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GELE Geodesy — Geodetic Datums and Coordinate SystemsCheat Sheet

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Cheat sheet for GELE Geodesy — Geodetic Datums and Coordinate Systems. Compact, printable, and organised around the concepts Professional Regulation Commission (PRC) — Board of Geodetic Engineering tests most frequently in the GELE 2026. Perfect for the week before exam day.

Exam context

The Geodetic Engineer Licensure Examination is conducted by Professional Regulation Commission (PRC) — Board of Geodetic Engineering and is scheduled for September 2026. The Geodesy subtest is marked as "Core" in the official pattern, and Geodetic Datums and Coordinate Systems appears in position 2nd of 6 in the GELE Geodesy review rotation. Passing mark: 70% weighted average, no sub-test below 50%. Recent GELE 2026 papers have drawn roughly a meaningful share of questions from this subject.

Geodetic Datums and Coordinate Systems - Cheat Sheet

Your last-minute revision companion for Geodetic Datums and Coordinate Systems. Master datum definitions, coordinate types, transformations, and Philippine systems (WGS84, PRS92, Clarke 1866). All formulas, key facts, and exam-critical concepts condensed for rapid recall.

Sections

Common Values

Value

6 378 137 m

Symbol

Quantity

WGS84 Semi-major axis

Value

1/298.257

Symbol

Quantity

WGS84 Flattening

Value

6 378 206.4 m

Symbol

Quantity

Clarke 1866 Semi-major axis

Value

1/294.9787

Symbol

Quantity

Clarke 1866 Flattening

Value

ΔX ≈ −127.6 m, ΔY ≈ −67.2 m, ΔZ ≈ −47.0 m

Symbol

ΔX, ΔY, ΔZ

Quantity

PRS92 ↔ WGS84 Typical shift (Cartesian)

Section Title

Datums Fundamentals

Important Facts

PRS92 origin station: Balanacan (Marinduque); based on Clarke 1866 ellipsoid (a = 6 378 206.4 m).
WGS84 origin: Earth's mass center; based on WGS84 ellipsoid (a = 6 378 137 m).
Luzon Datum 1911 and PRS92 use same ellipsoid (Clarke 1866) but different datums.
GPS naturally outputs WGS84 — must transform to PRS92 for Philippine surveys.
Local datums cannot be extended beyond their region without large systematic errors.
Datum shift is typically 100–200 m in Cartesian coordinates between WGS84 and PRS92.
Ellipsoidal height ≠ orthometric height (MSL); difference varies by location and geoid model.

Key Definitions

Term

Geodetic Datum

Example

WGS84 (geocentric, GPS); PRS92 (local, Philippines)

Definition

A reference ellipsoid with defined origin, orientation, and scale; fixes mathematical ellipsoid to physical Earth for coordinate meaning.

Term

Geocentric (Global) Datum

Example

WGS84, ITRF series — standard for GNSS/GPS

Definition

Ellipsoid centered at Earth's mass center; valid worldwide; mass-based origin from satellite data.

Term

Local (Regional) Datum

Example

PRS92 (Philippines, Clarke 1866, origin Balanacan); Luzon Datum 1911 (same ellipsoid)

Definition

Ellipsoid best-fitting one region with defined origin station; valid in that region only.

Term

Reference Ellipsoid

Example

Clarke 1866: a = 6 378 206.4 m, f = 1/294.9787; WGS84: a = 6 378 137 m, f = 1/298.257

Definition

Mathematical surface (rotational ellipsoid) approximating Earth's shape; defined by semi-major axis *a*, semi-minor axis *b*, and flattening *f*.

Term

Ellipsoidal Height (h)

Example

GPS outputs WGS84 ellipsoidal heights; ~86 m higher than orthometric height in Philippines

Definition

Height of a point above the reference ellipsoid surface, measured along the normal.

Diagrams To Know

Ellipsoid geometry: equatorial radius (a), polar radius (b), flattening formula.
Datum vs. Ellipsoid vs. Coordinate System (Venn diagram).
Geocentric (XYZ origin at Earth's center) vs. Local (origin at station) positioning.

