Community hub
Recent from talks
Contribute something
Nothing was collected or created yet.
Projected coordinate system
View on Wikipedia
| Geodesy |
|---|
 |
A projected coordinate system – also called a projected coordinate reference system, planar coordinate system, or grid reference system – is a type of spatial reference system that represents locations on Earth using Cartesian coordinates (x, y) on a planar surface created by a particular map projection.[1] Each projected coordinate system, such as "Universal Transverse Mercator WGS 84 Zone 26N," is defined by a choice of map projection (with specific parameters), a choice of geodetic datum to bind the coordinate system to real locations on the earth, an origin point, and a choice of unit of measure.[2] Hundreds of projected coordinate systems have been specified for various purposes in various regions.
When the first standardized coordinate systems were created during the 20th century, such as the Universal Transverse Mercator, State Plane Coordinate System, and British National Grid, they were commonly called grid systems; the term is still common in some domains such as the military that encode coordinates as alphanumeric grid references. However, the term projected coordinate system has recently become predominant to clearly differentiate it from other types of spatial reference system. The term is used in international standards such as the EPSG and ISO 19111 (also published by the Open Geospatial Consortium as Abstract Specification 2), and in most geographic information system software.[3][2]
History
[edit].jpg)
The map projection and the geographic coordinate system (GCS, latitude and longitude) date to the Hellenistic period, proliferating during the Enlightenment Era of the 18th century. However, their use as the basis for specifying precise locations, rather than latitude and longitude, is a 20th century innovation.
Among the earliest was the State Plane Coordinate System (SPCS), which was developed in the United States during the 1930s for surveying and engineering, because calculations such as distance are much simpler in a Cartesian coordinate system than the three-dimensional trigonometry of GCS. In the United Kingdom, the first version of the British National Grid was released in 1938, based on earlier experiments during World War I by the Army and the Ordnance Survey.[4]
During World War II, modern warfare practices required soldiers to quickly and accurately measure and report their location, leading to the printing of grids on maps by the U.S. Army Map Service (AMS) and other combatants.[5] Initially, each theater of war was mapped in a custom projection with its own grid and coding system, but this resulted in confusion. This led to the development of the Universal Transverse Mercator coordinate system, possibly adopted from a system originally developed by the German Wehrmacht.[6] To facilitate unambiguous reporting, the alphanumeric Military Grid Reference System (MGRS) was then created as an encoding scheme for UTM coordinates to make them easier to communicate.[5]
After the War, UTM gradually gained users, especially in the scientific community. Because UTM zones do not align with political boundaries, several countries followed the United Kingdom in creating their own national or regional grid systems based on custom projections. The use and invention of such systems especially proliferated during the 1980s with the emergence of geographic information systems. GIS requires locations to be specified as precise coordinates and performs numerous calculations on them, making Cartesian geometry preferable to spherical trigonometry when computing power was at a premium. In recent years, the rise of global GIS datasets and satellite navigation, along with an abundance of processing speed in personal computers, have led to a resurgence in the use of GCS. That said, projected coordinate systems are still very common in the GIS data stored in the spatial data infrastructures (SDI) of local areas, such as cities, counties, states and provinces, and small countries.
System specification
[edit]Because the purpose of any coordinate system is to accurately and unambiguously measure, communicate, and perform calculations on locations, it must be defined precisely. The EPSG Geodetic Parameter Dataset is the most common mechanism for publishing such definitions in a machine-readable form, and forms the basis for many GIS and other location-aware software programs.[3] A projected SRS specification consists of three parts:
- An abstract two-dimensional Cartesian coordinate system that allows for the measurement of each location as a tuple (x, y), which are also called the easting and northing in many systems such as UTM. Any coordinate system definition must include a planar surface, an origin point, a set of orthogonal axes to define the direction of each measurement, and a unit of measure (usually the meter or US foot).
- A choice of map projection that creates a planar surface for the coordinate system that is connected to locations on the Earth. In addition to the general type of projection (e.g., Lambert conformal conic, transverse Mercator), a coordinate system definition will specify the parameters to be used, such as a center point, standard parallels, scale factor, false origin, and such. With these parameters, the underlying formulas of the projection convert latitude and longitude directly into the (x, y) coordinates of the system.
