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

Exam Tips

• Remember: Geoid = physical gravity surface ≈ mean sea level; Ellipsoid = mathematical model for computation.
• RA 8560 = PRS92 law; PD 1529 = Property Registration Decree; CA 141 = Public Land Act — know which law governs which survey activity.
• The formula H = h − N appears frequently: H is orthometric height (from leveling), h is GNSS ellipsoidal height, N is the geoid undulation.
• In the Philippines, geoid undulations (N) typically range from about +5 m to +20 m above the WGS84 ellipsoid.

Key Points

• The Earth is an oblate spheroid (ellipsoid of revolution): flattened at the poles, bulging at the equator.
• A sphere is only a first approximation; the equatorial radius exceeds the polar radius by about 21 km.
• The geoid is the equipotential surface of Earth's gravity that coincides with mean sea level — it is irregular and cannot be used directly for computation.
• The reference ellipsoid is a mathematically smooth surface that closely approximates the geoid; all geodetic coordinates (φ, λ, h) are referenced to it.
• The separation between the geoid and the ellipsoid at any point is the geoid undulation N (also called geoid height).
• PRS92 uses the WGS84 ellipsoid with its origin at the Earth's center of mass (geocentric datum), replacing the old Luzon Datum (Clarke 1866 ellipsoid).
• RA 8560 (Philippine Geodetic Reference System Act of 1998) mandates the use of PRS92 for all geodetic surveys in the Philippines.

Definitions

Section Title

1. The Shape of the Earth — From Sphere to Ellipsoid

Common Mistakes

• Confusing the geoid with the ellipsoid — they are different surfaces; the geoid is physical, the ellipsoid is mathematical.
• Using 'elevation' and 'ellipsoidal height' interchangeably — elevations (orthometric heights) are referenced to the geoid, not the ellipsoid.
• Thinking PRS92 uses the Clarke 1866 ellipsoid — that was the old Luzon Datum; PRS92 uses WGS84.
• Assuming the Earth is a perfect sphere in computation problems — always use ellipsoidal formulas unless explicitly told to approximate.

Formulas

Exam Tips

• Memorize WGS84: a = 6 378 137 m, 1/f = 298.257223563, e² = 0.00669438.
• Quick check: e² ≈ 2f (since f² is negligible), so e² ≈ 2/298.257 ≈ 0.00671 — a useful sanity check.
• If only a and b are given, use e² = 1 − (b/a)² to avoid computing f as an intermediate step.
• Board exams sometimes use the Clarke 1866 ellipsoid — know its a and b values.

Key Points

• An ellipsoid is fully defined by two parameters; the standard pair is the semi-major axis a and the flattening f.
• Semi-major axis a = equatorial radius (longest); semi-minor axis b = polar radius (shortest).
• WGS84 defining parameters: a = 6 378 137 m (exact), f = 1/298.257223563.
• All other ellipsoid quantities are derived from a and f.
• The first eccentricity squared e² and the second eccentricity squared e'² are the most frequently used derived quantities.
• Flattening f ≈ 1/298, meaning the Earth is only about 0.335% flatter than a sphere — very nearly spherical.
• The Clarke 1866 ellipsoid (used by the old Luzon Datum) has a = 6 378 206.4 m, b = 6 356 583.8 m — still tested in board exams as a comparison reference.

Definitions

Section Title

2. Ellipsoid Parameters and Their Relationships

Common Mistakes

• Using f instead of 1/f — the flattening f ≈ 0.00335, not 298. In problems stating '1/f = 298.257', you must compute f = 1/298.257 first.
• Computing e instead of e² — all curvature formulas use e² directly; avoid the extra square-root step.
• Using e'² where e² is required (or vice versa) in curvature formulas — identify the formula form first.
• Rounding a and b too early — always carry at least 4–6 significant figures in intermediate steps to avoid accumulated error.

Formulas

Exam Tips

• At φ = 0° (equator): N = a = 6 378 137 m; M = a(1−e²) = 6 335 439 m — memorize these equatorial values.
• At φ = 90° (poles): N = M = a²/b = 6 399 594 m — the polar radius of curvature.
• N > M at all latitudes (except poles); at mid-latitudes the difference is about 40–50 km.
• For the Philippines (φ ≈ 6°–20°N), M ≈ 6 334 000–6 337 000 m and N ≈ 6 378 500–6 380 000 m.
• Board exam problems often give φ and ask for N, M, or R — practice the substitution systematically.