Formulas

Formula

Geodetic to Cartesian: X = (N + h) cos φ cos λ; Y = (N + h) cos φ sin λ; Z = (N(1 − e²) + h) sin φ

Meaning

φ = latitude, λ = longitude, h = ellipsoidal height, N = radius of curvature in prime vertical, e = eccentricity

Watch Out

Use correct ellipsoid (WGS84 vs. Clarke 1866); e² ≠ eccentricity; N depends on φ and ellipsoid.

When To Use

Convert geographic (lat/lon/height) to 3D Cartesian (X, Y, Z) for datum transformations or GNSS processing.

Formula

Cartesian to Geodetic: iterative methods (Heikkinen, Bowring) or approximate atan2(Y, X) = λ

Meaning

Inverse of above; usually computed iteratively because Z, φ coupling is nonlinear.

Watch Out

Non-unique; use atan2 to get correct quadrant for λ; iteration may not converge if h is extreme; different algorithms vary in speed.

When To Use

Convert GNSS X, Y, Z output back to lat/lon/h for reporting and mapping.

Formula

Radius of Curvature in Prime Vertical: N = a / √(1 − e² sin² φ)

Meaning

N = radius of curvature; a = ellipsoid semi-major axis; e² = first eccentricity squared; φ = latitude

Watch Out

N ≠ radius of Earth; N ≥ a always; do NOT confuse with radius of curvature in meridian (M).

When To Use

Computing Cartesian coordinates or scale corrections; N varies with latitude.

Formula

First Eccentricity Squared: e² = (a² − b²) / a²; Second: e'² = (a² − b²) / b²

Meaning

e² characterizes ellipsoid shape; both depend on flattening f = (a − b) / a

Watch Out

e² is NOT the eccentricity e; for WGS84, e² ≈ 0.00669; always use e², not e, in standard formulas.

When To Use

Geodetic ↔ Cartesian conversions; scale and projection formulas.

Common Values

Value

0.006694380004260827

Symbol

e²

Quantity

WGS84 First Eccentricity Squared

Value

0.006768657

Symbol

e²

Quantity

Clarke 1866 First Eccentricity Squared

Value

6 378 137 m

Symbol

Quantity

Earth's Mean Equatorial Radius (WGS84)

Value

6 356 752.3 m

Symbol

Quantity

Earth's Mean Polar Radius (WGS84)

Section Title

Coordinate Types

Important Facts

Three main coordinate representations: geodetic (φ, λ, h), Cartesian (X, Y, Z), and projected (E, N).
All three describe the same point; choice depends on application (GNSS, mapping, surveying).
Conversions use ellipsoid parameters (a, b, e²) and must specify datum (WGS84 vs. PRS92).
Cartesian coordinates are datum-specific: same lat/lon on WGS84 and PRS92 yield different X, Y, Z.
Geodetic latitude (φ) = angle from equator to normal; NOT geocentric latitude.
Ellipsoidal height (h) can be obtained from GPS; orthometric height (H) = h − N (where N = geoid undulation).
Projected coordinates allow area/distance calculations; scale corrections and grid convergence apply.

Key Definitions

Term

Geodetic Coordinates

Example

Manila: φ ≈ 14.600°N, λ ≈ 120.982°E (WGS84)

Definition

Latitude (φ), longitude (λ), and ellipsoidal height (h); curvilinear; used in geographic/navigation contexts.

Term

Cartesian Coordinates (Geocentric)

Example

Manila: X ≈ −3 188 188 m, Y ≈ 5 305 735 m, Z ≈ 1 532 921 m (WGS84)

Definition

X, Y, Z measured from Earth's center; Z-axis toward North Pole, X-axis toward Greenwich meridian intersection with equator.

Term

Plane/Projected Coordinates

Example

PPCS Zone III (Philippines): Central meridian 122°E; false easting 500 000 m

Definition

Easting (E), Northing (N) on a 2D map projection (e.g., UTM, PPCS); computed from geodetic via projection formulas.

Term

Latitude (φ)

Example

Cebu: φ ≈ 10.316°N; Davao: φ ≈ 7.073°N

Definition

Angle from equatorial plane to ellipsoid normal; ranges −90° to +90°; positive North.

