Advanced and Geodetic Surveying - Cheat Sheet
Your 30-minute exam-ready reference for triangulation, trilateration, stadia measurement, and geodetic fundamentals. Covers formulas, definitions, critical values, and exam pitfalls.
STADIA MEASUREMENT — Core Formulas
Formulas
D = Ks + C
Meaning
D = horizontal distance (m); K = stadia interval factor (usually 100); s = stadia intercept on rod (m); C = additive constant (≈ 0 for internal-focusing instruments)
Watch out
C is often neglected but can be 0.2–0.5 m for external-focusing telescopes — always check the instrument specs. Do NOT confuse K with the focal length.
When to use
Horizontal sights or as baseline for inclined corrections; the most common stadia formula on board exams
D_H = Ks cos²α
Meaning
D_H = horizontal distance for inclined sight (m); α = vertical angle of inclination (degrees or radians)
Watch out
Common error: using cos α instead of cos²α. The vertical angle correction uses cos²α, NOT cos α. Vertical angle positive upslope, negative downslope.
When to use
Inclined stadia sights — this corrects for the angle of the rod relative to the line of sight
V = (1/2) Ks sin(2α)
Meaning
V = vertical distance component (m); sin(2α) = sine of double the vertical angle
Watch out
Must use sin(2α), not sin(α) — this is the double-angle formula. If V is positive, target is above instrument; negative, below.
When to use
Finding elevation difference between instrument and target in stadia measurement
s = (D − C) / K
Meaning
Rearrangement to solve for stadia intercept; used when distance is known and rod spacing is checked
Watch out
Rarely asked directly; more for internal checks. Ensure C is subtracted first.
When to use
Verification: given a known distance, predict the stadia reading to check instrument calibration
Common values
Value
100
Symbol
K
Quantity
Stadia interval factor (standard)
Value
0 (or very small, ~0.05 m)
Symbol
C
Quantity
Additive constant (internal-focusing)
Value
0.2–0.5 m
Symbol
C
Quantity
Additive constant (external-focusing)
Value
10–300 m
Symbol
D
Quantity
Practical stadia range
Important facts
- Stadia method is rapid and does not require electronic distance measurement (EDM); useful for reconnaissance and rough detail surveys.
- Accuracy improves with larger intercept s; minimum practical distance ≈ 10 m, maximum ≈ 300 m (depending on rod visibility and refraction).
- For inclined sights, D_H = Ks cos²α is always less than D = Ks (the uncorrected reading) — the cosine correction accounts for the slant distance.
- Vertical component V = (1/2) Ks sin(2α) = Ks sin α cos α (alternative form); maximum vertical sensitivity at α = 45°.
- For small angles (α < 5°), cos²α ≈ 1 and sin(2α) ≈ 2α (in radians), so D_H ≈ Ks and V ≈ Ks·α (approximate forms for quick checks).
- Stadia is NOT as accurate as EDM or modern GNSS but remains a valid method for spot checks and when electronic instruments are unavailable.
- Rod reading must be taken perpendicular to the line of sight; tilted rods produce systematic errors.
- Refraction and heat shimmer (mirage) increase error at long distances or over water; stadia over 200 m requires extra caution.
Key definitions
Stadia interval factor (K)
Example
If K = 100 and s = 0.80 m, then D = 80 m (for horizontal sight).
Ratio of focal length to stadia hair separation; typically 100 for modern telescopes, meaning 1 m rod intercept = 100 m distance.
Additive constant (C)
Example
External telescope, C = 0.30 m, s = 1.20 m: D = 100(1.20) + 0.30 = 120.30 m.
Constant distance from the instrument's focal point to its trunnion axis; approximately 0 for internal-focusing telescopes, 0.2–0.5 m for external-focusing.
Vertical angle (α)
Example
Looking at a target 10° above horizontal: α = +10°.
Angle between the horizontal and the line of sight; positive upward, negative downward.
Stadia intercept (s)
Example
Rod reading: upper hair = 2.50 m, lower hair = 1.65 m → s = 0.85 m.
Distance on the rod (or staff) between the upper and lower stadia hairs; measured perpendicular to the rod axis.
