GELE Geodesy — Figure of the Earth and the Reference EllipsoidCheat Sheet
Figure of the Earth and the Reference Ellipsoid cheat sheet — the reference card you wish you had on exam day. Condensed from the full study notes, this is the high-yield core of Figure of the Earth and the Reference Ellipsoid for GELE Geodesy. Download, print, revise.
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 Figure of the Earth and the Reference Ellipsoid appears in position 1st 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.
Figure of the Earth and the Reference Ellipsoid - Cheat Sheet
Your last-minute revision companion for the ellipsoid parameters, radii of curvature, and coordinate transformations essential for geodetic engineering. Master these formulas and relationships in 30 minutes.
Sections
Formulas
Formula
f = (a − b) / a
Meaning
f = flattening; a = semi-major axis (equatorial radius); b = semi-minor axis (polar radius)
Watch Out
Flattening is VERY SMALL (~0.00335 for WGS84), NOT 0.335 or 0.298. Use full precision: f = 1/298.257223563
When To Use
When converting between a, b and flattening; always the first step in ellipsoid problems
Formula
b = a(1 − f)
Meaning
Semi-minor axis derived from semi-major axis and flattening
Watch Out
Do NOT write b = a − f × a without the parentheses; order of operations matters. Always compute 1 − f first
When To Use
Finding b when a and f are given; essential for all ellipsoid calculations
Formula
e² = 2f − f²
Meaning
First eccentricity squared; the flattening expansion form for numerical stability
Watch Out
This is NOT e² = (a² − b²) / a² in direct form — use the flattening form for accuracy
When To Use
Computing e² from flattening; preferred in programming and high-precision work
Formula
e² = (a² − b²) / a²
Meaning
First eccentricity squared; explicit geometric definition
Watch Out
This form is accurate but for high-precision work use e² = 2f − f² instead
When To Use
When a and b are both known and no flattening given; verification calculations
Formula
e'² = (a² − b²) / b² = e² / (1 − e²)
Meaning
Second eccentricity squared; related to the minor axis
Watch Out
e'² ≠ e²; they are related but distinct. Always check which one your formula needs
When To Use
Some coordinate conversion formulas use e'² instead of e²; know both forms
Common Values
Value
6,378,137 m
Symbol
a
Quantity
WGS84 semi-major axis
Value
6,356,752.31 m
Symbol
b
Quantity
WGS84 semi-minor axis
Value
1/298.257223563
Symbol
f
Quantity
WGS84 flattening (as fraction)
Value
0.00335281
Symbol
f
Quantity
WGS84 flattening (decimal)
Value
0.00669438
Symbol
e²
Quantity
WGS84 first eccentricity squared
Value
0.00673949
Symbol
e'²
Quantity
WGS84 second eccentricity squared
Value
6,378,206.4 m
Symbol
a
Quantity
Clarke 1866 semi-major axis
Value
6,356,583.8 m
Symbol
b
Quantity
Clarke 1866 semi-minor axis
Section Title
Ellipsoid Definition & Fundamental Parameters
Important Facts
- WGS84 (World Geodetic System 1984) is the GNSS ellipsoid and the standard for Philippine digital mapping (RA 8560)
- PRS92 (Philippine Reference System 1992) is based on WGS84 with the same ellipsoid parameters
- The difference a − b = 21,384.69 m for WGS84; this 21 km polar flattening is the source of all curvature variation
- Flattening f ≈ 1/298 means the Earth is compressed by only ~0.33% — nearly spherical but the 21 km difference is huge for surveys