Term

Longitude (λ)

Example

Manila: λ ≈ 120.982°E; West/negative East: λ < 0

Definition

Angle from prime meridian to meridian through point; ranges −180° to +180°; positive East.

Diagrams To Know

Ellipsoid cross-section: equator, pole, semi-major axis (a), semi-minor axis (b).
Cartesian axes: X toward Greenwich, Y toward 90°E, Z toward North Pole.
Latitude/longitude grid on sphere vs. ellipsoid; true/magnetic north distinction.

Formulas

Formula

3-Parameter (Molodensky-Badekas Simple): X_new = X_old + ΔX; Y_new = Y_old + ΔY; Z_new = Z_old + ΔZ

Meaning

ΔX, ΔY, ΔZ = translation components (in metres); applied to Cartesian coordinates.

Watch Out

3-parameter ignores rotation and scale — only valid if datums have same orientation/scale; PRS92 ↔ WGS84 often uses this.

When To Use

Quick transformation between datums over small areas (< 100 km); sufficient for many regional surveys.

Formula

7-Parameter (Helmert): X_new = (1 + s) R X_old + ΔX; where R = rotation matrix, s = scale factor ppm

Meaning

s = scale (parts per million); R = 3×3 rotation matrix (rotations rx, ry, rz about axes); ΔX, ΔY, ΔZ = translations.

Watch Out

Scale factor given in ppm; multiply by 10⁻⁶ for dimensionless form; small rotations (arcsec) must be converted to radians (÷ 206265); order of rotation application matters.

When To Use

High-precision transformation over large areas (> 100 km); accounts for datum rotation and scale mismatch.

Formula

Scale Effect on Distance: ΔL = s × L × 10⁻⁶; where s = scale in ppm, L = baseline length (m)

Meaning

Scale factor changes baseline length; +ppm lengthens, −ppm shortens.

Watch Out

ppm conversion: s_ppm ÷ 10⁶ = dimensionless factor; a 2.5 ppm scale on 10 km = 25 mm change.

When To Use

Estimating effect of 7-parameter transformation on measured baselines.

Formula

Rotation (small angle approximation): atan(arcsec) ≈ arcsec / 206265 radians

Meaning

Convert rotation angles from arcseconds to radians for matrix computation.

Watch Out

Only valid for rotations < 1°; use exact trig for larger angles; 1 arcsec = 1/3600 degree ≈ 4.85 × 10⁻⁶ radians.

When To Use

7-parameter transformations; typical datums rotate by 1–5 arcsec.

Common Values

Value

−127.6 m

Symbol

ΔX

Quantity

PRS92 ↔ WGS84 ΔX (3-parameter)

Value

−67.2 m

Symbol

ΔY

Quantity

PRS92 ↔ WGS84 ΔY (3-parameter)

Value

−47.0 m

Symbol

ΔZ

Quantity

PRS92 ↔ WGS84 ΔZ (3-parameter)

Value

206265

Symbol

—

Quantity

Arcseconds per radian

Value

1 ppm = 10⁻⁶

Symbol

—

Quantity

Scale ppm to dimensionless factor

Section Title

Datum Transformation (Helmert Methods)

Important Facts

Published datum shifts differ by institution (NGA, IGRF); use official Philippine government sources (BIR/NAMRIA) for PRS92 ↔ WGS84.
PRS92 ↔ WGS84: typical 3-parameter shift is ΔX ≈ −127.6 m, ΔY ≈ −67.2 m, ΔZ ≈ −47.0 m.
Rotations in 7-parameter transform are typically very small (< 5 arcsec); convert to radians before applying.
Scale factor s in ppm: a +2.5 ppm scale increases all distances by 2.5 millimetres per kilometre.
3-parameter sufficient if regions/epochs have same orientation; 7-parameter needed if orientation differs.
Datum shift is cumulative — transforming A → B → C ≠ directly A → C; use direct published parameters.
Reverse transformation (B → A) uses negative parameters (−ΔX, −ΔY, −ΔZ, −rx, −ry, −rz, −s).
Transformation accuracy: 3-param typically ±1–2 m; 7-param ±0.1–0.5 m (depends on number/quality of control points).

