (This projection is sometimes used for long-shaped regions, like as continents of
Americas
The Americas, which are sometimes collectively called America, are a landmass comprising the totality of North and South America. The Americas make up most of the land in Earth's Western Hemisphere and comprise the New World.
Along with th ...
or
Japanese archipelago
The Japanese archipelago (Japanese: 日本列島, ''Nihon rettō'') is a group of 6,852 islands that form the country of Japan, as well as the Russian island of Sakhalin. It extends over from the Sea of Okhotsk in the northeast to the East Chin ...
.)
*
Stereographic projection (Conformal azimuthal projection. Every circle on the earth is drawn as a circle or a straight line on the map.)
**Miller Oblated Stereographic Projection (Modified stereographic projection for continents of
Africa
Africa is the world's second-largest and second-most populous continent, after Asia in both cases. At about 30.3 million km2 (11.7 million square miles) including adjacent islands, it covers 6% of Earth's total surface area ...
and
Europe
Europe is a large peninsula conventionally considered a continent in its own right because of its great physical size and the weight of its history and traditions. Europe is also considered a subcontinent of Eurasia and it is located entirel ...
.)
**
GS50 projection (This projection are made from a stereographic projection with an adjustment by a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example ...
on
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s.)
*
Littrow projection (conformal retro-azimuthal projection)
*Lagrange projection (a polyconic projection, and a composition of a Lambert conformal conic projection and a
Möbius transformation.)
**August epicycloidal projection (a composition of Lagrange projection of sphere in circle and a polynomial of degree 3 on complex numbers.)
*Application of
elliptic function
**
Peirce quincuncial projection
The Peirce quincuncial projection is the conformal map projection from the sphere to an unfolded square dihedron, developed by Charles Sanders Peirce in 1879. Each octant projects onto an isosceles right triangle, and these are arranged into a ...
(This projects the earth into a square conformally except at four singular points.)
**
Lee conformal projection of the world in a tetrahedron
Applications
Large scale
Many large-scale maps use conformal projections because figures in large-scale maps can be regarded as small enough. The figures on the maps are nearly similar to their physical counterparts.
A non-conformal projection can be used in a limited domain such that the projection is locally conformal. Glueing many maps together restores roundness. To make a new sheet from many maps or to change the center, the body must be re-projected.
Seamless online maps can be very large
Mercator projections, so that any place can become the map's center, then the map remains conformal. However, it is difficult to compare lengths or areas of two far-off figures using such a projection.
The
Universal Transverse Mercator coordinate system
The Universal Transverse Mercator (UTM) is a map projection system for assigning coordinates to locations on the surface of the Earth. Like the traditional method of latitude and longitude, it is a horizontal position representation, which means i ...
and the
Lambert system in France are projections that support the trade-off between seamlessness and scale variability.
For small scale
Maps reflecting directions, such as a
nautical chart
A nautical chart is a graphic representation of a sea area and adjacent coastal regions. Depending on the scale of the chart, it may show depths of water and heights of land ( topographic map), natural features of the seabed, details of the co ...
or an
aeronautical chart
An aeronautical chart is a map designed to assist in the navigation of aircraft, much as nautical charts do for watercraft, or a roadmap does for drivers. Using these charts and other tools, pilots are able to determine their position, safe alt ...
, are projected by conformal projections. Maps treating values whose gradients are important, such as a
weather map
A weather map, also known as synoptic weather chart, displays various meteorological features across a particular area at a particular point in time and has various symbols which all have specific meanings. Such maps have been in use since the m ...
with
atmospheric pressure
Atmospheric pressure, also known as barometric pressure (after the barometer), is the pressure within the atmosphere of Earth. The standard atmosphere (symbol: atm) is a unit of pressure defined as , which is equivalent to 1013.25 millibars, ...
, are also projected by conformal projections.
Small scale maps have large scale variations in a conformal projection, so recent world maps use other projections. Historically, many world maps are drawn by conformal projections, such as Mercator maps or hemisphere maps by
stereographic projection.
Conformal maps containing large regions vary scales by locations, so it is difficult to compare lengths or areas. However, some techniques require that a length of 1 degree on a meridian = 111 km = 60
nautical miles. In non-conformal maps, such techniques are not available because the same lengths at a point vary the lengths on the map.
In Mercator or stereographic projections, scales vary by
latitude
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 pol ...
, so bar scales by latitudes are often appended. In complex projections such as of oblique aspect. Contour charts of scale factors are sometimes appended.
See also
*
List of map projections
Notes
References
*
490
Further reading
*
*
*
*
* Chapters also published i
''The Canadian Cartographer''. 13 (1). 1976.*
*
*
*
*
{{Map projection
Conformal projections