Geodetic coordinates
Geodetic coordinates are a curvilinear orthogonal coordinate system employed in geodesy to specify the position of points on or above the Earth's surface relative to a reference ellipsoid that models the planet's oblate spheroid shape.[1] This system comprises three primary components: geodetic latitude (φ), which measures the angle between the ellipsoid normal at the point and the equatorial plane (ranging from -90° to 90°); geodetic longitude (λ), the angle eastward from the prime meridian to the meridian containing the normal (ranging from -180° to 180° or 0° to 360°); and ellipsoidal height (h), the perpendicular distance along the normal from the ellipsoid surface.[2] Unlike geocentric coordinates, which use Cartesian (X, Y, Z) measurements from the Earth's center of mass, geodetic coordinates align with the local vertical defined by the ellipsoid normal, providing a more intuitive framework for surface-based applications.[1] Historically, geodetic coordinates evolved from spherical approximations to ellipsoidal models in the 18th century, with modern systems like the International Terrestrial Reference Frame (ITRF2020, released in 2021 and current as of 2025) integrating the ellipsoid's parameters (semimajor axis a and flattening f) with the Earth's gravity field and orientation to achieve global consistency in positioning.[3][4] The reference ellipsoid approximates the geoid—the equipotential surface of Earth's gravity field coinciding with mean sea level—but geodetic height (h) differs from orthometric height (H) by the gravimetric geoid undulation (N), related via the formula h = H + N.[5] This distinction is critical for applications requiring separation between geometric and physical heights, such as in satellite altimetry and gravity modeling.[6] In practice, geodetic coordinates underpin modern geospatial technologies, including the Global Positioning System (GPS), where receivers compute positions in real-time relative to datums like WGS 84 (updated to align with ITRF2020 as of 2024), enabling accuracies on the order of centimeters for surveying, navigation, and environmental monitoring.[7] They facilitate transformations between local and global frames, supporting national geodetic surveys and international standards for mapping and disaster response.[1] Ongoing refinements, driven by space-based observations from missions like GRACE-FO and networks such as GNSS, continue to enhance the precision of these systems to account for Earth's dynamic deformations.[8][9]Introduction
Definition and Components
Geodetic coordinates form a curvilinear orthogonal system used in geodesy to specify positions on or near the Earth's surface relative to a reference ellipsoid, consisting of three primary components: geodetic latitude (\phi), geodetic longitude (\lambda), and ellipsoidal height (h). This system approximates the Earth's irregular shape with an oblate spheroid model, providing a mathematically precise framework for global positioning that accounts for the planet's equatorial bulge and polar flattening.[1][10] Geodetic latitude (\phi) is defined as the angle between the equatorial plane and the normal to the reference ellipsoid at the point of interest, ranging from $0^\circ at the equator to $90^\circ at the poles (positive north, negative south). This differs from simpler spherical models by following the ellipsoid's curvature along the meridian, ensuring more accurate representation of surface positions.[1][2] Geodetic longitude (\lambda) measures the angular distance east or west of the Prime Meridian, which passes through Greenwich, England, in the equatorial plane, with values from -180^\circ to $180^\circ or $0^\circ to $360^\circ. It is determined by the angle between the plane of the Prime Meridian and the plane of the meridian passing through the point on the ellipsoid.[11][12] Ellipsoidal height (h) represents the orthogonal distance from the reference ellipsoid surface to the point, measured along the direction of the ellipsoid normal (positive upward, negative downward), typically in meters. This height provides the vertical component essential for three-dimensional positioning, distinct from orthometric heights tied to mean sea level.[13][14] Geometrically, these coordinates visualize the Earth as an oblate spheroid—a flattened sphere with a semi-major axis along the equator and a shorter semi-minor axis at the poles—allowing latitude and longitude to trace curved paths on the ellipsoid while height extends perpendicularly outward, closely modeling the geoid's undulations for practical applications in surveying and navigation.[15][16]Historical Development
The concept of geodetic coordinates originated in ancient times with approximations of the Earth as a sphere. In the 3rd century BCE, Eratosthenes of Cyrene calculated the Earth's circumference to within about 2% accuracy by measuring the angle of the Sun's rays at Alexandria and Syene (modern Aswan) on the summer solstice, assuming a spherical Earth and using the known distance between the cities.[17] This measurement laid foundational principles for latitude and longitude systems, though spherical assumptions limited precision for large-scale mapping.[18] Advancements in the 17th and 18th centuries shifted focus to the Earth's oblateness, driven by theoretical insights into gravity and rotation. Isaac Newton, in his 1687 Philosophiæ Naturalis Principia Mathematica, predicted the Earth as an oblate spheroid due to centrifugal forces at the equator, estimating a polar flattening of about 1:230.[19] Christiaan Huygens independently arrived at a similar conclusion in 1690, calculating a flattening of 1:240 based on pendulum observations, which spurred early ellipsoidal models to better represent the planet's shape.[20] These theories prompted French expeditions in the 1730s to Peru and Lapland, confirming oblateness through arc measurements and paving the way for ellipsoid-based coordinates.[19] In the 19th century, regional surveys refined ellipsoidal parameters for national purposes. Friedrich Wilhelm Bessel developed the Bessel ellipsoid in 1841, derived from arc measurements across Europe and Asia, with a semi-major axis of 6,377,396 meters and flattening of 1:299.15, which became widely used in central Europe and colonial networks.[21] Alexander Ross Clarke proposed multiple spheroids, including the Clarke 1866 (semi-major axis 6,378,206 meters, flattening 1:294.98) for U.S. surveys and the Clarke 1880 for British and international arcs, supporting triangulation networks like the Principal Triangulation of Britain.