GELE Geodesy — Figure of the Earth and the Reference EllipsoidSummary

The mathematical model of Earth's shape, formed by rotating an ellipse about its minor axis (polar axis). It is defined by two axes: the semi-major axis (a) at the equator and the semi-minor axis (b) at the poles. The equatorial radius is approximately 6,378,137 m, while the polar radius is approximately 6,356,752 m—a difference of about 21,385 m. This flattening occurs because Earth's rotation creates centrifugal forces that cause the equatorial bulge. The oblate form is essential for accurate geodetic computations; using a sphere would introduce errors of several hundred meters in position over long distances.

Flattening quantifies how much Earth deviates from a perfect sphere. It is the ratio of the difference between the equatorial and polar radii to the equatorial radius: f = (a − b) / a. For WGS84, f = 1/298.257223563 ≈ 0.003353. This small value (less than 1%) confirms Earth is only slightly flattened, but this small difference is absolutely critical in geodesy. The reciprocal 1/f = 298.257223563 is often used in geodetic references. Flattening is used to derive all other ellipsoid parameters, including eccentricity and the radii of curvature.

The semi-major axis (a) is the equatorial radius of the ellipsoid—the distance from Earth's center to the equator. The semi-minor axis (b) is the polar radius—the distance from Earth's center to either pole. They are related by: b = a(1 − f). For WGS84: a = 6,378,137 m (exact) and b = 6,356,752.31447 m (derived). These are not arbitrary values; they are selected based on modern satellite geodesy and GNSS measurements to best fit Earth's actual shape. The difference (a − b) ≈ 21,385 m represents Earth's equatorial bulge.

Eccentricity squared (first eccentricity squared) measures the elongation of the ellipsoid and is derived from the semi-major and semi-minor axes: e² = (a² − b²) / a² = 2f − f². It can also be expressed as e² = (2a − f)f (when f is small). For WGS84, e² = 0.00669438. The quantity (1 − e²sin²φ) appears frequently in geodetic formulas, particularly in expressions for radii of curvature. A smaller e² indicates a rounder shape; e² = 0 represents a perfect sphere. The eccentricity itself (without squaring) is e = √(e²) ≈ 0.0818, but geodetic formulas almost always use e² directly.

The prime vertical radius (N) is the radius of curvature in the plane perpendicular to the meridian at any point on the ellipsoid. It is oriented in the east–west direction. The formula is: N = a / √(1 − e²sin²φ), where φ is geodetic latitude. At the equator (φ = 0°), N = a. As latitude increases, N increases monotonically until at the poles (φ = 90°), N = a² / b = a / (1 − f) ≈ 6,389,272 m for WGS84. The prime vertical radius is always ≥ the meridian radius (N ≥ M). In Philippine surveying and GPS processing, N values are critical for converting ellipsoidal coordinates (φ, λ, h) to Cartesian coordinates and vice versa, and for calculating distances in the east–west direction.

The meridian radius (M) is the radius of curvature in the meridian plane (containing the longitude lines), oriented north–south. The formula is: M = a(1 − e²) / (1 − e²sin²φ)^(3/2). At the equator (φ = 0°), M = a(1 − e²) ≈ 6,335,439 m for WGS84. At the poles (φ = 90°), M = a² / b. The meridian radius increases with latitude and equals N only at the poles. For most of Earth, M < N. The meridian radius controls the curvature of meridian lines and is essential for calculating distances along north–south directions on the ellipsoid.

The radii of curvature N and M both depend on latitude φ. Their relationship is: N ≥ M for all latitudes, with equality only at the poles. At the equator (φ = 0°): N = a and M = a(1 − e²). At 45° latitude, the values are intermediate. At the poles (φ = 90°): both N and M equal a² / b ≈ 6,389,272 m. The ratio N / M increases from the equator toward the poles. This variation with latitude reflects the shape of the ellipsoid: the equatorial sections are flatter (larger radius), while the polar sections are more tightly curved. This latitude dependence is why geodetic formulas must be applied locally—a single value of curvature cannot be used across an entire region.

The mean radius of curvature (Gaussian mean radius) is defined as R = √(MN). This geometric mean provides a single scalar representation of curvature at a given latitude. At the equator, R = √(a × a(1 − e²)) = a√(1 − e²) ≈ 6,356,752 m. The mean radius varies with latitude but increases monotonically toward the poles. For small regions (typically < 50 km × 50 km), the ellipsoid may be approximated as a sphere of radius R, significantly simplifying calculations. This is the basis for many classical surveying methods that treat Earth as a local sphere.