Key Points

• Unlike a sphere, an ellipsoid has different curvatures in different directions at any given surface point.
• The two principal radii of curvature at any point are: M (meridian, north–south) and N (prime vertical, east–west).
• Both M and N are functions of the geodetic latitude φ and the ellipsoid parameters a and e².
• N ≥ M at all latitudes; they are equal only at the poles.
• At the equator (φ = 0°): N = a and M = a(1 − e²); at the poles (φ = 90°): N = M = a²/b = a/√(1−e²).
• The Gaussian mean radius R = √(MN) is used for converting small ellipsoidal areas to a sphere and in local geodetic approximations.
• For arc-to-chord corrections and scale factor computations in PPCS/UTM, M and N are used directly.

Definitions

Section Title

3. Radii of Curvature of the Reference Ellipsoid

Common Mistakes

• Swapping M and N — N is the prime vertical (E–W) radius; M is the meridian (N–S) radius. The mnemonic: 'N for Normal (prime vertical Normal section)'.
• Using degrees without converting to radians in sin functions — always set your calculator to DEGREE mode when computing sinφ.
• Forgetting to raise the denominator to the 3/2 power for M but only to the 1/2 power for N.
• Assuming M = N = a (sphere approximation) without checking whether the problem allows it.
• Not squaring the sine: the formula uses sin²φ, not sinφ.

Formulas

Exam Tips

• Systematic approach: (1) Compute sin²φ → (2) Compute W = 1−e²sin²φ → (3) Compute N = a/√W → (4) Compute M = a(1−e²)/W^(3/2) → (5) Compute R = √(MN).
• For multiple-choice questions, use the value of N to quickly eliminate wrong answers: N is always between 6 378 137 m (equator) and 6 399 594 m (pole).
• If asked for 'radius of curvature in the prime vertical,' that is N — always.
• Always double-check your φ is in degrees and your calculator is in DEGREE mode before evaluating sin φ.

Key Points

• Always identify: what is given (a, b, f, e², φ), what is asked (N, M, R, f, e², b).
• Write the formula, substitute values carefully, and compute step-by-step.
• Keep at least 4 decimal places in intermediate steps.
• Verify your answer using a quick-check: N should be close to 6 378 000–6 380 000 m for Philippine latitudes.

Section Title

4. Worked Board-Style Problems

Common Mistakes

• Forgetting to compute W = 1 − e²sin²φ before computing N and M separately.
• Using the wrong exponent: N uses W^(1/2); M uses W^(3/2).
• Computing (1 − e²sin²φ)^(3/2) by cubing the square root twice — use the direct exponent: W^(3/2) = W × √W.
• Mixing up which formula applies to which direction (M = N–S; N = E–W).

Formulas

Exam Tips

• When the problem says 'Philippine Reference System' or 'PRS92,' always use WGS84 parameters.
• RA 8560 amended RA 4374 — know both; RA 8560 is the current controlling law for geodetic engineering practice.
• PD 1529 governs title registration; CA 141 governs public land; both require geodetic surveys tied to official control.
• PPCS/UTM: the Philippines spans Zones I (116.5°–121°E) to V (125°–127.5°E) — Zone III (121°–123°E) covers most of Luzon.

Key Points

• Know both ellipsoids: WGS84 is current (PRS92); Clarke 1866 is historic (Old Luzon Datum) — both appear in board exams.
• The primary ellipsoid for all current Philippine geodetic work is WGS84 under PRS92 (RA 8560).
• All PPCS/UTM grid coordinates in the Philippines are projected from the WGS84/PRS92 ellipsoid.
• For cadastral and titling purposes under PD 1529 and CA 141, geodetic control points are tied to PRS92.
• RA 4374 (Geodetic Engineering Act, as amended by RA 8560) governs the practice of geodetic engineering in the Philippines.

Definitions

Section Title

5. Summary of Key Ellipsoid Parameters — WGS84 vs. Clarke 1866

Common Mistakes

• Applying Clarke 1866 parameters in a problem that specifies WGS84/PRS92.
• Ignoring the legal framework — exam questions sometimes ask about which law mandates a particular datum or survey requirement.
• Forgetting that PPCS Zone I–V cover the Philippines from Mindanao to Batanes — zone selection depends on longitude bands.