Key Definitions

Term

Helmert Transformation

Example

PRS92 ↔ WGS84; Luzon Datum ↔ WGS84

Definition

General 7-parameter rigid-body transformation (3 translations, 3 rotations, 1 scale) mapping coordinates between two datums.

Term

3-Parameter Transformation

Example

Often sufficient for local surveys; faster, fewer parameters to publish.

Definition

Simplified Helmert assuming no rotation/scale; only translation (ΔX, ΔY, ΔZ) applied.

Term

7-Parameter Transformation

Example

WGS84 to national datums over continental scales.

Definition

Full Helmert: 3 translations + 3 rotations (rx, ry, rz) + 1 scale factor; high precision for large regions.

Term

Scale Factor (s)

Example

s = +2.5 ppm means a 1 000 m baseline becomes 1 000.0025 m.

Definition

Dimensionless multiplier (or in ppm units) applied to Cartesian coordinates; accounts for ellipsoid/epoch scale differences.

Term

Rotation Matrix (R)

Example

Used in 7-parameter transformation; very small angles (arcsec) for nearby datums.

Definition

3×3 orthogonal matrix encoding three successive rotations (rx, ry, rz about x-, y-, z-axes).

Diagrams To Know

3D Cartesian axes with datum ellipsoids: local ellipsoid shifted from geocentric.
Rotation angles (rx, ry, rz) about cardinal axes; small-angle visualization.
Scale bar showing baseline before/after scale transformation.
Flowchart: raw GPS (WGS84) → 7-param Helmert → PRS92 adjusted coordinates.

Reactions Or Equations

Note

Positive s = lengthening; negative s = shortening. Always apply scale before rotation for stability.

Equation

ΔL = s × L × 10⁻⁶ (scale effect on baseline)

Conditions

s in ppm; L in metres; result in same units as L

Note

206265 = 1 radian in arcseconds (180 × 3600 / π). Use for rx, ry, rz in Helmert 7-param.

Equation

Arcsec to radians: θ_rad = θ_arcsec / 206265

Conditions

θ_arcsec = rotation in arcseconds

Common Values

Value

6 378 206.4 m

Symbol

Quantity

PRS92 Ellipsoid (Clarke 1866) Semi-major axis

Value

1/294.9787

Symbol

Quantity

PRS92 Ellipsoid Flattening

Value

13°46'22.178'' N

Symbol

Quantity

Balanacan Latitude

Value

121°47'08.850'' E

Symbol

Quantity

Balanacan Longitude

Section Title

Philippine Datums: PRS92 & Luzon Datum 1911

Important Facts

PRS92 is the official datum for the Philippines; all government surveys and land titles reference it.
Established in 1992 via satellite observations; refined from Luzon Datum 1911 (classical triangulation).
Clarke 1866 ellipsoid: semi-major axis 6 378 206.4 m; differs from WGS84 (6 378 137 m) by ~69 m.
Balanacan origin: fixed at φ ≈ 13°46'22.178'' N, λ ≈ 121°47'08.850'' E in WGS84 terms.
GPS/GNSS outputs WGS84; must transform via 3- or 7-parameter Helmert to align with PRS92 monuments.
Historical coordinates on Luzon Datum 1911 and PRS92 differ due to modern satellite adjustments; direct comparison requires transformation.
Philippine laws (RA 4374, RA 8560, PD 1529) mandate use of PRS92 for official surveys and land registration.
PPCS (Philippine Plane Coordinate System) is projected onto PRS92; different zones use different central meridians on Clarke 1866.

Key Definitions

Term

PRS92 (Philippine Reference System 1992)

Example

Official datum for PPCS, land titles, cadastral surveys; established by NAMRIA via satellite & classical obs.

Definition

National geodetic datum; based on Clarke 1866 ellipsoid; origin at Balanacan, Marinduque; valid across Philippines.

Term

Luzon Datum 1911

Example

Legacy surveys, old maps; coordinate values differ slightly from PRS92 due to adjustment methods.