- A choice of geodetic datum, which includes a choice of earth ellipsoid. This binds the coordinate system to actual locations on the Earth by controlling the measurement framework for latitude and longitude (GCS). Thus, there will be a significant difference between the coordinate of a location in "UTM NAD83 Zone 14N" and for the same location in "UTM NAD27 Zone 14N", even though the UTM formulas are identical, because the underlying latitude and longitude values are different. In some GIS software, this part of the definition is called the choice of a particular geographic coordinate system.
Projections
[edit]To establish the position of a geographic location on a map, a map projection is used to convert geodetic coordinates to plane coordinates on a map; it projects the datum ellipsoidal coordinates and height onto a flat surface of a map. The datum, along with a map projection applied to a grid of reference locations, establishes a grid system for plotting locations. Conformal projections are generally preferred. Common map projections include the transverse Mercator (used in Universal Transverse Mercator, the British National Grid, the State Plane Coordinate System for some states), Lambert conformal conic (some states in the SPCS), and Mercator (Swiss coordinate system).
Map projection formulas depend on the geometry of the projection as well as parameters dependent on the particular location at which the map is projected. The set of parameters can vary based on the type of project and the conventions chosen for the projection. For the transverse Mercator projection used in UTM, the parameters associated are the latitude and longitude of the natural origin, the false northing and false easting, and an overall scale factor.[7] Given the parameters associated with particular location or grin, the projection formulas for the transverse Mercator are a complex mix of algebraic and trigonometric functions.[7]: 45–54
Easting and northing
[edit]Every map projection has a natural origin, e.g., at which the ellipsoid and flat map surfaces coincide, at which point the projection formulas generate a coordinate of (0,0).[7] To ensure that the northing and easting coordinates on a map are not negative (thus making measurement, communication, and computation easier), map projections may set up a false origin, specified in terms of false northing and false easting values, that offset the true origin. For example, in UTM, the origin of each northern zone is a point on the equator 500 km west of the central meridian of the zone (the edge of the zone itself is just under 400 km to the west). This has the desirable effect of making all coordinates within the zone positive values, being east and north of the origin. Because of this, they are often referred to as the easting and northing.
Grid north
[edit]Grid north (GN) is a navigational term referring to the direction northwards along the grid lines of a map projection. It is contrasted with true north (the direction of the North Pole) and magnetic north (the direction in which a compass needle points). Many topographic maps, including those of the United States Geological Survey and Great Britain's Ordnance Survey, indicate the difference between grid north, true north, and magnetic north.[8]
The grid lines on Ordnance Survey maps divide the UK into one-kilometre squares, east of an imaginary zero point in the Atlantic Ocean, west of Cornwall. The grid lines point to a Grid North, varying slightly from True North. This variation is zero on the central meridian (north-south line) of the map, which is at two degrees west of the Prime Meridian, and greatest at the map edges. The difference between grid north and true north is very small and can be ignored for most navigation purposes. The difference exists because the correspondence between a flat map and the round Earth is necessarily imperfect.
At the South Pole, grid north conventionally points northwards along the Prime Meridian.[9] Since the meridians converge at the poles, true east and west directions change rapidly in a condition similar to gimbal lock. Grid north solves this problem.
Grid reference encodings
[edit]Locations in a projected coordinate system, like any cartesian coordinate system, are measured and reported as easting/northing or (x, y) pairs. The pair is usually represented conventionally with easting first, northing second. For example, the peak of Mount Assiniboine (at 50°52′10″N 115°39′03″W / 50.86944°N 115.65083°W on the British Columbia/Alberta border in Canada) in UTM Zone 11 is at (0594934mE, 5636174mN), meaning that is almost 600km east of the false origin for Zone 11 (95km east of the true central meridian at 117°W) and 5.6 million meters north of the equator.
While such precise numbers are easy to store and calculate in GIS and other computer databases, they can be difficult for humans to remember and communicate. Thus, since the mid 20th century, there have been alternative encodings that shorten the numbers or convert the numbers into some form of alphanumeric string.