Diagrams to know
- Stadia telescope cross-hairs (upper, middle, lower hairs) and their relationship to the rod.
- Inclined stadia ray diagram showing vertical angle α, slant distance D, horizontal D_H, and vertical component V.
- Geometric relationship: right triangle with hypotenuse = slant distance, adjacent side = D_H, vertical side = V.
TRIANGULATION & TRILATERATION — Network Methods
Formulas
Law of Sines: (a / sin A) = (b / sin B) = (c / sin C)
Meaning
a, b, c = side lengths; A, B, C = opposite angles in a triangle
Watch out
All angles must sum to 180° in plane triangles. If they don't, there is a systematic error in angle measurement (e.g., theodolite misalignment).
When to use
Triangulation: given baseline and angles, solve for unknown sides; or verify angles in closed triangles.
Law of Cosines: c² = a² + b² − 2ab cos C
Meaning
c = side opposite angle C; a, b = adjacent sides
Watch out
Ensure the angle C is in radians (or convert from degrees). Small angle errors lead to large side errors (sensitive near C = 0° or 180°).
When to use
Trilateration: given all three side lengths, solve for angles. Or, in triangulation, check closure.
Triangle Closure Error: e = Σ angles − 180°
Meaning
e = closure error in degrees (or minutes); Σ angles = sum of three measured angles
Watch out
Closure error > ±5 arc-minutes suggests instrument calibration issues or environmental refraction. Re-observe angles if closure is poor.
When to use
Check for gross errors or systematic bias in angle measurement during triangulation.
Relative Error in Trilateration: r = |(D_measured − D_theoretical) / D_theoretical|
Meaning
r = relative error (dimensionless); D = distance
Watch out
A small absolute error on a short side can give a large relative error; prioritize closure on longer sides.
When to use
Quality control in trilateration networks; acceptable relative error typically 1:5000 to 1:10000 for engineering surveys.
Common values
Value
±5 arc-minutes for local; ±1 arc-minute for precise surveys
Symbol
e
Quantity
Acceptable angle closure error (plane survey)
Value
1:5000 to 1:10000
Symbol
r
Quantity
Typical relative accuracy (trilateration, local)
Value
1:100000 or better
Symbol
r
Quantity
Typical relative accuracy (GNSS, modern)
Important facts
- Triangulation was the backbone of national control networks (e.g., NAMRIA Geodetic Reference System) before GNSS became dominant.
- In plane surveying, Σ angles = 180° exactly; in geodetic surveying over large areas, spherical excess = (E in arc-seconds) ≈ (area in km²) / 30 must be added.
- Angle measurement in triangulation typically uses a theodolite; modern traverse or control uses total station (combining angles and distances).
- Trilateration is now preferred for local networks because EDM is fast and accurate, and requires no vertical angle corrections.
- Triangle chains are formed by overlapping triangles; error propagates along the chain. Shorter baselines and more frequent re-observation reduce closure errors.
- Atmospheric refraction causes vertical angles to have systematic errors; refraction correction ≈ 0.0675 D² (m) for D in km (geodetic surveys only).
Key definitions
Triangulation
Example
Baseline AB = 500 m, angles at A, B, and C measured; solve for sides AC and BC.
Control network established by measuring angles in a series of connected triangles from a known baseline; side lengths are computed using the law of sines.
Trilateration
Example
EDM measures sides AB, BC, CA; solve for angles A, B, C.
Control network established by measuring all side distances (usually with EDM) and computing angles using the law of cosines.
Baseline
Example
A baseline of 1500 m is measured with a steel tape and corrections; all subsequent sides are computed from this baseline.
A precisely measured side that serves as the reference for all other measurements in a triangulation or trilateration network.
Triangle closure
Example
Angles 45°30′, 60°15′, 74°15′ sum to 180°00′ — perfect closure.
Sum of the three interior angles in a measured triangle; should equal 180° (or 180° plus spherical excess for large triangles on Earth's surface).
Diagrams to know
- Simple triangle with baseline, measured angles, and computed sides.
- Triangle chain or network diagram showing how multiple triangles overlap to extend control.
- Law of sines proportionality diagram.
