Geodetic and Cartesian Coordinates - Cheat Sheet
Your last-minute reference guide for converting between geodetic (φ, λ, h) and Cartesian (X, Y, Z) coordinates. Master the formulas, avoid the pitfalls, and ace the PRC exam.
Core Reference Ellipsoids (WGS84 & PRS92)
Formulas
N = a / √(1 − e²sin²φ)
Meaning
N = prime vertical radius; a = semi-major axis; e = eccentricity; φ = latitude
Watch out
N changes with latitude — must recalculate if φ updates; do NOT use constant N
When to use
Every forward conversion and in latitude iteration formulas
Common values
Value
6,378,137 m
Symbol
a
Quantity
WGS84 semi-major axis
Value
6,356,752.3 m
Symbol
b
Quantity
WGS84 semi-minor axis
Value
1 / 298.257223563
Symbol
f
Quantity
WGS84 flattening
Value
0.00669438
Symbol
e²
Quantity
WGS84 eccentricity squared
Important facts
- WGS84: a = 6,378,137 m; b = 6,356,752.3 m; f = 1/298.257223563; e² = 0.00669438
- PRS92 ellipsoid parameters are identical to WGS84 (1992 adjustment)
- Eccentricity e² = 2f − f² relates flattening to ellipsoid shape
- Cartesian origin is Earth's center of mass; axes: X through Greenwich meridian, Z through north pole
- Geodetic coordinates tied to ellipsoid; NOT orthometric (levelled) heights
Key definitions
WGS84
Example
All GNSS receivers output positions in WGS84 Cartesian or geodetic
World Geodetic System 1984; global datum; a = 6,378,137 m; e² = 0.00669438; used by GNSS/GPS
PRS92
Example
Land titles and RA 4374/RA 8560 property descriptions reference PRS92
Philippine Reference System 1992; national datum for PH; based on WGS84; used in cadastral surveys
Ellipsoidal height (h)
Example
GPS gives h; geoid separation N converts to orthometric height: H = h − N
Vertical distance from ellipsoid surface to point; NOT orthometric height; always positive above ellipsoid
Prime vertical (N)
Example
At φ = 0°: N ≈ a; at poles: N = a/(1 − e²)
Radius of curvature in meridian plane at latitude φ; varies from equator to poles
Diagrams to know
- Ellipsoid cross-section (a, b, f, e²)
- Geocentric frame: X, Y, Z axes and Greenwich meridian
- Prime vertical plane showing N, φ, h relationship
Forward Conversion: Geodetic (φ, λ, h) → Cartesian (X, Y, Z)
Formulas
X = (N + h)·cosφ·cosλ
Meaning
Cartesian X-coordinate; λ measured from Greenwich meridian
Watch out
cosφ multiplied TWICE — once for radius reduction, once for East component. Forgetting second cosφ is exam suicide
When to use
Always; eastern hemisphere gives positive X, western negative
Y = (N + h)·cosφ·sinλ
Meaning
Cartesian Y-coordinate; λ in radians or degrees converted properly
Watch out
Same structure as X: (N + h)·cosφ MUST multiply both sinλ and cosλ. sinλ is negative for western hemisphere
When to use
Always; eastern hemisphere (λ > 0) gives positive Y
Z = [N(1 − e²) + h]·sinφ
Meaning
Vertical component; (1 − e²) accounts for ellipsoid flattening; NO second h inside bracket
Watch out
CRITICAL: (1 − e²) MUST be there. Dropping it turns ellipsoid into sphere — instant fail. h is OUTSIDE the bracket
When to use
Always; northern hemisphere positive, southern negative
N = a / √(1 − e²sin²φ)
Meaning
Prime vertical radius at latitude φ
Watch out
sin²φ inside (no cosφ), and 1 − NOT 1 + . φ must be in radians if using sin/cos directly
When to use
Calculate FIRST before X, Y, Z
Important facts
- Three equations are COUPLED: all depend on N(φ)
- Height h is ellipsoidal, measured perpendicular to ellipsoid surface
- At h = 0 (ellipsoid surface): X² + Y² + Z² = (N·cosφ)² + [N(1 − e²)·sinφ]²
- Longitude λ ranges from −180° to +180° (or 0° to 360°); convert to radians for trig functions
- Formula valid for both hemispheres if signs are handled correctly
Key definitions
Forward conversion
Example
GNSS receiver outputs (X, Y, Z); surveyor converts to (φ, λ, h) for map plotting (PRS92 or PPCS)