For example, a truncated grid reference may be used where the general location is already known to participants and may be assumed.[10] Because the (leading) most significant digits specify the part of the world and the (trailing) least significant digits provide a precision that is not needed in most circumstances, they may be unnecessary for some uses. This permits users to shorten the example coordinates to 949-361 by concealing 05nnn34 56nnn74, assuming the significant digits (3,4, and 5 in this case) are known to both parties.[11]
Alphanumeric encodings typically use codes to replace the most significant digits by partitioning the world up into large grid squares. For example, in the Military Grid Reference System, the above coordinate is in grid 11U (representing UTM Zone 11 5xxxxxx mN), and grid cell NS within that (representing the second digit 5xxxxxmE x6xxxxxm N), and as many remaining digits as are needed are reported, yielding an MGRS grid reference of 11U NS 949 361 (or 11U NS 9493 3617 or 11U NS 94934 36174).
The Ordnance Survey National Grid (United Kingdom) and other national grid systems use similar approaches. In Ordnance Survey maps, each Easting and Northing grid line is given a two-digit code, based on the British national grid reference system with an origin point just off the southwest coast of the United Kingdom. The area is divided into 100 km squares, each of which is denoted by a two-letter code. Within each 100 km square, a numerical grid reference is used. Since the Eastings and Northings are one kilometre apart, a combination of a Northing and an Easting will give a four-digit grid reference describing a one-kilometre square on the ground. The convention is the grid reference numbers call out the lower-left corner of the desired square. In the example map above, the town Little Plumpton lies in the square 6901, even though the writing which labels the town is in 6802 and 6902, most of the buildings (the orange boxed symbols) are in square 6901.
Precision
[edit] |
The more digits added to a grid reference, the more precise the reference becomes. To locate a specific building in Little Plumpton, a further two digits are added to the four-digit reference to create a six-digit reference. The extra two digits describe a position within the 1-kilometre square. Imagine (or draw or superimpose a Romer) a further 10x10 grid within the current grid square. Any of the 100 squares in the superimposed 10×10 grid can be accurately described using a digit from 0 to 9 (with 0 0 being the bottom left square and 9 9 being the top right square).
For the church in Little Plumpton, this gives the digits 6 and 7 (6 on the left to right axis (Eastings) and 7 on the bottom to top axis (Northings). These are added to the four-figure grid reference after the two digits describing the same coordinate axis, and thus our six-figure grid reference for the church becomes 696017. This reference describes a 100-metre by 100-metre square, and not a single point, but this precision is usually sufficient for navigation purposes. The symbols on the map are not precise in any case, for example the church in the example above would be approximately 100x200 metres if the symbol was to scale, so in fact, the middle of the black square represents the map position of the real church, independently of the actual size of the church.
Grid references comprising larger numbers for greater precision could be determined using large-scale maps and an accurate Romer. This might be used in surveying but is not generally used for land navigating for walkers or cyclists, etc. The growing availability and decreasing cost of handheld GPS receivers enables determination of accurate grid references without needing a map, but it is important to know how many digits the GPS displays to avoid reading off just the first six digits. A GPS unit commonly gives a ten-digit grid reference, based on two groups of five numbers for the Easting and Northing values. Each successive increase in precision (from 6 digit to 8 digit to 10 digit) pinpoints the location more precisely by a factor of 10. Since, in the UK at least, a 6-figure grid reference identifies a square of 100-metre sides, an 8-figure reference would identify a 10-metre square, and a 10-digit reference a 1-metre square. In order to give a standard 6-figure grid reference from a 10-figure GPS readout, the 4th, 5th, 9th and 10th digits must be omitted, so it is important not to read just the first 6 digits.
Examples of projected CRS
[edit]
- Universal Transverse Mercator (UTM): not a single coordinate system, but a series of 60 zones (each being a gore 6° wide), each a system with its own Transverse Mercator projection.
- Universal Polar Stereographic (UPS): a pair of coordinate systems covering the Arctic and Antarctica using a Stereographic projection.
- Ordnance Survey National Grid (OSNG): a transverse mercator projection centered on 2°W that covers Great Britain with its own encoding scheme.
- State Plane Coordinate System (SPCS): another composite system of more than 120 coordinate systems (zones), each covering a state of the United States or a portion thereof.