[22] These models emphasized local fitting over global uniformity, reflecting the era's focus on continental-scale accuracy.[23] The 20th century saw efforts toward international standardization amid growing needs for global consistency. John Fillmore Hayford's 1909 ellipsoid, based on U.S. Coast and Geodetic Survey data, featured a semi-major axis of 6,378,388 meters and flattening of 1:297, and was adopted by the International Union of Geodesy and Geophysics in 1924 as the International Ellipsoid of Reference, facilitating cross-border surveys.[24] This marked a transition from regional to more unified systems, though variations persisted until satellite technology enabled precision.[25] Post-1960s satellite geodesy revolutionized the field, allowing global measurements free from terrestrial limitations. The Geodetic Reference System 1980 (GRS80), adopted by the International Association of Geodesy, defined an ellipsoid with a semi-major axis of 6,378,137 meters and flattening of 1:298.257, incorporating Doppler and laser ranging data for unprecedented accuracy.[26] This system underpinned the World Geodetic System 1984 (WGS84), developed by the U.S. Department of Defense using satellite observations, which standardized global coordinates for navigation and became the foundation for GPS with its near-identical parameters to GRS80.[27]Coordinate Systems
Components of Geodetic Coordinates
Geodetic coordinates are defined by three primary components: geodetic latitude (φ), longitude (λ), and ellipsoidal height (h), which together specify a point's position relative to a reference ellipsoid.[28] The standard notation uses the Greek letter φ for latitude and λ for longitude, while h denotes height; these symbols are widely adopted in geodetic literature and standards.[29][30] Geodetic latitude φ measures the angle between the equatorial plane of the reference ellipsoid and the normal to the ellipsoid surface at the point, ranging from -90° at the South Pole to +90° at the North Pole, with positive values indicating northern latitudes and negative values southern latitudes.[29] Longitude λ represents the angle east or west from the prime meridian (typically Greenwich), with two common conventions: -180° to +180° (positive east, negative west) or 0° to 360° (eastward only).[31][32] Ellipsoidal height h is the distance along the normal from the reference ellipsoid surface to the point, measured in meters and typically positive upward.[33][30] A key distinction exists between geodetic latitude φ, which is defined by the perpendicular to the ellipsoid, and parametric latitude (also called reduced latitude, denoted β), an auxiliary angle used in ellipsoidal computations that relates to the projection onto a confocal sphere via the ellipsoid's flattening.[29][34] On the ellipsoid surface, β is always less than or equal to φ in magnitude except at the equator and poles, where they coincide, providing a means to simplify certain geometric transformations without altering the underlying reference surface.[29] Angular components φ and λ are expressed in decimal degrees for computational precision, though the degrees-minutes-seconds (DMS) format—such as 40° 45' 30" N—remains common for human-readable representation, where 1° = 60' and 1' = 60".[35][36] Precision typically reaches 6-8 decimal places for latitude and longitude in modern systems, corresponding to sub-meter accuracy, while h uses meters with similar decimal precision.[37][38] Alternative units like radians (for mathematical computations) or grads (0 to 400 gon, where 1 gon = 0.9°) are occasionally employed but are less standard than degrees.[29] Special cases arise at the poles, where φ = ±90° and λ becomes undefined due to the convergence of all meridians, rendering any assigned longitude arbitrary for that exact point.[39][40] At the equator, φ = 0°, and λ fully defines the east-west position without such ambiguities.[31][32]Reference Ellipsoids
A reference ellipsoid is a mathematical model of the Earth in the form of an oblate spheroid that approximates the irregular surface of mean sea level, providing a smooth, standardized surface for geodetic computations.[41] This oblate spheroid is generated by rotating an ellipse about its minor axis, resulting in an equatorial bulge and polar flattening that better matches the Earth's overall shape than a perfect sphere.[42] The ellipsoid is defined by two primary parameters: the semi-major axis a, which represents the equatorial radius, and the flattening f, a dimensionless value quantifying the compression at the poles. The polar radius, or semi-minor axis b, is derived as b = a(1 - f). For modern global ellipsoids, a is approximately 6378 km and f \approx 1/298.257, yielding a b of about 6357 km and an equatorial bulge of roughly 21 km.[41] Reference ellipsoids are categorized as local or global, depending on their intended scope of application. Local ellipsoids, such as Clarke 1866, are optimized for specific regions like North America, where they minimize deviations from the local geoid.[43] In contrast, global ellipsoids like WGS84 are designed for worldwide use, aligning closely with the Earth's center of mass and supporting applications such as satellite navigation.[44] Selection of a reference ellipsoid involves criteria focused on achieving the best possible fit to the geoid, thereby minimizing geoid undulations—the vertical separations between the ellipsoid and the geoid surface—across the target area.[43] Modern ellipsoids integrate satellite observations, gravity measurements, and least-squares adjustments of global data to refine parameters and ensure compatibility with datum transformations.[45] Prominent examples include the World Geodetic System 1984 (WGS84), with a = 6378137 m and f = 1/298.257223563, adopted as the standard for GPS and global geospatial reference.[44] The Geodetic Reference System 1980 (GRS80), defined by the International Association of Geodesy with a = 6378137 m and f = 1/298.257222101, serves as a conventional ellipsoid for geodetic and geophysical computations.[45] These differ subtly in the precision of f, with GRS80 emphasizing theoretical consistency from satellite-derived gravity models and WGS84 prioritizing practical alignment for defense and navigation systems, resulting in undulation differences under 1 meter globally.[43]| Ellipsoid | Semi-Major Axis a (m) | Flattening f | Primary Use |
|---|---|---|---|
| Clarke 1866 | 6378206.4 | 1/294.9786982 | Local (North America)[46] |
| GRS80 | 6378137 | 1/298.257222101 | Global geodetic standard[45] |
| WGS84 | 6378137 | 1/298.257223563 | Global navigation (GPS)[44] |