Transform from geodetic (φ, λ, h) to geocentric Cartesian (X, Y, Z) using ellipsoid parameters
Prime-vertical radius N
Example
At φ = 14° N (Manila): N ≈ 6,379,387 m (used in Example 1)
Radius of curvature perpendicular to meridian; maximum at equator, minimum at poles
Diagrams to know
- Forward conversion flowchart: φ, λ, h → N → X, Y, Z
- Prime vertical plane: angle φ, radius N, height h
- Cartesian frame showing X, Y (equatorial), Z (polar)
Reactions or equations
X² + Y² = (N + h)²cos²φ = p²
Note
This identity is used in inverse conversion to recover h and φ
Conditions
p = cylindrical radius (distance from Z-axis)
Inverse Conversion: Cartesian (X, Y, Z) → Geodetic (φ, λ, h)
Formulas
λ = atan2(Y, X)
Meaning
Longitude from Cartesian coordinates; atan2 resolves quadrant automatically
Watch out
atan2(Y, X) — order matters (Y first, X second). arctan(Y/X) alone gives only ±90° range; misses quadrants III & IV
When to use
ALWAYS use atan2, NOT tan⁻¹(Y/X). Direct calculation; no iteration needed
p = √(X² + Y²)
Meaning
Cylindrical radius (distance from Z-axis); needed for latitude iteration
Watch out
p is NOT the ellipsoidal radius. p is strictly the horizontal distance; always positive
When to use
First step in inverse conversion; precalculate before φ iteration
φ₀ = atan(Z / [p(1 − e²)])
Meaning
Initial latitude estimate; starting point for iteration
Watch out
Denominator is p(1 − e²), NOT p. Missing (1 − e²) gives wrong convergence path
When to use
Compute once at start of inverse; only valid as approximation
φᵢ₊₁ = atan{ Z / p · [1 − e²N(φᵢ) / (N(φᵢ) + h)]⁻¹ }
Meaning
Iterative latitude update; N and h depend on previous φᵢ
Watch out
h must be updated each iteration: h = p/cosφ − N. Forgetting to update h stops convergence dead
When to use
Each iteration step; continue until |φᵢ₊₁ − φᵢ| < 10⁻¹² rad
h = p / cosφ − N
Meaning
Ellipsoidal height from corrected latitude; N must match final φ
Watch out
h = p/cosφ − N, NOT p − N. Division by cosφ is critical; cosφ → 1 at equator, → 0 at poles (pole singularity)
When to use
After φ converges; recalculate N(φ_final) then compute h
Important facts
- Longitude λ has no iteration — direct from atan2(Y, X); quadrant resolved automatically
- Latitude and height are COUPLED through N(φ); cannot solve independently
- Typical convergence: 2–3 iterations for mm accuracy; 4–5 for sub-mm
- At poles (p ≈ 0), singularity: use alternative algorithm (e.g., Bowring) or recognize pole explicitly
- Height h = 0 on ellipsoid surface; above = positive, below = negative (rare for Earth surface)
Key definitions
Inverse conversion
Example
GNSS solution (X, Y, Z) → convert to (φ, λ, h) for cadastral entry (RA 4374 property description)
Transform from geocentric Cartesian (X, Y, Z) to geodetic (φ, λ, h); λ is direct, φ and h require iteration or closed-form
Iteration (inverse φ)
Example
φ₀ → φ₁ → φ₂ → φ₃; stop when change < 1e-12 rad
Successive refinement of latitude φ because N depends on φ; typically converges in 2–4 steps
Bowring's closed-form method
Example
Used in high-speed computations; beyond PRC licensure scope (but worth knowing exists)
Algebraic solution for φ, h without iteration; more complex formula but guaranteed single pass
Diagrams to know
- Inverse conversion iteration flowchart: X, Y, Z → λ, p → φ₀ → iterate φ → final h
- N(φ) curve: shape showing how prime vertical varies with latitude
- Convergence diagram: φ error vs iteration count
Reactions or equations
N(φ) = a / √(1 − e²sin²φ)
Note
If N is not updated, iteration diverges or stalls
Conditions
Must be recalculated each iteration with updated φ
Convergence criterion: |φᵢ₊₁ − φᵢ| < 1 × 10⁻¹² rad
Note
Loose tolerance (e.g., 1e-6) risks mm-level errors in resulting coordinates
Conditions