- Swiss coordinate system (LV95): covers Switzerland, using a Mercator projection.
- Irish Transverse Mercator (ITM): jointly created by the Republic of Ireland and United Kingdom to cover the island of Ireland.
- Bermuda National Grid
- Hellenic Geodetic Reference System 1987 (Greece)
- Israeli Transverse Mercator (NIG)
- Swedish grid (RT90)
See also
[edit]- Discrete global grid (DGG)
- East north up
- Geocodes
- Geodetic datum
- Geographical distance
- Graticule (cartography)
- Horizontal plane
- Lattice graph (grid as mathematical abstraction)
- Map projection
- Spatial reference system
- Spatial grid
References
[edit]- ^ Chang, Kang-tsung (2016). Introduction to Geographic Information Systems (9th ed.). McGraw-Hill. p. 34. ISBN 978-1-259-92964-9.
- ^ a b "OGC Abstract Specification Topic 2: Referencing by coordinates Corrigendum". Open Geospatial Consortium. Retrieved 2018-12-25.
- ^ a b "Using the EPSG geodetic parameter dataset, Guidance Note 7-1". EPSG Geodetic Parameter Dataset. Geomatic Solutions. Retrieved 15 December 2021.
- ^ Russell, Don. "Understanding Maps: The British National Grid". Uncharted 101. Retrieved 21 December 2021.
- ^ a b Raisz, Erwin (1948). General Cartography. McGraw-Hill. pp. 225–229.
- ^ Buchroithner, Manfred; Pfahlbusch, René (2017). "Geodetic grids in authoritative maps – new findings about the origin of the UTM Grid". Cartography and Geographic Information Science. 44 (3): 186–200. doi:10.1080/15230406.2015.1128851. S2CID 131732222.
- ^ a b c "Geomatics Guidance Note Number 7, part 2 Coordinate Conversions and Transformations including Formulas" (PDF). International Association of Oil and Gas Producers (OGP). pp. 9–10. Archived from the original (PDF) on 6 March 2014. Retrieved 5 March 2014.
- ^ Estopinal, Stephen V. (2009). A Guide to Understanding Land Surveys. John Wiley & Sons. p. 35. ISBN 978-0-470-23058-9.
- ^ "Moving the South Pole". Archived 2011-07-16 at the Wayback Machine, NASA Quest
- ^ "Truncated Grid References". Bivouac.com – Canadian Mountain Encyclopedia. 2006-11-17.
- ^ "Grids and Reference Systems". National Geospatial-Intelligence Agency. Retrieved 4 March 2014.
| International | |
|---|---|
| National | |
| Other | |
Projected coordinate system
View on GrokipediaFundamentals
Definition and Purpose
A projected coordinate system (PCS), also known as a projected coordinate reference system, is a type of coordinate reference system derived from a two-dimensional geodetic coordinate reference system by applying a map projection, which converts ellipsoidal coordinates (latitude and longitude) into Cartesian coordinates (x, y) on a planar surface. This projection process mathematically transforms the curved three-dimensional surface of the Earth onto a two-dimensional plane, enabling the representation of geographic positions using linear units such as meters. The primary purpose of a PCS is to facilitate accurate measurements of distances, areas, and shapes on flat maps, which is essential for applications in surveying, navigation, geographic information systems (GIS), and large-scale mapping, as it minimizes distortions that occur when using angular spherical coordinates directly. By providing a planar framework, PCS allows for straightforward Euclidean calculations that are impractical on the Earth's spheroid, thereby supporting precise spatial analysis and data integration in fields requiring metric consistency. Key components of a PCS include a geodetic datum, which defines the reference ellipsoid and ties the system to the Earth's surface (e.g., WGS 84); projection parameters such as the method (e.g., Transverse Mercator), false origin (to avoid negative coordinates), central meridian, latitude of origin, and scale factor; and linear units for the resulting coordinates. A full specification might be denoted as "UTM Zone 32N / WGS 84," where UTM refers to the Universal Transverse Mercator projection applied to the WGS 84 datum, with coordinates expressed in meters easting and northing. PCS definitions and specifications are standardized through frameworks like ISO 19111, which provides the conceptual schema for coordinate referencing, and the EPSG (European Petroleum Survey Group) registry, a public dataset of over 10,000 coordinate systems and transformations maintained by the International Association of Oil & Gas Producers (IOGP).Comparison to Geographic Coordinate Systems