- For WGS84: e² = 0.00669438 (NOT 0.0669 or 0.669); this small value is critical in all radius-of-curvature formulas
- The relationship e² = 2f − f² is numerically stable because f² is extremely small (f² ≈ 1.1 × 10⁻⁵)
Key Definitions
Term
Reference Ellipsoid
Example
WGS84 with a = 6,378,137 m and f = 1/298.257223563
Definition
A mathematical model of Earth's shape: an oblate ellipsoid of revolution defined by two parameters (usually a and f)
Term
Semi-major axis (a)
Example
WGS84: a = 6,378,137 m; PRS92: a = 6,378,137 m
Definition
The equatorial radius of the ellipsoid; Earth's maximum distance from centre
Term
Semi-minor axis (b)
Example
WGS84: b ≈ 6,356,752.31 m; difference a − b ≈ 21.4 km
Definition
The polar radius of the ellipsoid; Earth's distance from centre at the poles
Term
Flattening (f)
Example
WGS84: f = 1/298.257223563 ≈ 0.00335281 or 1:298
Definition
The fractional compression at the poles: f = (a − b) / a; always positive and very small
Term
Oblate Ellipsoid
Example
All modern reference ellipsoids including WGS84, PRS92, Clarke 1866
Definition
An ellipsoid flattened at the poles (not at the equator), matching Earth's actual shape
Term
First Eccentricity (e)
Example
WGS84: e² = 0.00669438; e ≈ 0.08182
Definition
A measure of the deviation of the ellipsoid from a sphere; e² = (a² − b²) / a²
Diagrams To Know
- Ellipsoid cross-section showing a, b, f, and the equatorial vs polar radii
- The oblate shape comparison: sphere (e = 0) vs Earth (e ≈ 0.082)
- Parameter relationships: a → b, a → f, f → e², and back
Formulas
Formula
N = a / √(1 − e² sin² φ)
Meaning
Prime vertical (or normal) radius of curvature at latitude φ; gives the E–W curvature radius
Watch Out
SIN φ MUST BE IN RADIANS or calculator in degree mode. At equator (φ = 0°): sin φ = 0, so N = a. At poles (φ = 90°): sin φ = 1, so N = a / √(1 − e²) = a / √(1 − 0.00669438) ≈ a / 0.99665 > a. N is ALWAYS ≥ a
When To Use
Every coordinate transformation, distance calculation, and elevation correction in geodesy
Formula
M = a(1 − e²) / (1 − e² sin² φ)^(3/2)
Meaning
Meridian radius of curvature at latitude φ; gives the N–S curvature radius
Watch Out
DO NOT confuse M and N. M has the (1 − e²) factor in the numerator; N does not. At equator: M = a(1 − e²) < a. At poles: M = a(1 − e²) = N
When To Use
Computing arc lengths along meridians; latitude-dependent distance reductions
Formula
R = √(M × N)
Meaning
Mean radius of curvature (Gaussian mean); geometric mean of M and N
Watch Out
This is NOT (M + N) / 2 (arithmetic mean); use the GEOMETRIC MEAN √(MN)
When To Use
Local sphere approximations for small areas; some projection formulas; spherical cap reductions
Formula
W = √(1 − e² sin² φ)
Meaning
Common auxiliary function; simplifies writing N and M
Watch Out
This is NOT eccentricity; it's a working variable. Always compute it first to avoid arithmetic errors in N and M
When To Use
Shorthand in long derivations; frequently appears in coordinate transformation code
Formula
1 − e² sin² φ = (1 − e²) / (1 − (1 − (1 − e²)) sin² φ)
Meaning
Algebraic identity relating φ-dependent terms
Watch Out
This is an identity, not a computing formula — use it only for algebra, not for numerical work
When To Use
Verifying derivations; converting between N/M formulas and geocentric radius