Definition

Historical local datum; same ellipsoid (Clarke 1866) and origin (Balanacan) as PRS92; predates modern satellite era.

Term

Balanacan (Origin Station)

Example

φ ≈ 13°46'22.178'' N, λ ≈ 121°47'08.850'' E (WGS84 equivalent); fundamental to all PRS92 coordinates.

Definition

Control point on Marinduque Island; serves as origin for both PRS92 and Luzon Datum 1911; fixed position.

Term

Clarke 1866 Ellipsoid

Example

Older system; Australian Colonial Observatory choice; now superseded by WGS84 globally but retained in Philippines for continuity.

Definition

Reference ellipsoid used by PRS92 and Luzon Datum; a = 6 378 206.4 m, f = 1/294.9787; slightly different from WGS84.

Diagrams To Know

Map of Philippines showing Balanacan (Marinduque) origin station location.
Ellipsoid comparison: Clarke 1866 vs. WGS84 (flattening, semi-major axis).
Timeline: Luzon Datum 1911 → PRS92 (1992); adjustment refinement.
PPCS zones (I–IV) with central meridians; all referenced to PRS92/Clarke 1866.

Formulas

Formula

UTM False Easting: FE = 500 000 m (500 km); False Northing: FN = 0 m (equator) or 10 000 000 m (south)

Meaning

All UTM eastings shifted by 500 km west to avoid negatives; northern hemisphere uses 0, southern uses 10 Mm north.

Watch Out

Do NOT confuse false easting with actual easting; subtract FE from reported easting to get true grid easting from central meridian.

When To Use

Interpreting UTM coordinates; easting always 200 000–800 000 m range; northing 0–10 000 000 m.

Formula

UTM Zone Number: Zone = ⌊(λ + 180) / 6⌋ + 1; Central Meridian λ_cm = 6(Zone − 1) − 180

Meaning

λ = longitude (−180 to +180); each 6° band is one zone; central meridian is mid-band.

Watch Out

Zone formula assumes longitude in −180 to +180 range; UTM zones are 6° wide; do NOT confuse with MGRS.

When To Use

Determining which UTM zone a point falls into; Philippines spans Zones 50, 51, 52.

Formula

PPCS Central Meridians: Zone I = 120°E, Zone II = 121°E, Zone III = 122°E, Zone IV = 123°E

Meaning

Each PPCS zone uses Clarke 1866 ellipsoid; false easting 500 000 m; false northing varies by convention.

Watch Out

PPCS is NOT UTM; PPCS uses PRS92 (Clarke 1866), UTM uses WGS84; different ellipsoids yield slightly different coordinates.

When To Use

All Philippine cadastral surveys, land titles, government mapping; narrower zones than UTM (1° vs 6°) for better scale factor.

Common Values

Value

0.9996

Symbol

k₀

Quantity

UTM Scale Factor (central meridian)

Value

500 000 m

Symbol

Quantity

UTM False Easting

Value

0 m

Symbol

Quantity

UTM False Northing (North)

Value

6°

Symbol

—

Quantity

UTM Zone Width

Value

1°

Symbol

—

Quantity

PPCS Zone Width

Value

120° E

Symbol

—

Quantity

PPCS Zone I Central Meridian

Value

122° E

Symbol

—

Quantity

PPCS Zone III Central Meridian

Section Title

PPCS & UTM Projections

Important Facts

UTM: global, 60 zones (6° wide each); Philippine zones: 50, 51, 52.
PPCS: Philippine-specific, 4 zones (1° wide each); much better scale accuracy than UTM over Philippines.
UTM scale factor k₀ = 0.9996; all UTM distances scaled by this on central meridian.
PPCS uses scale factor k₀ = 1.0 (sometimes); varies by zone and standard.
False easting UTM: 500 000 m; false northing: 0 (NH) or 10 000 000 m (SH).
Philippines entirely in Northern Hemisphere; northing 0–10 000 000 m range.
Grid convergence: zero on central meridian, ±3° at zone edges; essential for bearing conversions.
Projection distortion: area shrinks near edges (k < 1 near edges for conformal projections); minimize by using PPCS for regional work.