The World Geodetic System 1984 (WGS84) is the global reference ellipsoid adopted by GPS/GNSS systems and is the foundation for international geodesy. Its defining parameters are: semi-major axis a = 6,378,137 m (exactly, by definition) and flattening f = 1/298.257223563. From these, the derived parameters are: semi-minor axis b = 6,356,752.31447 m, first eccentricity squared e² = 0.00669438002900925, and reciprocal flattening 1/f = 298.257223563. WGS84 was established by fitting an ellipsoid to global satellite laser ranging (SLR) and Doppler observations. It is the standard reference system for all GNSS positioning (GPS, GLONASS, Galileo, BeiDou, etc.) and is now the standard for Philippine coordinate systems through the Philippine Reference System (PRS92), which is effectively WGS84 for practical purposes.

The Philippine Reference System 1992 (PRS92) is the official geodetic datum for the Philippines, established under RA 4374 (Geodetic Engineering Act). PRS92 uses the WGS84 ellipsoid (semi-major axis a = 6,378,137 m, flattening f = 1/298.257223563) but is physically realized through a network of Ground Control Points (GCPs) established by the National Mapping and Resource Information Authority (NAMRIA). The PRS92 datum is based on observations through GNSS (primarily GPS). Coordinates are expressed as geodetic latitude (φ), geodetic longitude (λ), and ellipsoidal height (h) above the WGS84 ellipsoid. PRS92 is used throughout the Philippines for mapping, cadastral surveying, engineering surveys, and all official geographic information systems (GIS).

Before WGS84, various nations and regions used different reference ellipsoids fitted to local surveys. Examples include: Clarke 1866 (a = 6,378,206.4 m, b = 6,356,583.8 m) used in North America and parts of the Philippines; Bessel 1841 (a = 6,377,397.2 m, b = 6,356,079.0 m) used in Europe and Asia; Hayford 1909 (a = 6,378,388 m, b = 6,356,912 m) used internationally. The Philippines historically used the Luzon datum based on Clarke 1866. As global GNSS systems became available, WGS84 became the universal standard. Today, converting old survey data from Clarke 1866 or Luzon datum to PRS92 (WGS84) requires applying datum transformation parameters (typically 7-parameter transformations: three translations, three rotations, and one scale factor). Understanding the differences between ellipsoids is necessary for interpreting historical surveys and understanding why modern coordinates differ from old surveyed values.

The second eccentricity squared (e'²) is an alternative measure of ellipsoid elongation, defined as: e'² = (a² − b²) / b² = e² / (1 − e²). For WGS84, e'² = 0.00673949675658. Unlike the first eccentricity (e²), which is referenced to the semi-major axis, the second eccentricity is referenced to the semi-minor axis. In some geodetic formulas—particularly in series expansions and in certain map projections—e'² appears instead of e². The two are related by: e'² = e² / (1 − e²) or e² = e'² / (1 + e'²). Students must be careful to use the correct form in formulas; using e² where e'² is required (or vice versa) produces significantly incorrect results.

Flattening f defines how much the ellipsoid deviates from a sphere. All other shape parameters—semi-minor axis (b), eccentricity squared (e²), second eccentricity (e'²), and the radii of curvature (N and M)—are derived from f and the semi-major axis (a). Understanding this hierarchical relationship shows why f is fundamental.

Neither N nor M is constant. Both vary with latitude φ through the factor (1 − e²sin²φ). This means that curvature is a function of position on the ellipsoid. The stronger this dependence, the more non-spherical the ellipsoid is. For a perfect sphere (e² = 0), both N and M would equal the radius everywhere.

Eccentricity squared measures how elongated the ellipsoid is. It can be expressed three ways: (1) in terms of axes: e² = (a² − b²)/a²; (2) in terms of flattening: e² = 2f − f²; (3) it directly controls the variation of N and M with latitude. A larger e² means greater variation in radius with latitude and stronger non-spherical behavior.

The mean radius R = √(MN) provides a single representative radius at each latitude. For small areas (< 50 km), the ellipsoid can be approximated as a sphere of radius R, greatly simplifying calculations. However, this approximation breaks down for larger areas where the variation of N and M becomes significant.

WGS84 has become the universal standard ellipsoid. PRS92 (Philippine standard) uses the WGS84 ellipsoid. All GNSS systems (GPS, GLONASS, Galileo, BeiDou) are referenced to WGS84. Understanding WGS84's specific parameter values is essential for modern geodetic practice and professional licensing.

Older surveys used ellipsoids like Clarke 1866 or Bessel 1841. Modern positioning (PRS92/WGS84) differs from these older datums. Converting between ellipsoids requires understanding both the original ellipsoid parameters AND the datum transformation parameters (translations, rotations, scale). This transformation process is necessary for integrating legacy survey data with modern systems.