Typical geodetic precision; stricter for high-accuracy work
Common Pitfalls & Exam Traps
Important facts
- TRAP 1: Forgetting (1 − e²) in Z formula → converts ellipsoid to sphere → exam failure
- TRAP 2: Using arctan(Y/X) instead of atan2(Y, X) → longitude off by 180° in wrong hemisphere
- TRAP 3: Not updating N(φ) during latitude iteration → convergence fails or diverges
- TRAP 4: Forgetting to divide h = p/cosφ − N by cosφ → height completely wrong
- TRAP 5: Mixing degrees and radians without conversion → all trig functions fail
- TRAP 6: Assuming h = 0 for all points (sea-level datum) → vertical errors of meters to hundreds of meters
- TRAP 7: Treating φ, λ as if they can be added/subtracted like Cartesian coordinates → nonsensical results
- TRAP 8: Not recognizing pole singularity → h calculation breaks at |φ| = 90°
Key definitions
Ellipsoidal height h vs. orthometric height H
Example
Manila: h ≈ 50 m (GPS), N_geoid ≈ −0.6 m → H ≈ 50.6 m (levelled height)
h = distance perpendicular to ellipsoid; H = elevation above geoid (what levelling measures); H = h − N_geoid
Quadrant resolution in longitude
Example
X < 0, Y > 0 → quadrant II → λ in (90°, 180°); arctan would miss this
atan2(Y, X) handles all four quadrants; arctan(Y/X) alone only gives [−90°, +90°]
Worked Board Examples (Exam-Style)
Important facts
- EXAMPLE 1 SETUP: φ = 14°00'00" N, λ = 121°00'00" E, h = 50 m (WGS84). Find X, Y, Z.
- EXAMPLE 1 ANSWER: X ≈ −3,188,055 m; Y ≈ 5,305,814 m; Z ≈ 1,532,994 m
- EXAMPLE 2 SETUP: Given X = −2,500,000 m, Y = −4,000,000 m. Find λ and resolve quadrant.
- EXAMPLE 2 ANSWER: atan2(−4e6, −2.5e6) → quadrant III → λ ≈ −238° ≡ 122° W (or −58° from reference)
- EXAMPLE 3 SETUP: X = 6,000,000 m, Y = 3,000,000 m, Z = 500,000 m. Iterate for φ, h to 1 mm accuracy.
- EXAMPLE 3 ANSWER: Requires 3–4 iterations; final φ ≈ 4.77°, h ≈ 1,345 m (verify convergence)
Philippine Context: RA 4374, RA 8560, PD 1529, CA 141
Important facts
- PH cadastral standard: PRS92 (geodetic φ, λ, h) or PPCS/UTM (projected Easting, Northing, Zone)
- Geodetic engineer MUST be able to convert GNSS output (WGS84 X,Y,Z) → PRS92 (φ, λ, h) → PPCS/UTM (E, N) for property survey
- RA 4374 professional seal certifies that coordinates are properly converted and referenced to approved datum
- RA 8560 compliance: if coordinates drift > specified tolerance (typically ±0.05 m for urban), survey must be re-executed
- Common exam scenario: 'GNSS recorded X, Y, Z in WGS84; convert to PRS92 geodetic for Title VI entry' — tests forward/inverse mastery
Key definitions
RA 4374 (Geodetic Engineer Licensure)
Example
PRC exam tests forward/inverse conversions as core competency for licensure
Profession regulation act; defines licensed geodetic engineer duties including coordinate conversions, cadastral surveys, geospatial data certification
RA 8560 (Real Property Registration Act of 1988)
Example
If surveyor delivers (X, Y, Z) from GNSS, must convert to PRS92 (φ, λ) or PPCS (UTM Easting, Northing) for title entry
Governs land titles and cadastral surveys; all property descriptions must use approved datums (PRS92, PPCS/UTM)
PD 1529 (Land Titles Decree)
Example
Property corners recorded in (φ, λ, h) on PRS92; geodetic engineer certifies accuracy under professional seal
Establishes cadastral boundary descriptions; references PRS92 datum and approved coordinate systems
CA 141 (Public Land Act)
Example
National grid system (PPCS) based on forward/inverse conversions from WGS84/PRS92
Defines public land administration; coordinates tied to approved geodetic datums for public surveys
Must remember
- 1. Z FORMULA MUST include (1 − e²): Z = [N(1 − e²) + h]sinφ. Dropping it is instant failure.
- 2. X and Y BOTH have cosφ multiplied: X = (N+h)cosφ cosλ, Y = (N+h)cosφ sinλ. Forgetting either is critical error.