Geographic coordinate systems (GCS) employ latitude and longitude measured in angular degrees on an ellipsoidal model of the Earth, providing a framework for global positioning that accounts for the planet's curvature.[4] These systems are inherently three-dimensional, relying on spherical or spheroidal geometry for accurate representation worldwide, but they necessitate complex spherical trigonometry to compute distances, areas, and directions, as the units of measurement vary with latitude—for instance, one degree of longitude spans approximately 111 km at the equator but only 55 km at 60° latitude.[4] In contrast, projected coordinate systems (PCS) transform this ellipsoidal framework onto a two-dimensional plane using a map projection, yielding linear coordinates such as easting and northing in meters, which enable straightforward Euclidean geometry for calculations.[5] The primary operational difference lies in their handling of Earth's curvature: GCS preserves the spheroidal shape without distortion in angular terms but introduces challenges for planar applications, while PCS flattens the surface to facilitate direct measurements, albeit at the cost of inevitable distortions in shape, area, distance, or direction depending on the projection method and region.[4] For example, in PCS, distances can be computed using simple Pythagorean theorem, whereas GCS requires geodesic algorithms to account for curvature, making PCS computations simpler and faster for local analyses.[5] However, PCS are geographically limited, often confined to specific zones to minimize distortion—such as the Universal Transverse Mercator (UTM) system's 6°-wide zones, where scale distortion remains below 0.1% within the zone but increases significantly beyond its boundaries, potentially exceeding 1 part in 1,000 at the edges.[6][7] PCS offer distinct advantages for regional engineering, surveying, and mapping tasks, where linear units support precise infrastructure design and resource management without curvature corrections, outperforming GCS in accuracy for scales larger than 1:1,000,000.[4] Their planar nature also simplifies overlay analysis and integration with CAD systems, reducing computational overhead for tasks like cadastral surveys.[5] Conversely, the zone-based limitations of PCS necessitate careful selection to avoid errors from projection-induced distortions, such as area exaggeration near zone edges in transverse Mercator projections like UTM, which can mislead global-scale interpretations.[6] GCS, while computationally intensive, excel in global contexts by maintaining positional fidelity across vast areas without such regional constraints, making them preferable for worldwide navigation or climate modeling.[4] Selection between the two depends on project scope: PCS are ideal for localized studies within approximately 6° of longitude, such as urban planning in a single UTM zone, where minimized distortion enhances reliability for measurements under 1,000 km.[7][6] For broader or worldwide applications, GCS provide the necessary global consistency, avoiding the fragmentation and re-projection issues inherent to PCS.[5]Historical Development
Early Map Projections
The origins of projected coordinate systems trace back to ancient cartographic efforts to represent the spherical Earth on flat surfaces. In the 2nd century BCE, the Greek astronomer Hipparchus is credited with developing foundational concepts, including the azimuthal equidistant projection, which preserved distances from the center and laid the groundwork for later azimuthal methods used in star maps and astronomical calculations.[8] Around 150 CE, Ptolemy advanced these ideas in his Geographia, describing cylindrical and conical projections that approximated the globe's curvature, such as the equidistant conic projection with straight meridians and parallels meeting at right angles, enabling the creation of world maps and regional representations despite distortions in larger areas.[9] These early techniques provided the conceptual basis for flattening spherical geography into usable grids, influencing subsequent developments in projected systems. During the Renaissance, innovations in navigation spurred significant advances. In 1569, Flemish cartographer Gerardus Mercator introduced his conformal cylindrical projection, designed specifically for maritime use by rendering rhumb lines as straight parallels to the equator, thereby preserving angles for compass-based sailing while accepting scale distortions that increased toward the poles, particularly affecting high-latitude landmasses. This projection revolutionized seafaring by allowing accurate course plotting on flat charts, though its size exaggerations at higher latitudes highlighted the trade-offs inherent in projecting a globe. Mercator's work built directly on ancient precedents, adapting them for practical European exploration and trade routes. The 19th century saw refinements addressing limitations in large-scale mapping, particularly meridional distortions. In 1772, Johann Heinrich Lambert proposed the transverse Mercator projection, a conformal variant that rotated the cylinder to align with a central meridian, maintaining true scale along that line and minimizing east-west distortions for north-south oriented regions, making it suitable for detailed topographic surveys.