Common Values
Value
N = 6,378,137 m = a
Symbol
N₀
Quantity
WGS84 prime vertical at equator (φ = 0°)
Value
M = 6,335,439 m ≈ a(1 − e²)
Symbol
M₀
Quantity
WGS84 meridian at equator (φ = 0°)
Value
N ≈ 6,383,200 m
Symbol
N_manila
Quantity
WGS84 prime vertical at Manila (φ = 14.6°)
Value
M ≈ 6,355,000 m
Symbol
M_manila
Quantity
WGS84 meridian at Manila (φ = 14.6°)
Value
R ≈ 6,371,000 m
Symbol
R
Quantity
WGS84 mean radius at 45°
Section Title
Radii of Curvature — The Cornerstone of Geodetic Calculations
Important Facts
- N ≥ M everywhere on the ellipsoid; N = M only at the equator and poles
- At the equator (φ = 0°): N = a and M = a(1 − e²); so M ≈ 0.9967a, meaning the N–S curvature is tighter at the equator
- At the poles (φ = 90°): N = M = a / √(1 − e²) ≈ 1.0034a; the two radii converge
- The ratio N / M = (1 − e² sin² φ) / ((1 − e²)(1 − e² sin² φ)^(−1/2)) increases as φ → 0, peaking at the equator
- M is always smaller than N because the meridional curvature is tighter (the pole is closer); the 21 km polar flattening concentrates curvature north–south
- For WGS84: M/N ratio at equator is ≈0.9967; at poles M/N = 1; at 45° they differ by ~1%
Key Definitions
Term
Prime Vertical Radius (N)
Example
At Manila (φ = 14.6°): N ≈ 6,383,200 m; slightly larger than equatorial radius
Definition
The radius of curvature in the east–west (perpendicular to the meridian) direction; always ≥ semi-major axis a
Term
Meridian Radius (M)
Example
At Manila (φ = 14.6°): M ≈ 6,355,000 m; significantly smaller than N
Definition
The radius of curvature in the north–south (along the meridian) direction; always ≤ semi-major axis a
Term
Latitude (φ or lat)
Example
Manila: φ ≈ 14.6° N; Davao: φ ≈ 7.1° N
Definition
Angular distance north or south of the equator, measured along a meridian; ranges from −90° (South Pole) to +90° (North Pole)
Term
Mean Radius of Curvature (R)
Example
For WGS84 at φ = 45°: R ≈ 6,371,000 m (close to Earth's nominal 'mean' radius)
Definition
The geometric mean R = √(MN), used to approximate the ellipsoid as a sphere for small areas
Diagrams To Know
- Ellipsoid profile: show N and M as radii from a point on the ellipsoid surface
- Variation of M and N with latitude (curves from equator to pole)
- Comparison of N > M throughout, converging at poles
Common Values
Value
120° E
Symbol
λ₀
Quantity
PPCS Zone 1 (Luzon) central meridian
Value
123° E
Symbol
λ₀
Quantity
PPCS Zone 2 (Luzon extension) central meridian
Value
126° E
Symbol
λ₀
Quantity
PPCS Zone 3 (Visayas) central meridian
Value
129° E
Symbol
λ₀
Quantity
PPCS Zone 4 (Mindanao) central meridian
Value
132° E
Symbol
λ₀
Quantity
PPCS Zone 5 (Mindanao extension) central meridian
Value
0.99999 (five nines)
Symbol
k₀
Quantity
PPCS standard scale factor
Section Title
PRS92 & Philippine Context
Important Facts
- PRS92 uses the WGS84 ellipsoid (a = 6,378,137 m, f = 1/298.257223563) — the SAME ellipsoid, not a different one
- RA 8560 became effective 1 January 1995; all old Clark 1866-based coordinates must be transformed to PRS92
- PPCS has five zones for convenience; each is a UTM-like transverse Mercator projection with scale factor k₀ = 0.99999
- UTM uses k₀ = 0.9996; PPCS uses k₀ = 0.99999 — slightly different scale factors affect distance reductions
- PD 1529 (Subdivision of Lands Act) and CA 141 (Public Land Act) reference survey standards; must comply with PRS92