- 3. Use atan2(Y, X) for longitude, NEVER arctan(Y/X). Quadrant resolution is automatic with atan2.
- 4. Update N(φ) EVERY iteration in inverse conversion: N = a/√(1 − e²sin²φ). Static N = divergence.
- 5. Height formula in inverse is h = p/cosφ − N (division by cosφ is essential, not optional).
- 6. Convergence in inverse: iterate until |φᵢ₊₁ − φᵢ| < 1e-12 rad; 2–4 passes typical; poles need special handling.
- 7. PRS92 parameters are IDENTICAL to WGS84; difference is reference frame realization. Convert GNSS (WGS84) → PRS92 by recognizing datum equivalence.
- 8. RA 8560 & PD 1529 require coordinates on PRS92 or PPCS/UTM for titles; surveyors MUST do conversion and seal certification.
- 9. Ellipsoidal height h ≠ orthometric/levelled height H; h is perpendicular to ellipsoid, H is perpendicular to geoid.
- 10. Board exams often combine: 'GNSS gives (X,Y,Z); convert to geodetic (φ,λ,h) and then to PPCS/UTM.' Master both directions seamlessly.
Last minute tips
- TIP 1: If you see a Z formula without (1 − e²), mark it wrong immediately. This factor distinguishes ellipsoid from sphere.
- TIP 2: Longitude from Cartesian? Use atan2. Every. Single. Time. No exceptions. Arctan will cost you points via quadrant errors.
- TIP 3: In inverse problems, always write out the iteration loop explicitly: compute p, λ, φ₀, then loop with N & h updates. Shows examiner you understand coupling.
- TIP 4: For Philippine property surveys: if answer is in WGS84 Cartesian, the examiner expects you to state 'must convert to PRS92 (φ,λ,h) or PPCS/UTM per RA 8560' to earn full marks.
- TIP 5: Pole singularity (p ≈ 0, φ ≈ ±90°) rarely appears in PRC exams, but if X ≈ Y ≈ 0 and |Z| ≈ 6,357,000, flag it and use explicit pole algorithm or Bowring method.
Comparison tables
Forward vs. Inverse Conversion: At a Glance
| Aspect | Forward (Geodetic → Cartesian) | Inverse (Cartesian → Geodetic) |
|---|---|---|
| Input | φ, λ, h | X, Y, Z |
| Output | X, Y, Z | φ, λ, h |
| Complexity | Direct: 3 equations, 1 pass | Longitude direct; φ, h iterative (2–4 passes) |
| Dependency on N | N computed once from input φ | N recomputed every iteration with updated φ |
| Singularities | None (valid 90°S to 90°N) | At poles: p → 0, h formula undefined; use closed-form |
| Exam frequency | 60% of inverse/forward questions | 40%; often combined in one problem |
WGS84 vs. PRS92: Parameters
| Parameter | WGS84 | PRS92 | Note |
|---|---|---|---|
| Semi-major axis (a) | 6,378,137 m | 6,378,137 m | Identical |
| Flattening (f) | 1/298.257223563 | 1/298.257223563 | Identical |
| Eccentricity (e²) | 0.00669438 | 0.00669438 | Identical |
| Use in Philippines | GNSS reference frame (global) | PH national datum (RA 8560, PD 1529) | PRS92 ≡ WGS84 (1992 realization) |
| Coordinate output | Usually X, Y, Z (geocentric) | Usually φ, λ, h or PPCS/UTM | Surveyors must convert for titles |
Height Types: Ellipsoidal vs. Orthometric
| Type | Ellipsoidal (h) | Orthometric (H) | Levelled (H_lev) |
|---|---|---|---|
| Reference surface | WGS84 ellipsoid | Geoid (mean sea level) | Levelling benchmarks |
| How measured | GNSS (X, Y, Z → h) | Levelling (benchmark datum) | Spirit or digital level |
| Relation to others | h (given by GNSS) | H = h − N_geoid | H ≈ H_lev (identical in PH) |
| Exam use | Input/output of coordinate conversions | Land title elevations, drainage design | BMs for levelling traverses |
Diagrams
Geodetic ↔ Cartesian Conversion Decision Tree
Decision flowchart for choosing forward or inverse conversion and step-by-step execution path.
Coordinate Systems: Relationships & Transformations
Comprehensive mind map showing relationships between geodetic and Cartesian coordinate systems, conversion pathways, and Philippine legal context.
Forward Conversion: Formula Application in Order
Linear flowchart showing exact step-by-step order for forward conversion calculations, with formula application sequence highlighted.