Key Definitions

Term

UTM (Universal Transverse Mercator)

Example

Philippines Zone 50 (120°–126°E), 51 (126°–132°E); Luzon spans zones 50–51.

Definition

Global projection system; 60 zones, each 6° wide; scale factor k₀ = 0.9996 on central meridian; false easting 500 km, false northing varies.

Term

PPCS (Philippine Plane Coordinate System)

Example

Zone III (central meridian 122°E) covers central Philippines.

Definition

National projection; 4 zones, each 1° wide; based on PRS92/Clarke 1866; narrower bands → better scale accuracy than UTM.

Term

Central Meridian

Example

UTM Zone 51: cm = 123°E; PPCS Zone III: cm = 122°E.

Definition

Meridian of zero convergence in projection; scale factor k₀ = 0.9996 (UTM) or 1.0 (PPCS); best accuracy near central meridian.

Term

Scale Factor (k)

Example

At edge of 6° UTM zone: k ≈ 1.0004 (0.04% stretch).

Definition

Ratio of grid distance to ground distance; k = 1 on central meridian; k > 1 away from it (expansion); k₀ = 0.9996 (UTM).

Term

Grid Convergence (γ)

Example

Important for compass/bearing conversions; can reach ±3° at zone edges.

Definition

Angle between true north (meridian) and grid north (projection y-axis); γ = 0 on central meridian, increases away from it.

Diagrams To Know

UTM zone map: 60 zones, 6° bands globally; Philippine zones 50, 51, 52 highlighted.
PPCS zone map: Zones I–IV with central meridians (120°, 121°, 122°, 123° E).
Scale factor graph: k vs. distance from central meridian; k₀ = 0.9996 (UTM).
Grid convergence: angle γ between true north and grid north; function of latitude and distance from cm.

Section Title

Philippine Laws & Standards

Important Facts

RA 4374 mandates PRS92 for all cadastral and land surveys in Philippines.
PRS92 is legal standard; WGS84 coordinates from GPS must be transformed to PRS92 for official use.
RA 8560 establishes geodetic engineer licensure; PRC exam covers datums, projections, surveys, laws.
NAMRIA (PD 1529) maintains official geodetic control, geoid models, transformation parameters.
All government surveys, titles, and engineering projects must reference PRS92.
CA 141 (Public Land Act) implies use of official datum (PRS92) for public land descriptions.
Transformation parameters (PRS92 ↔ WGS84) published by NAMRIA; use official values for legal compliance.
Violations of datum/projection standards can invalidate survey or land title.

Key Definitions

Term

RA 4374 (Cadastral Law, as amended)

Example

All land titles in Registry of Deeds must reference PRS92 coordinates (or PPCS projections thereof).

Definition

Philippines law mandating use of PRS92 for all cadastral surveys and land registration; establishes PRS92 as official datum.

Term

RA 8560 (Geodetic Engineering Act of 1998)

Example

Geodetic engineers must comply with RA 8560 to perform cadastral, engineering, hydrographic surveys.

Definition

Regulates practice of geodetic engineering; establishes PRC licensure exam; requires professional standards for surveys.

Term

PD 1529 (Creation of NAMRIA, 1978)

Example

NAMRIA publishes official geodetic control point data, geoid models, transformation parameters.

Definition

Presidential Decree establishing National Mapping and Resource Information Authority; custodian of PRS92 and geodetic control.

Term

CA 141 (Public Land Act, as amended)

Example

Original title descriptions; survey monuments on public lands.

Definition

Core Philippine law governing public lands; coordinates and datums for land surveys must comply with cadastral law (RA 4374).

Diagrams To Know

Timeline: CA 141 (public lands) → PD 1529 (NAMRIA) → RA 4374 (cadastral/PRS92) → RA 8560 (geodetic profession).
Organizational chart: NAMRIA role as custodian of PRS92, control networks, geoid.

