Conformal map projection

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In cartography, a conformal map projection is one in which every angle between two curves that cross each other on Earth (a sphere 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.

Contents

Properties

We can define a conformal projection 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, even by conformal projections.

In a conformal projection, any small figure is similar to the image, but the ratio of similarity (scale) varies by location. This 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.

List of conformal projections

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 and the Lambert system in France are projections that support the trade-off between seamlessness and scale variability.

For small scale

A contour chart of scale factors of GS50 projection GS-50 projection with lines of constant scale.svg
A contour chart of scale factors of GS50 projection

Maps reflecting directions, such as a nautical chart or an aeronautical chart, are projected by conformal projections. Maps treating values whose gradients are important, such as a weather map with atmospheric pressure, 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 mile s. 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, so bar scales by latitudes are often appended. In complex projections such as of oblique aspect. Contour charts of scale factors are sometimes appended.

Sources

Related Research Articles

Latitude geographic coordinate specifying north–south position

In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the geodetic latitude as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six auxiliary latitudes which are used in special applications.

Mercator projection Map projection for navigational use that distorts areas far from the equator

The Mercator projection is a cylindrical map projection presented by Flemish geographer and cartographer Gerardus Mercator in 1569. It became the standard map projection for navigation because of its unique property of representing any course of constant bearing as a straight segment. Such a course, known as a rhumb or, mathematically, a loxodrome, is preferred by navigators because the ship can sail in a constant compass direction to reach its destination, eliminating difficult and error-prone course corrections. Linear scale is constant on the Mercator in every direction around any point, thus preserving the angles and the shapes of small objects and fulfilling the conditions of a conformal map projection. As a side effect, the Mercator projection inflates the size of objects away from the equator. This inflation is very small near the equator, but accelerates with latitude to become infinite at the poles. So, for example, landmasses such as Greenland and Antarctica appear far larger than they actually are relative to landmasses near the equator, such as Central Africa.

Geographic coordinate system Coordinate system

A geographic coordinate system is a coordinate system that enables every location on Earth to be specified by a set of numbers, letters or symbols. The coordinates are often chosen such that one of the numbers represents a vertical position and two or three of the numbers represent a horizontal position; alternatively, a geographic position may be expressed in a combined three-dimensional Cartesian vector. A common choice of coordinates is latitude, longitude and elevation. To specify a location on a plane requires a map projection.

Map projection Systematic representation of the surface of a sphere or ellipsoid onto a plane

In cartography, a map projection is a way to flatten a globe's surface into a plane in order to make a map. This requires a systematic transformation of the latitudes and longitudes of locations from the surface of the globe into locations on a plane. All projections of a sphere on a plane necessarily distort the surface in some way and to some extent. Depending on the purpose of the map, some distortions are acceptable and others are not; therefore, different map projections exist in order to preserve some properties of the sphere-like body at the expense of other properties. Every distinct map projection distorts in a distinct way, by definition. The study of map projections is the characterization of these distortions. There is no limit to the number of possible map projections. Projections are a subject of several pure mathematical fields, including differential geometry, projective geometry, and manifolds. However, "map projection" refers specifically to a cartographic projection.

Stereographic projection particular mapping that projects a sphere onto a plane

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it preserves angles at which curves meet. It is neither isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.

Rhumb line 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 or magnetic north.

Transverse Mercator projection The transverse Mercator projection is the transverse aspect of the standard (or Normal) Mercator projection

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 UTM. When paired with a suitable geodetic datum, the transverse Mercator delivers high accuracy in zones less than a few degrees in east-west extent.

Scale (map) 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.

Oblique Mercator projection The oblique Mercator projection is the oblique aspect of the standard (or Normal) Mercator projection

The oblique Mercator map projection is an adaptation of the standard Mercator projection. The oblique version is sometimes used in national mapping systems. When paired with a suitable geodetic datum, the oblique Mercator delivers high accuracy in zones less than a few degrees in arbitary directional extent.

Universal Transverse Mercator coordinate system coordinate system

The Universal Transverse Mercator (UTM) is a 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 it ignores altitude and treats the earth as a perfect ellipsoid. However, it differs from global latitude/longitude in that it divides earth into 60 zones and projects each to the plane as a basis for its coordinates. Specifying a location means specifying the zone and the x, y coordinate in that plane. The projection from spheroid to a UTM zone is some parameterization of the transverse Mercator projection. The parameters vary by nation or region or mapping system.

Universal polar stereographic coordinate system coordinate system

The universal polar stereographic (UPS) coordinate system is used in conjunction with the universal transverse Mercator (UTM) coordinate system to locate positions on the surface of the earth. Like the UTM coordinate system, the UPS coordinate system uses a metric-based cartesian grid laid out on a conformally projected surface. UPS covers the Earth's polar regions, specifically the areas north of 84°N and south of 80°S, which are not covered by the UTM grids, plus an additional 30 minutes of latitude extending into UTM grid to provide some overlap between the two systems.

Tissots indicatrix

In cartography, a Tissot's indicatrix is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize local distortions due to map projection. It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the resulting diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map.

Lambert conformal conic projection map projection

A Lambert conformal conic projection (LCC) is a conic map projection used for aeronautical charts, portions of the State Plane Coordinate System, and many national and regional mapping systems. It is one of seven projections introduced by Johann Heinrich Lambert in his 1772 publication Anmerkungen und Zusätze zur Entwerfung der Land- und Himmelscharten.

Space-oblique Mercator projection map 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 ground track to optimize its representation on the map. The oblique Mercator projection, on the other hand, optimizes for a given geodesic.

Guyou hemisphere-in-a-square projection

The Guyou hemisphere-in-a-square projection is a conformal map projection for the hemisphere. It is an oblique aspect of the Peirce quincuncial projection.

Cylindrical equal-area projection

In cartography, the cylindrical equal-area projection is a family of cylindrical, equal-area map projections.

Web Mercator projection Mercator projection variant

Web Mercator, Google Web Mercator, Spherical Mercator, WGS 84 Web Mercator or WGS 84/Pseudo-Mercator is a variant of the Mercator projection and is the de facto standard for Web mapping applications. It rose to prominence when Google Maps adopted it in 2005. It is used by virtually all major online map providers, including Google Maps, Mapbox, Bing Maps, OpenStreetMap, Mapquest, Esri, and many others. Its official EPSG identifier is EPSG:3857, although others have been used historically.

In geodesy, a map projection of the tri-axial ellipsoid maps Earth or some other astronomical body modeled as a tri-axial ellipsoid to the plane. Such a model is called the reference ellipsoid. In most cases, reference ellipsoids are spheroids, and sometimes spheres. Massive objects have sufficient gravity to overcome their own rigidity and usually have an oblate ellipsoid shape. However, minor moons or small solar system bodies are not under hydrostatic equilibrium. Usually such bodies have irregular shapes. Furthermore, some of gravitationally rounded objects may have a tri-axial ellipsoid shape due to rapid rotation or unidirectional strong tidal forces.

Stereographic projection in cartography

The stereographic projection, also known as the planisphere projection or the azimuthal conformal projection, is a conformal map projection whose use dates back to antiquity. Like the orthographic projection and gnomonic projection, the stereographic projection is an azimuthal projection, and when on a sphere, also a perspective projection.

References

  1. "Miller Oblated Stereographic Projection".