GEODETIC SURVEYING — Earth Curvature & Corrections
Formulas
h_cr = 0.0675 D²
Meaning
h_cr = curvature-refraction correction (m); D = sight distance (km); accounts for both Earth curvature (curves sight downward) and atmospheric refraction (curves it upward).
Watch out
The formula h_cr = 0.0675 D² applies when D is expressed in kilometers, NOT meters. If D = 5 km, then h_cr = 0.0675 × 25 = 1.688 m. Common error: using meters directly without conversion.
When to use
Long lines of sight in geodetic surveying (D > 1 km); plane surveying ignores this for D < 0.5 km.
Spherical Excess (E) = (Area in km²) / 30 (arc-seconds)
Meaning
E = angular excess in a spherical triangle on Earth's surface; accounts for the fact that angles sum to >180°.
Watch out
Spherical excess is very small for local surveys but grows rapidly with area. For a 100 km² triangle area, E ≈ 3.3 arc-seconds.
When to use
Large triangles (sides > 10 km) in geodetic surveys; add E to the plane-triangle angle sum to get the actual geometric sum.
Refraction Coefficient (k) = 0.13 (approximate standard)
Meaning
k = fraction of curvature effect that is offset by refraction; typical range 0.10–0.16 depending on temperature gradient and humidity.
Watch out
Refraction coefficient varies with weather; use k = 0.13 unless local studies show otherwise. In desserts or over water, k can be 0.20 or higher.
When to use
Refined geodetic calculations; the net effect is h_cr = (0.5 − k) R (D/R)² ≈ 0.0675 D² for the standard k = 0.13 and R = 6371 km.
Convergence (γ) = Δλ sin(φ_avg) (arc-seconds)
Meaning
γ = convergence angle between meridians; Δλ = difference in longitude (arc-seconds); φ_avg = average latitude
Watch out
Convergence is not the same as grid bearing correction; must account for both convergence AND scale factor when converting between true bearings and grid bearings.
When to use
Converting bearings between projection zones (e.g., UTM); relevant only when working across multiple zones.
Common values
Value
6371 km
Symbol
R
Quantity
Earth radius (mean)
Value
0.13
Symbol
k
Quantity
Refraction coefficient (standard)
Value
0.0675 m ≈ 6.75 cm
Symbol
h_cr
Quantity
Curvature-refraction at 1 km sight
Value
1.688 m
Symbol
h_cr
Quantity
Curvature-refraction at 5 km sight
Value
−30 to −45 m
Symbol
N
Quantity
Philippine Geoid Height (approximate range)
Important facts
- For plane surveying, use curvature correction only if D > 1 km; below 1 km, the effect is < 0.07 m and often ignored.
- Geodetic surveys require ellipsoidal heights (heights above the ellipsoid) and ellipsoidal coordinates; conversion to orthometric heights (above geoid) requires the geoid height (N) from a geoid model.
- In the Philippines, the Philippine Geoid Model (PGM) is used to convert between ellipsoidal (WGS84) and orthometric heights; N typically ranges from −30 to −45 m.
- Atmospheric refraction is strongest in layers near the ground and weakest at high altitudes; temperature gradients and humidity cause daily variations.
- Spherical trigonometry (spherical law of sines, law of cosines for spheres) applies to large triangles on the ellipsoid; for local surveys, plane trigonometry is sufficient.
- Distortion due to map projection (e.g., UTM) introduces scale factors that grow away from the central meridian; for UTM, scale factor ≈ 1.0004 at ±3° from the central meridian.
Key definitions
Geodetic surveying
Example
NAMRIA Geodetic Reference System (NGRS) is a geodetic control network covering the Philippines using WGS84 ellipsoid.
Survey method that accounts for Earth's curvature and uses ellipsoidal coordinates (latitude, longitude); required for areas > 100 km² or national networks.
Plane surveying
Example
A city lot survey or building site survey uses plane surveying.
Survey method that treats Earth as a flat plane; acceptable for areas < 100 km² and local engineering projects.
Curvature-refraction correction
Example
For a 10 km sight, h_cr = 0.0675 × 100 = 6.75 m (target appears 6.75 m higher than geometric calculation predicts).