Section Title

Worked Examples & Board-Style Problems

Important Facts

Example 1 (3-param datum shift): Point with local Cartesian coords (X, Y, Z) + official shift (ΔX, ΔY, ΔZ) → WGS84 coords.
Example 2 (GNSS datum identification): GPS → WGS84; monuments on PRS92 → must transform.
Example 3 (scale factor in 7-param): s (ppm) on a baseline L (m) → ΔL = s × L × 10⁻⁶.
Example 4 (coordinate conversion): lat/lon/height ↔ Cartesian via ellipsoid N and e².
Example 5 (UTM zone determination): given longitude, compute zone number; find central meridian.
Example 6 (PPCS vs. UTM): same point → different easting/northing on PPCS (Zone III) vs. UTM (Zone 51).
Example 7 (grid convergence): angle between true north and grid north as function of position.
Example 8 (datum transformation verification): apply 3-param, verify result matches published tables.

Must Remember

A **datum** = ellipsoid + origin + orientation. **WGS84** is geocentric (Earth's center), valid globally, used by GPS. **PRS92** is local (origin at Balanacan, Marinduque), based on Clarke 1866, legal standard for Philippine surveys.
**3-parameter Helmert** (ΔX, ΔY, ΔZ) is quick datum shift; valid for small regions. **7-parameter** adds rotations + scale for high precision over large areas. PRS92 ↔ WGS84 typical shift: ΔX ≈ −127.6 m, ΔY ≈ −67.2 m, ΔZ ≈ −47.0 m.
**Clarke 1866** ellipsoid (a = 6,378,206.4 m, f = 1/294.9787) used by both **PRS92 and Luzon Datum 1911**. **WGS84** ellipsoid differs (a = 6,378,137 m); never mix ellipsoids in conversions without transforming.
**Geodetic coordinates** (φ, λ, h) are most intuitive. **Cartesian (X, Y, Z)** needed for transformations. **Projected (E, N)** used for mapping and area calculations. Same point → different numbers on each system.
**UTM** (60 zones, 6° wide, WGS84 datum, k₀ = 0.9996) is global standard. **PPCS** (4 zones, 1° wide, PRS92 datum) is Philippine legal system. Use PPCS for all Philippine cadastral work (RA 4374).
**Scale factor** in ppm: s (ppm) × L (metres) × 10⁻⁶ = change in baseline length. Example: +2.5 ppm on 10 km = 25 mm lengthening.
**Grid convergence (γ)** = angle between true north and grid north; zero on central meridian, ±3° at zone edges. Critical for bearing/azimuth conversions in surveys.
**GPS outputs WGS84**; Philippine monuments are on **PRS92**. Must apply 3- or 7-parameter Helmert transformation before comparing or adjusting GPS to control networks.
**Philippine Laws**: **RA 4374** mandates PRS92; **RA 8560** establishes geodetic licensing; **PD 1529** created NAMRIA (custodian of PRS92); **CA 141** implies compliance with official datum.
**Ellipsoidal height (h)** from GPS ≠ **orthometric height (H)**. Relation: H = h − N (where N = geoid undulation, ~80–90 m in Philippines). GNSS gives h; geoid model needed for MSL-referenced H.

Last Minute Tips

**Always specify datum and ellipsoid.** A coordinate without datum is meaningless. If converting WGS84 → PRS92, always use published shift parameters (ΔX, ΔY, ΔZ or 7-param); do NOT guess.
**Scale factor in ppm: multiply by 10⁻⁶.** A +2.5 ppm scale does NOT mean 2.5%; it means +(2.5 × 10⁻⁶) × baseline, or 2.5 mm per km. Forgetting the 10⁻⁶ will cost marks.
**3-param vs. 7-param: know when to use which.** Exam may ask why 3-param is insufficient (answer: different orientation/epoch requires rotation/scale). Conversely, 3-param is simpler/faster if orientation is same.
**PPCS ≠ UTM for Philippines.** PPCS uses PRS92/Clarke 1866; UTM uses WGS84. Same point yields *different* easting/northing. Exam may ask to convert between them — know PPCS is legal standard.
**Recognize Balanacan.** NAMRIA's control network origin is Balanacan, Marinduque (PRS92 and Luzon Datum 1911). If exam mentions 'Philippine datum origin' or 'monumento,' it is likely Balanacan.

