GELE Geodesy — Figure of the Earth and the Reference EllipsoidStudy Notes

The Earth's true shape is an irregular geoid—a gravitational equipotential surface that would be occupied by seawater if the oceans covered the entire planet. However, for practical geodetic and surveying work, we model the Earth as a mathematical surface called the reference ellipsoid (also called the Earth ellipsoid or geodetic ellipsoid). This ellipsoid is a surface of revolution created by rotating an ellipse about its minor axis (the polar axis). The resulting shape is an oblate spheroid, slightly flattened at the North and South Poles and bulged outward at the equator due to Earth's rotation. The difference between the equatorial and polar radii is only about 21.4 km on a planet with a 6378 km equatorial radius—a remarkably small difference (about 0.335%), yet this small deviation is critically important in high-precision surveying and GNSS applications. The geoid deviates from the ellipsoid by as much as ±100 m, but the ellipsoid provides the best mathematical approximation for computational purposes. In the Philippines, the Philippine Reference System 1992 (PRS92) is based on the WGS84 ellipsoid, which is the standard for all GNSS positioning, land surveys, and official mapping under RA 4374 and PD 1529. Understanding why we use an ellipsoid rather than a sphere is crucial: a sphere cannot represent Earth's mass distribution and rotational characteristics as accurately, leading to significant errors in positioning, especially near the poles.

A reference ellipsoid is uniquely defined by two independent geometric parameters. The most common choice is the semi-major axis (equatorial radius) denoted a, and the flattening denoted f. From these two parameters, all other ellipsoid properties can be derived mathematically. The semi-minor axis b is the polar radius, smaller than a by the amount a·f. Flattening f is defined as the relative difference between the equatorial and polar radii: f = (a − b)/a. For Earth, f is very small, approximately 1/298.257223563 for WGS84. The first eccentricity squared (e²) measures the deviation of the ellipse from a circle and is derived from a and b: e² = (a² − b²)/a² = 2f − f². There is also a second eccentricity e'² = (a² − b²)/b² used in some formulas. The relationship e² = 2f − f² shows how these parameters connect; since f is small, e² ≈ 2f for practical purposes. For WGS84, the standard ellipsoid of the Philippines' PRS92 system: a = 6,378,137 m exactly, 1/f = 298.257223563 (which means f ≈ 0.00335281), e² ≈ 0.00669438, and b ≈ 6,356,752.31 m. These precise values are fixed by international agreement and appear in all GNSS receivers and cadastral software used in the Philippines. Memorizing the WGS84 values is essential for exam success.

Although the ellipsoid is a smooth, well-defined mathematical surface, its curvature is not uniform—it varies with latitude. This variation is crucial for geodetic calculations because many surveying operations depend on local radii of curvature. Two principal radii are used: the prime vertical radius of curvature (N, also called the radius of curvature in the prime vertical) and the meridian radius of curvature (M, also called the meridian radius). The prime vertical N represents the radius of curvature in the east–west direction (along a line of constant latitude). It is computed as N = a / √(1 − e²sin²φ), where φ is the geodetic latitude in radians or degrees. The meridian radius M represents the radius of curvature in the north–south direction (along a meridian, or line of constant longitude). It is computed as M = a(1 − e²) / (1 − e²sin²φ)^(3/2). A key observation is that N ≥ M everywhere on the ellipsoid. At the equator (φ = 0°), N equals a (the equatorial radius), while M = a(1 − e²) (the smallest meridian radius). At the poles (φ = ±90°), both N and M become infinite, but their ratio approaches 1. The mean radius of curvature R = √(M·N), often called the Gaussian radius, is used when approximating the ellipsoid locally as a sphere for simplified calculations. In Philippine surveying, these radii are essential for computing the convergence of meridians, scale factors in map projections (especially the Universal Transverse Mercator—UTM—system used in the Philippine Plane Coordinate System), and for reducing measured distances to the ellipsoid. Many surveying equations, particularly those for geodetic distance calculations and arc-length computations along parallels or meridians, require accurate knowledge of these radii at the specific latitude of the project.