Net vertical correction to line of sight due to Earth curvature (curves downward) and atmospheric refraction (curves upward), resulting in h_cr = 0.0675 D² m.
Spherical excess
Example
Triangle on Earth's surface with area 50 km²: E ≈ 50 / 30 ≈ 1.67 arc-seconds.
The amount by which the sum of angles in a spherical triangle exceeds 180°; grows with triangle area.
Meridian convergence
Example
At 15°N latitude, 1° difference in longitude ≈ 0.966 sin(15°) ≈ 0.25° convergence.
Angle between a true meridian (north–south line) and a grid meridian (e.g., UTM zone meridian); varies with latitude and longitude difference.
Diagrams to know
- Diagram showing how Earth curvature causes line of sight to curve downward, but refraction curves it upward; net effect = 0.0675 D² m.
- Convergence diagram: true meridians converging toward a pole vs. parallel grid lines in a map projection.
- Spherical triangle on Earth's surface with sides greater than plane triangle sides of the same angles.
Reactions or equations
h_cr = (1 − 2k) D² / (2R) ≈ 0.0675 D² (k = 0.13, R = 6371 km, D in km)
Note
The 0.0675 coefficient is derived from (1 − 2 × 0.13) / (2 × 6371 × 1000) in consistent units. Different sources use k = 0.14 or 0.15, giving slightly different coefficients.
Conditions
Standard atmosphere, typical refraction coefficient k = 0.13, Earth radius R ≈ 6371 km
COORDINATE SYSTEMS & POSITION FIXING
Formulas
UTM Easting = E₀ + k₀ Δx; UTM Northing = N₀ + k₀ Δy
Meaning
E₀, N₀ = false easting/northing (500,000 m E, 0 m or 10,000,000 m N); k₀ = scale factor (0.9996); Δx, Δy = plane coordinates relative to central meridian
Watch out
False northing varies: 0 m for northern hemisphere, 10,000,000 m for southern. Always check the UTM zone (zone number = 31 + floor(longitude/6) for eastern hemisphere).
When to use
Converting geographic coordinates to UTM grid coordinates; Philippines uses UTM zones 51N, 52N, 53N depending on location.
Scale Factor (k) = k₀ [1 + (Δx / R)² / 2] (approximate)
Meaning
k = local scale factor; k₀ = central meridian scale (0.9996 for UTM); Δx = distance from central meridian (m); R = Earth radius (m)
Watch out
At ±3° from central meridian, k ≈ 1.0004. Distances measured 100 km from central meridian are increased by ~4 mm/km; must account for scale when comparing field and grid distances.
When to use
Correcting measured distances in the field to grid coordinates; scale factor grows away from central meridian.
Distance Correction = D_field × k / k₀
Meaning
D_field = measured distance; k / k₀ = ratio of local scale factor to central meridian scale factor
Watch out
This correction is often neglected in local surveys but becomes critical for distances > 50 km from central meridian or high-precision work.
When to use
Converting field distances to grid distances (or vice versa) for precise positioning in UTM.
Common values
Value
0.9996
Symbol
k₀
Quantity
UTM scale factor at central meridian
Value
500,000 m
Symbol
E₀
Quantity
UTM false easting
Value
0 m
Symbol
N₀
Quantity
UTM false northing (N hemisphere)
Value
10,000,000 m
Symbol
N₀
Quantity
UTM false northing (S hemisphere)
Value
123°E (zone 51N), 129°E (zone 52N), 135°E (zone 53N)
Symbol
λ₀
Quantity
Philippines central meridians
Important facts
- Philippines uses WGS84 ellipsoid as the standard (NGRS = NAMRIA Geodetic Reference System).
- UTM zone number = INT(longitude / 6) + 31 for eastern hemisphere; always verify the correct zone for a given longitude.
- Scale factor at central meridian (k₀ = 0.9996) is less than 1 to reduce distortion at zone edges; points on central meridian are measured shorter than true distance by 0.04%.
- Convergence (angle between true north and grid north) is zero at the central meridian and grows as cos(latitude) × longitude difference (in appropriate units).
- For engineering surveys within a single UTM zone and away from zone boundaries, plane surveying methods and UTM coordinates give sufficient accuracy (error < 1:5000).