Key Definitions
Term
PRS92 (Philippine Reference System 1992)
Example
All survey work and mapping in the Philippines must reference PRS92 since 1995
Definition
The official geodetic reference system for the Philippines, mandated by RA 8560; based on WGS84 ellipsoid and datum
Term
Philippine Standard Coordinate System (PPCS)
Example
PPCS Zones 1–5 cover the entire Philippines; each zone has central meridian and false easting/northing
Definition
The projection system for Philippine mapping; uses PRS92 as geodetic reference with transverse Mercator (UTM-like) zones
Term
RA 8560 (Geocentric Datum of the Philippines Act)
Example
Compliance required for all cadastral surveys, title deeds, and public works projects
Definition
The Philippine law mandating use of WGS84/PRS92 for all government surveys, maps, and geographic data
Diagrams To Know
- Map of Philippines with five PPCS zones and their central meridians
- Relationship: WGS84 ellipsoid → PRS92 datum → PPCS projection → survey coordinates
Section Title
Worked Board Problems for Exam Preparation
Important Facts
- PROBLEM 1: Given WGS84 a = 6,378,137 m and f = 1/298.257223563, find b and e². SOLUTION: f = 0.00335281; b = 6,378,137 × (1 − 0.00335281) = 6,356,752.31 m; e² = 2(0.00335281) − (0.00335281)² = 0.00669438. CHECK: (6,378,137² − 6,356,752.31²) / 6,378,137² = 0.00669438 ✓
- PROBLEM 2: At Manila (φ = 14.6° = 0.25472 rad), find N on WGS84. SOLUTION: sin 14.6° = 0.25242; sin² 14.6° = 0.06372; 1 − 0.00669438 × 0.06372 = 0.99574; N = 6,378,137 / √0.99574 = 6,378,137 / 0.99786 = 6,395,149 m
- PROBLEM 3: At φ = 14.6°, find M on WGS84. SOLUTION: 1 − e² = 0.99330562; (1 − 0.00669438 × 0.06372)^1.5 = (0.99574)^1.5 = 0.99361; M = 6,378,137 × 0.99330562 / 0.99361 = 6,356,750 m (much less than N)
- PROBLEM 4: Find e'² for WGS84. SOLUTION: e'² = e² / (1 − e²) = 0.00669438 / 0.99330562 = 0.00673949 OR e'² = (a² − b²) / b² = (6,378,137² − 6,356,752.31²) / 6,356,752.31² = 0.00673949 ✓
- PROBLEM 5 (Clarke 1866): Given a = 6,378,206.4 m, b = 6,356,583.8 m, find f and e². SOLUTION: f = (6,378,206.4 − 6,356,583.8) / 6,378,206.4 = 21,622.6 / 6,378,206.4 = 0.00338814 = 1/295.3; e² = 1 − (6,356,583.8 / 6,378,206.4)² = 0.00676866
Must Remember
- WGS84 PARAMETERS: a = 6,378,137 m, f = 1/298.257223563, e² = 0.00669438. These WILL appear in exam questions.
- FLATTENING IS SMALL: f ≈ 0.00335 (or 1:298), NOT 0.298 or 0.335. The polar flattening is only 21.4 km on a 6,378 km radius.
- FORMULAS DISTINGUISH M AND N: N = a / √(1 − e² sin² φ) (East–West); M = a(1 − e²) / (1 − e² sin² φ)^(3/2) (North–South). DO NOT SWAP.
- N ≥ M ALWAYS: Prime vertical radius is always greater than or equal to meridian radius. At equator N > M; at poles N = M.
- ECCENTRICITY SQUARED: Formulas use e², NOT e. For WGS84: e² = 0.00669438; e ≈ 0.0818. Always compute e² first.
- SIN φ IN CORRECT MODE: φ must be in radians OR calculator in degree mode. A mistake here propagates through N and M.
- PRS92 IS BASED ON WGS84: PRS92 ellipsoid parameters are identical to WGS84. They are NOT different ellipsoids.
- RA 8560 COMPLIANCE: All Philippine surveys since 1995 must use PRS92; Clarke 1866 is obsolete for official work.
- AUXILIARY FUNCTION: W = √(1 − e² sin² φ) appears in both N and M. Compute W once, use twice to save arithmetic.