Comparison Tables

Rows

Values

Geocentric (global)
Local (regional)
Local (regional)

Property

Type

Values

WGS84 (a = 6378137 m)
Clarke 1866 (a = 6378206.4 m)
Clarke 1866 (a = 6378206.4 m)

Property

Ellipsoid

Values

Earth's mass center
Balanacan, Marinduque
Balanacan, Marinduque

Property

Origin

Values

Reference (zero shift)
ΔX ≈ −127.6 m, ΔY ≈ −67.2 m, ΔZ ≈ −47.0 m
~Similar to PRS92, minor differences

Property

Datum Shift from WGS84

Values

Satellite-based (GPS era)
Satellite (1992 adjustment)
Classical triangulation (1911)

Property

Source

Values

Reference, GPS output
Official datum (RA 4374)
Legacy/historical

Property

Legal Use (Philippines)

Values

GNSS, global positioning
All cadastral & engineering surveys in PH
Old surveys, legacy monuments

Property

Typical Application

Columns

Attribute
WGS84
PRS92
Luzon Datum 1911

Table Title

WGS84 vs. PRS92 vs. Luzon Datum 1911

Rows

Values

ΔX, ΔY, ΔZ (translations only)
ΔX, ΔY, ΔZ, rx, ry, rz, s (translations + rotations + scale)

Property

Parameters

Values

X_new = X_old + ΔX (similarly Y, Z)
X_new = (1+s)R·X_old + ΔX

Property

Formula

Values

±1–2 m (regional areas)
±0.1–0.5 m (high precision, large areas)

Property

Typical Accuracy

Values

Small regions, same orientation/epoch
Large regions, different epochs, rotation/scale needed

Property

When to Use

Values

Simple addition of shifts
Matrix multiplication; need arcsec → radian conversion

Property

Complexity

Values

PRS92 ↔ WGS84 quick conversions
Continental/global datum links (ITRF)

Property

Common Application

Columns

Aspect
3-Parameter
7-Parameter

Table Title

3-Parameter vs. 7-Parameter Helmert Transformation

Rows

Values

60 zones (6° wide each)
4 zones (1° wide each, Philippines-specific)

Property

Zone Count

Values

WGS84
PRS92 (Clarke 1866)

Property

Datum

Values

Zone 50: 123°E; Zone 51: 129°E; Zone 52: 135°E
Zone I: 120°E; Zone II: 121°E; Zone III: 122°E; Zone IV: 123°E

Property

Central Meridians (PH)

Values

k₀ = 0.9996 (all zones)
k₀ = 1.0 or ~0.99996 (varies by standard)

Property

Scale Factor

Values

500,000 m
500,000 m (sometimes 200,000 m)

Property

False Easting

Values

0 m (NH), 10,000,000 m (SH)
0 m (sometimes 1,000,000 m)

Property

False Northing

Values

Global reference, international maps
Philippine cadastral surveys, land titles, government projects (RA 4374)

Property

Best For

Values

±0.04% at zone edges (6° bands wide)
~±0.01% at zone edges (1° bands narrower)

Property

Scale Distortion (PH)

Columns

Feature
UTM
PPCS

Table Title

UTM vs. PPCS

Rows

Values

φ (lat), λ (lon), h (height)
φ: −90° to +90°; λ: −180° to +180°; h: metres
Navigation, GNSS output, simple reference
Yes (WGS84, PRS92, etc.)

Property

Geodetic (Geographic)

Values

X, Y, Z from Earth's center
X, Y, Z: ~±6.4 million m
Satellite positioning, datum transformations
Yes (different X,Y,Z per datum)

Property

Cartesian (Geocentric)

Values

E (Easting), N (Northing) on map
E, N: depends on false easting/northing; m
Mapping, surveying, area calculations, legal descriptions
Yes (via projection equations and datum)

Property

Projected (Grid/Plane)

Columns

Type
Variables
Range/Units
Use Case
Datum Dependent

Table Title

Geodetic vs. Cartesian vs. Projected Coordinates

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Figure of the Earth and the Reference Ellipsoid

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Geodetic and Cartesian Coordinates

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