- Converting to/from geodetic (latitude/longitude) requires iterative calculations or lookup tables; standard formulas involve long series approximations.
Key definitions
Universal Transverse Mercator (UTM)
Example
A point in Metro Manila (~121°E, 14°N) falls in UTM zone 51N; its UTM coordinates are approximately (500,000 m E, 1,546,000 m N).
Projected coordinate system dividing Earth into 60 zones (each 6° wide in longitude); Philippines spans zones 51N, 52N, 53N.
Scale factor
Example
At the central meridian (±0°), k₀ = 0.9996; at ±3° away, k ≈ 1.0004.
Ratio of grid distance to geodetic distance; varies across a map projection (< 1 near central meridian, > 1 away from it).
Central meridian
Example
UTM zone 51N has central meridian at 123°E; zone 52N at 129°E.
The longitude line in the center of a UTM zone where the scale factor is minimum (0.9996); all UTM zones have a central meridian 6° apart.
False Easting / False Northing
Example
A point 200 km west of central meridian has raw easting ≈ 300,000 m; with 500,000 m false easting, UTM easting = 800,000 m.
Arbitrary offset added to all grid coordinates to ensure positive values; UTM uses 500,000 m false easting and 0 m (N hemisphere) or 10,000,000 m (S hemisphere) false northing.
Diagrams to know
- UTM zone map of Philippines showing zones 51N, 52N, 53N and central meridians.
- Scale factor graph: k vs. distance from central meridian (showing k₀ = 0.9996 at center and k > 1 away from center).
- Convergence diagram showing angle between true north and grid north at different latitudes and longitudes.
GNSS & MODERN POSITIONING
Formulas
Pseudorange = c × (satellite time − receiver time) + c × dT
Meaning
Pseudorange = observed distance (m); c = speed of light (3 × 10⁸ m/s); dT = receiver clock offset (s)
Watch out
A 1 nanosecond (10⁻⁹ s) clock error equals ~0.3 m pseudorange error; this is why GNSS receivers need accurate clocks or dual-frequency data.
When to use
Understanding GNSS code-based positioning; the receiver clock error dT is one of four unknowns solved in GNSS navigation.
Dilution of Precision (DOP) = 1 / √(trace of geometry matrix)
Meaning
DOP = dimensionless factor; higher DOP = weaker satellite geometry = larger positioning error for a given measurement noise.
Watch out
DOP depends on satellite positions in the sky, not the measurement accuracy itself. Even with perfect receivers, poor satellite geometry (e.g., satellites all in one direction) gives large error.
When to use
Assessing GNSS positioning quality; PDOP < 4 is good, PDOP > 10 is poor.
Horizontal Error ≈ HDOP × code noise (m)
Meaning
HDOP = horizontal DOP; code noise ≈ 0.3–1 m for standard C/A code; total horizontal error
Watch out
This is a rough estimate; real-world GNSS error also includes ionospheric delay, tropospheric delay, multipath, and receiver noise. RTK or DGPS can achieve cm-level accuracy.
When to use
Quick estimate of GNSS horizontal accuracy; HDOP < 3 gives accuracy < 1 m with C/A code.
Common values
Value
3 × 10⁸ m/s (exact)
Symbol
c
Quantity
Speed of light
Value
~10 m horizontal (1σ standard deviation)
Symbol
σ
Quantity
GPS C/A code accuracy
Value
0.02 m horizontal, 0.03 m vertical
Symbol
σ
Quantity
RTK accuracy (cm-level)
Value
~11.97 hours (medium Earth orbit)
Symbol
T
Quantity
GNSS satellite orbital period
Value
4 (3 for position, 1 for clock offset)
Symbol
n
Quantity
Minimum satellites for 3D positioning
Important facts
- Standard GPS (C/A code) achieves ~10 m horizontal accuracy in open sky; selective availability was turned off in 2000, improving civilian accuracy from ~100 m.
- RTK requires a reference station within ~20 km (for L1 corrections) to ~50 km (with advanced networks) of the rover receiver.
- GNSS signals are degraded in dense urban canyons, dense forest, and underwater; the Philippine Geoid Model (PGM) converts ellipsoidal heights from GNSS to orthometric (sea-level-relative) heights.