[8] Later in the century, Carl Friedrich Gauss analyzed and extended this in 1822 with ellipsoidal formulations, which were further developed by Louis Krüger into the Gauss-Krüger system in 1912, enhancing accuracy for national geodetic networks by incorporating Earth's ellipsoidal shape and zoning meridians into manageable strips.[10] These advancements shifted focus from global navigation to precise, localized coordinate frameworks. A key transition to grid-based coordinates emerged in 18th-century France through the Cassini projection, employed by César-François Cassini de Thury for the first national topographic survey (Carte de Cassini, initiated 1744). This transverse equidistant cylindrical method preserved distances along the central meridian with straight lines for both meridians and parallels, facilitating military and colonial surveying by overlaying rectangular grids on projected maps for systematic land measurement across the kingdom.[8] Such applications demonstrated how early projections evolved into structured coordinate systems, paving the way for modern projected coordinate systems by integrating grids for quantifiable positioning.Modern Standardization Efforts
In the early 20th century, national efforts to standardize projected coordinate systems emerged to support precise surveying and mapping within defined territories. The U.S. State Plane Coordinate System (SPCS), developed in the 1930s by Oscar S. Adams of the U.S. Coast and Geodetic Survey, provided a conformal framework for transforming latitudes and longitudes into plane coordinates tailored to individual states, minimizing distortion for engineering and mapping applications.[11] Similarly, the British National Grid was introduced in 1938 as part of the Retriangulation of Great Britain, adopting a Transverse Mercator projection to enhance accuracy across the country and replace earlier inconsistent systems.[12] World War II and its aftermath accelerated global standardization driven by military imperatives. The Universal Transverse Mercator (UTM) system, formulated in the 1940s by the U.S. Army and formalized in 1947, divided the world into zones using a secant Transverse Mercator projection to facilitate consistent large-scale topographic mapping and navigation for joint operations.[10] Complementing these developments, the International Map of the World (IMW), proposed in 1891 by Albrecht Penck and advanced through international conferences by 1913, established uniform sheetline and projection standards that influenced subsequent grid conventions in national mapping programs.[13] The proliferation of Geographic Information Systems (GIS) in the 1980s prompted the creation of centralized registries for coordinate definitions. This led to the EPSG Geodetic Parameter Dataset, initiated in the early 1990s by the European Petroleum Survey Group to catalog parameters for projected coordinate systems and ensure interoperability in geospatial data handling.[14] Building on this, the ISO 19111 standard, first issued in 2003 and revised in 2019, defined a conceptual schema for spatial referencing by coordinates, encompassing projected systems to support standardized descriptions in geographic information applications.[15] Post-2020, the EPSG registry has incorporated enhancements for Global Navigation Satellite System (GNSS) integration, such as codes for advanced terrestrial reference frames like ETRF2020 and ITRF2020, improving alignment with real-time positioning data without fundamental overhauls to core structures.[16] These refinements have bolstered adoption of projected coordinate systems in emerging domains, including high-definition mapping for autonomous vehicles, where local planar grids enable precise localization and path planning, and climate modeling, where they facilitate accurate spatial simulations of regional environmental changes.[17][18]Technical Specifications
Projection Methods
A projected coordinate system relies on map projections to transform three-dimensional coordinates on an ellipsoidal Earth model into a two-dimensional Cartesian plane, enabling linear measurements in units like meters. This transformation mathematically projects the curved surface onto a developable surface—typically a cylinder, cone, or plane—that can be unrolled without tearing or stretching, though distortion in shape, area, distance, or direction is inevitable except at specific points or lines. Projections are classified by their geometric properties: conformal projections preserve local angles and shapes, making them suitable for navigation and detailed mapping; equal-area projections maintain accurate relative sizes of regions, ideal for thematic maps like population density; and equidistant projections preserve distances from a central point or along specified lines, useful for polar or radial analyses.