- SECOND ECCENTRICITY: e'² = e² / (1 − e²) = 0.00673949 for WGS84. Some formulas (especially older references) use e'² instead of e².
Last Minute Tips
- CALCULATOR SETUP: Before any calculation, ensure your calculator is in DEGREE mode for latitude (φ), unless you've explicitly converted to radians. One wrong mode = wrong N, M, everything.
- DOUBLE-CHECK SMALL NUMBERS: Flattening f ≈ 0.00335, e² ≈ 0.00669. If your answers are off by 100× (e.g., f = 0.335 instead of 0.00335), you've made a magnitude error. Always verify units and decimal places.
- N AND M ROUGH ESTIMATES: At any latitude in Philippines, expect N ≈ 6,380,000–6,395,000 m and M ≈ 6,355,000–6,360,000 m. If your N < 6,370,000 m or M > 6,370,000 m, something is wrong.
- RECOGNIZE QUESTION TYPES: If given a and f, use b = a(1 − f) and e² = 2f − f². If given a and b, use e² = (a² − b²) / a². If φ is involved, N and M are needed. Identify the type first.
- PROOF CHECK: In 2–3 minutes, verify that N × (1 − e²) ≈ M at any latitude. This is a quick sanity check. If the relationship breaks, recalculate.
Comparison Tables
Rows
Values
- East–West (perpendicular to meridian)
- North–South (along meridian)
Property
Direction
Values
- N = a / √(1 − e² sin² φ)
- M = a(1 − e²) / (1 − e² sin² φ)^(3/2)
Property
Formula
Values
- N = a = 6,378,137 m
- M = a(1 − e²) = 6,335,439 m
Property
At Equator (φ = 0°)
Values
- N = a / √(1 − e²)
- M = a / √(1 − e²) = N
Property
At Poles (φ = 90°)
Values
- N ≥ a always
- M ≤ a always
Property
Relative Size
Values
- N ≥ M everywhere; N > M at equator
- M ≤ N everywhere; M = N only at poles
Property
Comparison
Values
- Radius of osculating circle in E–W plane
- Radius of osculating circle in N–S plane
Property
Physical Meaning
Columns
- Property
- Prime Vertical N
- Meridian M
Table Title
Radius of Curvature: N vs M
Rows
Values
- 6,378,137 m
- 6,378,206.4 m
Property
Semi-major axis (a)
Values
- 6,356,752.31 m
- 6,356,583.8 m
Property
Semi-minor axis (b)
Values
- 1/298.257223563 = 0.00335281
- 1/295.30 = 0.00338814
Property
Flattening (f)
Values
- 0.00669438
- 0.00676866
Property
First Eccentricity (e²)
Values
- Current standard (RA 8560, PRS92)
- Obsolete (pre-1995)
Property
Philippine Status
Values
- YES — this is the GNSS ellipsoid
- NO — transformation required
Property
GNSS Compatible
Columns
- Parameter
- WGS84 / PRS92
- Clarke 1866
Table Title
Ellipsoid Parameters: WGS84 vs PRS92 vs Clarke 1866
Rows
Values
- b = a(1 − f)
- WGS84: b = 6,378,137 × 0.99664719 = 6,356,752.31 m
Property
Find b from a and f
Values
- e² = 2f − f² (preferred for precision)
- e² = 0.00669438 for WGS84
Property
Find e² from a and f
Values
- e² = (a² − b²) / a²
- Verification after finding b
Property
Find e² from a and b
Values
- N = a / √(1 − e² sin² φ)
- N at Manila (14.6°) ≈ 6,395,149 m
Property
Find N at a given latitude
Values
- M = a(1 − e²) / (1 − e² sin² φ)^(3/2)
- M at Manila (14.6°) ≈ 6,356,750 m
Property
Find M at a given latitude
Values
- R = √(MN)
- Approximation when curvature variation is negligible
Property
Find local sphere radius for small area
Columns
- Problem Type
- Use This Formula
- Example
Table Title
Common Formulas: When to Use Which
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