- GNSS epoch time uses GPS weeks and seconds-of-week, not calendar time; GPS week zero started January 6, 1980; the GPS week number rolled over in 1999 and 2019 (must account for this in legacy systems).
- Atmospheric delays (ionosphere, troposphere) are the dominant errors for single-frequency GNSS over distances > 100 km; dual-frequency receivers can largely eliminate ionospheric delay.
- GNSS is now the standard method for establishing control networks in surveying; triangulation and trilateration are rarely used except in areas with poor GNSS coverage (tunnels, heavily forested areas, or underground).
Key definitions
Global Navigation Satellite System (GNSS)
Example
GPS (USA), GLONASS (Russia), Galileo (EU), BeiDou (China), and QZSS (Japan) are all GNSS systems.
Generic term for satellite positioning systems (GPS, GLONASS, Galileo, BeiDou); provides global position, velocity, and time 24/7.
Differential GNSS (DGPS)
Example
Coast Guard operates DGPS stations in US ports; a ship receiving these corrections achieves ~2 m accuracy vs. 10 m with standard GPS.
Method using a stationary reference receiver (base station) to broadcast corrections to mobile receivers, reducing error to 1–5 m.
Real-Time Kinematic (RTK)
Example
Surveyor with RTK receiver can position points to ±0.02 m (2 cm) horizontal, ±0.03 m vertical, in real-time without post-processing.
GNSS technique using carrier-phase observations and a nearby reference station to achieve cm-level positioning in real-time.
Dilution of Precision (DOP)
Example
PDOP = 5 with ±1 m code noise gives ~±5 m positioning error; PDOP = 2 gives ~±2 m error.
Dimensionless measure of how satellite geometry affects positioning accuracy; higher DOP means larger error for the same measurement noise.
Diagrams to know
- GNSS constellation diagram: satellite orbits, signal paths to ground receiver, atmospheric delays.
- DOP degradation vs. satellite geometry: ideal pyramid of satellites vs. all satellites in one direction.
- RTK network architecture: base station, rover receiver, and real-time radio/internet link.
Must remember
- Stadia horizontal distance formula: D_H = Ks cos²α (NOT cos α). This is the #1 mistake on exams. For inclined sights, you MUST square the cosine.
- Vertical component in stadia: V = (1/2) Ks sin(2α), not Ks sin(α). Use the double-angle formula sin(2α); maximum sensitivity is at α = 45°.
- Curvature-refraction correction h_cr = 0.0675 D² applies when D is in kilometers. For D = 5 km, h_cr = 1.688 m. If you use D in meters, the coefficient changes completely.
- Law of Sines for triangulation: (a/sin A) = (b/sin B) = (c/sin C). Baseline is the known side; solve for unknown sides given measured angles. Sum of angles must equal 180° in plane geometry.
- Triangle closure: Σ(angles) should equal 180°. If closure error > ±5 arc-minutes, suspect systematic error (e.g., theodolite tilt, refraction) — do NOT ignore.
- Triangulation measures angles from a baseline; trilateration measures all distances. Modern surveys use trilateration (EDM) or GNSS. Triangulation is rarely used today.
- Scale factor in UTM: k₀ = 0.9996 at central meridian; k increases away from it. At ±3° from central meridian, k ≈ 1.0004. Ignore this for local surveys < 50 km from central meridian.
- Spherical excess E = (Area in km²) / 30 (arc-seconds). Add this to 180° for large triangles on Earth's surface. For areas < 10 km², E is usually negligible.
- GNSS RTK achieves ±0.02 m (2 cm) horizontal accuracy; standard GPS is ~10 m. GNSS is now the standard method for establishing control. Triangulation/trilateration are backup only.
- Philippine coordinates use WGS84 ellipsoid with geoid height N ≈ −30 to −45 m. GNSS gives ellipsoidal heights; subtract N to get orthometric heights (elevations above mean sea level).
Last minute tips
- Identify the angle in stadia formulas carefully. α is the vertical angle (from horizontal to the line of sight), NOT the angle of inclination from the rod. If the problem says 'elevation angle 5°', then α = 5°, and cos²(5°) ≈ 0.9924.