[19][10] Among common projection methods for projected coordinate systems, the transverse Mercator stands out as a conformal cylindrical projection, particularly effective for minimizing distortion along a central meridian in north-south oriented zones. It projects the ellipsoid onto a cylinder rotated 90 degrees from the standard Mercator, resulting in straight meridians that converge toward the poles and parallels as complex curves, with scale true along the central meridian and low distortion within narrow zones. Key parameters include the central meridian (defining the zone's reference longitude), latitude of origin (often 0° for equatorial zones), a constant scale factor (typically 0.9996 to reduce overall distortion), and false easting (e.g., 500,000 m) and false northing (e.g., 0 m in the Northern Hemisphere) to ensure positive coordinates. The scale factor varies with position, approximated in the east-west direction as , where is the angular distance from the central meridian, and incorporating ellipsoidal effects through the term , with as latitude and as the ellipsoid's eccentricity; full ellipsoidal formulations use series expansions for forward and inverse transformations, such as and , where is the radius of curvature in the prime vertical, is the meridian arc length, and , , are intermediate terms. This method underpins the Universal Transverse Mercator (UTM) system, where 6°-wide zones limit scale distortion to less than 0.1% (1 part in 1,000) across the zone.[10][8][20] The Lambert conformal conic projection, another conformal method suited to mid-latitude regions with east-west extents, uses a secant cone tangent along two standard parallels to balance distortion across broader latitudinal bands. Meridians project as straight lines converging at the apex, while parallels form concentric arcs; it excels in preserving shapes for aeronautical charts and regional mapping. Essential parameters are the two standard parallels (e.g., 33°N and 45°N for the conterminous U.S.), central meridian, latitude of origin, and false easting/northing (e.g., 0 m and 600,000 m in some zones) to avoid negative values, with the scale factor often near 1.0 along the standards. The ellipsoidal formulation involves , , , , where is the radial distance from the apex, is the cone constant , , , and ; the scale factor is , true (1.0) along the standards and varying minimally between them. This projection forms the basis for many zones in the State Plane Coordinate System (SPCS), such as California's Zone 1 with standards at 40°00'N and 41°40'N, central meridian at 122°00'W, and .[10][21][8] For applications requiring area preservation, such as continental-scale thematic mapping, the Albers equal-area conic projection employs a secant cone to ensure no net distortion in regional sizes, though shapes elongate away from the standard parallels. It features straight converging meridians and unequally spaced parallel arcs, with scale true along two standards and varying elsewhere to compensate for area. Parameters include the standard parallels (e.g., 29°30'N and 45°30'N for the U.S.), central meridian (e.g., 96°W), latitude of origin (e.g., 23°N), and false easting/northing (often 0 m). The ellipsoidal equations are , , , , , , , , where at the latitude of origin , and subscripts 1 and 2 denote values at the standard parallels. Scale factors along meridians () and parallels () differ but their product remains 1, yielding near-unity values between the standards. Widely adopted for U.S. national maps at 1:2,500,000 and smaller scales, it supports accurate area-based analyses like resource distribution.[10][8]Easting, Northing, and Axes
In projected coordinate systems, the X-axis, known as easting, measures the linear distance eastward from a designated origin or central meridian, while the Y-axis, referred to as northing, measures the linear distance northward from the equator or a standard parallel.[22][23] These coordinates provide a Cartesian framework on the projected plane, replacing angular geographic coordinates with straight-line measurements for easier distance and area calculations.[22] To prevent negative values and ensure all coordinates within a zone are positive, false origins are introduced by adding arbitrary offsets to the true projected positions. For instance, in the Universal Transverse Mercator (UTM) system, the central meridian of each zone receives a false easting of 500,000 meters, and the equator is set at a false northing of 0 meters for northern hemisphere zones.[24][25] The final position is then computed as (x, y) = (false_easting + projected_easting, false_northing + projected_northing), where projected_easting and projected_northing incorporate the effects of the map projection's scale factor along the axes.[25] These coordinates are expressed in linear units such as meters or feet, depending on the system's specification, and reflect distortions introduced by the projection process.[26] In the UTM system, for example, southern hemisphere zones assign a false northing of 10,000,000 meters at the equator to distinguish them from northern zones and avoid overlap or negative values.[27] This hemisphere-specific adjustment maintains continuity across the global grid while accommodating the Earth's curvature in the projected framework.[27]Grid North and Orientations