The World Geodetic System 1984 (WGS84) is the standard reference ellipsoid for all Global Positioning System (GPS) and modern GNSS receivers worldwide, including those used throughout the Philippines. The WGS84 ellipsoid is defined by two primary parameters: the semi-major axis a = 6,378,137 m (exact, by definition) and the inverse flattening 1/f = 298.257223563 (exact, by definition). From these, all other parameters are derived: f ≈ 0.00335281, b ≈ 6,356,752.31 m, and e² ≈ 0.00669438. In 1992, the Philippines officially adopted the Philippine Reference System 1992 (PRS92), which is explicitly based on the WGS84 ellipsoid. This was mandated under Republic Act 4374 (an act adopting the PRS as the standard datum for all mapping and surveying in the Philippines) and further reinforced by RA 8560 (the Property Boundaries Law) and Presidential Decree 1529 (the Property Registration Decree). All official cadastral maps, digital land records, infrastructure projects (roads, utilities, airports), and GNSS-based surveys in the Philippines must reference WGS84/PRS92. This ensures that a surveyor in Luzon and another in Mindanao are using the same geometric reference frame, critical for nationwide coordination, property disputes, and infrastructure planning. Any surveyor failing to use WGS84/PRS92 for official work is in violation of Philippine law. The PRC Geodetic Engineer Licensure Examination tests thorough knowledge of WGS84 parameters and their application to Philippine mapping and surveying standards. It is essential to memorize the WGS84 values and understand their role in the Philippine legal framework.

The reference ellipsoid and its derived parameters are not abstract theoretical concepts—they directly impact every surveying and mapping project in the Philippines. In cadastral surveying under RA 8560, surveyors use GNSS receivers that output WGS84 coordinates. These coordinates are then processed using N and M values at the local latitude to reduce measured distances to the ellipsoid and compute accurate grid coordinates in the Philippine Plane Coordinate System (PPCS). The PPCS is based on the Universal Transverse Mercator (UTM) projection, which divides the Philippines into zones. For each zone, a central meridian is selected, and the scale factor k at any point depends on the distance from the central meridian and the local radius of curvature N. Without accurate knowledge of N, the scale factor cannot be computed correctly, leading to errors in cadastral maps. For infrastructure projects (highways, railways, utilities), surveying networks must be established and adjusted using the WGS84 ellipsoid as the reference frame. Height measurements from GNSS are ellipsoidal heights (heights above the WGS84 ellipsoid), which differ from orthometric heights (heights above mean sea level) by the geoid undulation. Surveyors must understand this distinction to avoid mistakes in vertical data. In topographic mapping and the production of official maps by agencies like the National Mapping and Resource Information Authority (NAMRIA), the ellipsoid parameters determine the accuracy and consistency of the entire map. Moreover, under RA 4374, all public maps and surveys must be consistent with the official PRS92 datum—the ellipsoid is the foundation of this nationwide legal requirement. Finally, when converting between old survey data (which may reference Clarke 1866 or other historical ellipsoids) and modern WGS84/PRS92 data, knowledge of both ellipsoids and their geometric differences is essential. This conversion is a common task when modernizing old cadastral records or integrating historical survey data with new GNSS observations.

The topic of the reference ellipsoid and its parameters is a frequent source of errors among surveying students and even practicing surveyors. Understanding these common pitfalls is crucial for success on the PRC Geodetic Engineer Licensure Examination. First, students often confuse eccentricity e with eccentricity squared e². The formulas use e² (e-squared), not e. For WGS84, e² = 0.00669438, so e = √0.00669438 ≈ 0.0818. Many incorrect calculations result from using e instead of e². Second, the flattening f is very small—approximately 1/298 or 0.00335 for Earth. Students sometimes misread this as f = 0.298 or 0.335, leading to wildly incorrect results. Always double-check: for WGS84, f = 1 ÷ 298.257223563 ≈ 0.00335281. Third, students confuse the prime vertical radius N (east–west curvature) with the meridian radius M (north–south curvature). Remember: N is along a parallel (constant latitude), and M is along a meridian (constant longitude). At the equator, N = a (the equatorial radius) and M = a(1 − e²) (less than a). At the poles, both approach infinity, but M approaches infinity faster. Fourth, when calculating sin²φ, ensure your calculator is in degree mode, not radian mode, if φ is given in degrees. A common exam mistake is computing sin²(14°) with the calculator in radian mode, giving a completely wrong value. Fifth, the semi-minor axis b is very close to the semi-major axis a, differing by only about 0.335%. When computing b from f, use b = a(1 − f), not b = a − f. The latter would give a completely wrong answer. Sixth, students sometimes forget that the formulas for N and M depend on latitude φ. They compute N or M at the equator and mistakenly use that value for all latitudes, ignoring the sin²φ term in the denominator. This is incorrect; N and M vary significantly with latitude. Seventh, when solving problems, always include units (meters, not millimeters or kilometers, unless specified otherwise in SI units). A distance of 6,378,137 without units is ambiguous; 6,378,137 m makes it clear. Finally, for the PRC exam, memorize the WGS84 parameters exactly: a = 6,378,137 m, 1/f = 298.257223563, e² = 0.00669438. These values appear in nearly every surveying calculation and are the foundation of the Philippine Reference System 1992. Examiners expect you to know these values by heart.