- Check units in curvature correction. The formula h_cr = 0.0675 D² ONLY works if D is in kilometers. If the problem gives D in meters, convert first: D(km) = D(m) / 1000.
- Verify triangle closure BEFORE solving. If Σ(angles) ≠ 180°, adjust angles proportionally using the closure error divided by 3. Do NOT proceed with an unbalanced triangle.
- In trilateration, use the Law of Cosines carefully. c² = a² + b² − 2ab cos(C) requires C to be the angle OPPOSITE side c. If you reverse this, you get the wrong answer.
- Always specify UTM zone when giving grid coordinates. A point in Zone 51N at 800,000 m E is 200 km WEST of the central meridian (123°E). Without the zone, the coordinate is meaningless.
Comparison tables
Triangulation vs. Trilateration vs. GNSS
| Method | Key Measurement | Primary Tool | Typical Accuracy | Setup Time | Modern Use |
|---|---|---|---|---|---|
| Triangulation | Angles (from baseline) | Theodolite + tape (baseline) | 1:5000–1:10000 | Several hours (if baseline is long) | Rare (replaced by EDM & GNSS) |
| Trilateration | Distances (all sides) | Total station or EDM | 1:5000–1:10000 | Minutes to hours (fast angle not needed) | Local control networks; verification |
| GNSS (RTK) | Satellite ranges (4+) | RTK receiver + base station | 0.02–0.05 m (cm-level) | Minutes (once base is set up) | Primary method for modern surveys; fast, accurate, no line-of-sight needed for clear sky |
Plane Surveying vs. Geodetic Surveying
| Aspect | Plane Surveying | Geodetic Surveying |
|---|---|---|
| Earth Shape | Treated as flat | Accounts for curvature; uses ellipsoid |
| Applicable Area | < 100 km² | > 100 km² or national networks |
| Coordinate System | Rectangular (x, y) or local grid | Latitude, longitude, then projected (e.g., UTM) |
| Curvature Correction | Ignored (or h_cr neglected if < 0.1 m) | Applied: h_cr = 0.0675 D² (m) for D in km |
| Typical Tool | Total station, transit tape | GNSS, theodolite, total station with atmospheric corrections |
| Angle Sum in Triangle | 180° exactly | 180° + spherical excess (a few arc-seconds for large areas) |
Stadia Measurement: Key Parameters & Effects
| Parameter | Symbol | Typical Value | Effect if Increased |
|---|---|---|---|
| Stadia interval factor | K | 100 | Distance increases proportionally; K = 50 gives half the range |
| Stadia intercept | s | 0.5–2 m | Distance increases proportionally; larger s = longer range; minimum practical ~0.05 m |
| Additive constant | C | 0 (internal) or 0.2–0.5 m (external) | Distance increases by constant offset; critical for external telescopes |
| Vertical angle | α | 0° (horizontal) | cos²α decreases, so D_H decreases; V increases; maximum at α = 45° |
UTM Zone Reference — Philippines
| Zone | Central Meridian | Longitude Range | Region Covered |
|---|---|---|---|
| 51N | 123°E | 120°–126°E | Western Philippines (Luzon, Visayas) |
| 52N | 129°E | 126°–132°E | Central Visayas, southern Mindanao |
| 53N | 135°E | 132°–138°E | Eastern Mindanao, eastern islands |
Diagrams
Stadia Measurement Decision Tree: Which Formula to Use
Quick decision tree to determine which stadia formula to apply based on sight geometry and desired output. Ensures students select the correct formula (horizontal vs. inclined, and horizontal distance vs. vertical component) before plugging in numbers.
Advanced Surveying Methods — Hierarchy & Key Formulas
Mind map showing the logical hierarchy of advanced surveying topics, with stadia at the detail level, triangulation/trilateration as classical methods, geodetic methods for large areas, and modern GNSS as the current standard. Each branch shows the key formula or concept students must remember.
Plane vs. Geodetic Surveying Decision: Which Method to Use
Decision flowchart to determine whether a given survey project should use plane surveying (flat-Earth approximation) or geodetic methods (accounting for curvature). Helps students quickly assess which formulas and corrections apply to their problem.