In projected coordinate systems, grid north denotes the direction parallel to the y-axis, aligning with the northing lines of the grid, and is inherently fixed by the properties of the map projection used. Unlike true north, which consistently points toward the Earth's geographic North Pole, grid north remains constant across the plane regardless of location, facilitating straightforward rectangular gridding for measurements. This fixed orientation simplifies computations in cartographic applications but introduces discrepancies when integrating with geographic or magnetic references.[28][29] The divergence between grid north and true north is quantified by the convergence angle, which measures the angular offset at any given point and arises due to the projection's geometry. For transverse Mercator-based systems, this angle γ is approximated by the formula where represents the rectifying latitude factor (accounting for ellipsoidal effects to ensure uniform scale along parallels), is the geodetic latitude, is the longitude, and is the central meridian longitude; the angle is zero directly on the central meridian and grows symmetrically eastward or westward. This convergence ensures conformality in the projection but requires correction for directional accuracy.[21][30] Grid magnetic declination addresses the practical need for compass-based navigation within these systems, defined as the angle between grid north and magnetic north, which is derived by adjusting the standard magnetic declination (the offset from true north) by the convergence angle. This adjustment varies spatially across a projection zone and temporally due to secular variations in the Earth's magnetic field, necessitating periodic updates from geomagnetic models for reliable use.[31][32] These angular references are critical for navigation and surveying, where azimuths referenced to true or magnetic north must be converted to grid north to align with map coordinates and avoid errors in bearing calculations. In the Universal Transverse Mercator (UTM) system, for instance, convergence reaches a maximum of approximately 3° at zone boundaries, about 3° of longitude from the central meridian, highlighting the need for zone-specific adjustments in large-scale operations.[20]Encoding and Precision
Grid Reference Formats
In projected coordinate systems, numeric formats typically represent positions using pairs of easting and northing values along the Cartesian axes, such as 500000 mE and 4000000 mN, where easting denotes the x-coordinate measured eastward from a reference meridian and northing the y-coordinate measured northward from an equator or baseline.[33] These full numeric coordinates provide precise locations within the projection's plane, often in meters for systems like the Universal Transverse Mercator (UTM). For coarser grid-based referencing, values are truncated to represent larger squares; for example, 500 4000 might indicate a 1 km square centered on the full coordinates 500000 mE 4000000 mN, facilitating manual plotting on maps without excessive digits.[34] Alphanumeric systems encode these coordinates more compactly by incorporating letters for grid zones or squares alongside numeric easting and northing values. The Military Grid Reference System (MGRS), derived from UTM and Universal Polar Stereographic (UPS) grids, uses a format like 32U BS 12345 67890, where "32" specifies the UTM zone, "U" the latitude band, "BS" a 100 km square identifier, and the following digits provide easting (12345 m) and northing (67890 m) offsets within that square.[35] Similarly, the British National Grid, based on the OSGB36 datum, employs letters for 100 km squares followed by numeric eastings and northings, as in SK 12345 67890, where "SK" denotes a specific 100 km tile in Great Britain and the digits give offsets in meters.[36] To specify the projection and datum, zone information is often prefixed to these references; for instance, "UTM Zone 11N 500000 4000000" indicates the northern hemisphere of UTM zone 11, using the WGS 84 datum by default. Standardized identifiers from ISO 19111, such as EPSG code 32611 for WGS 84 / UTM zone 11N, provide machine-readable references to the full projected coordinate system definition, ensuring interoperability across datasets.[37] In geographic information systems (GIS), projected coordinate systems are defined using conventions like Well-Known Text (WKT), a standardized string format from the Open Geospatial Consortium. A typical WKT for UTM Zone 11N begins withPROJCS["WGS 84 / UTM zone 11N", GEOGCS["WGS 84", ...], PROJECTION["Transverse_Mercator"], ...], encapsulating parameters for the datum, projection method, and units to enable precise transformations and rendering.[38]