Condition Equations and Figure Adjustment - Cheat Sheet
Your last-minute revision companion for condition equations, misclosure calculation, and equal/weighted distribution in geodetic figure adjustment. Master the core formulas, common pitfalls, and exam strategies in 30 minutes.
Core Concept: Condition Equations
Formulas
Misclosure = Σ(observed values) − (required total)
Meaning
Misclosure = sum of all measured observations minus the theoretical/required sum that geometry demands
Watch out
Sign matters! Positive misclosure = observations exceed requirement; negative = observations fall short. The correction is opposite in sign to the misclosure.
When to use
Every time you measure angles in a triangle, horizon, or loop — calculate misclosure FIRST
Correction per observation (equal weights) = −(Misclosure) / n
Meaning
n = number of observations being corrected; negative sign reverses the misclosure direction
Watch out
The correction is NEGATIVE of the misclosure divided by count. If misclosure is +12″, correction per angle is −12″/3 = −4″ per angle. Apply this correction to each observation.
When to use
When all observations have equal reliability (equal weights or equal precision)
Corrected observation = Original observation + Correction
Meaning
Add the calculated correction (which carries the negative sign) to each measured value
Watch out
Use signed arithmetic. If correction is −4″ and original angle is 60°00'10″, result is 60°00'06″.
When to use
After calculating per-observation correction, apply to all raw measurements
Common values
Value
180°00'00″ (or 180° in decimal, 3.14159 rad)
Symbol
None; use as reference total
Quantity
Plane triangle angle sum (required)
Value
360°00'00″ (or 360° in decimal, 2π rad)
Symbol
None; use as reference total
Quantity
Horizon (station) angle sum (required)
Value
1″ to 10″ (varies with area and latitude); ε = A/(2R²) in radians, then convert to seconds
Symbol
ε
Quantity
Spherical excess (typical large triangles)
Important facts
- Condition equations enforce exact geometry; they are not approximations.
- Misclosure = observed sum − required sum; correction = −misclosure/n (for equal weights).
- Equal-weight adjustment: divide misclosure equally among all observations (opposite sign).
- Weighted adjustment: distribute misclosure proportional to observation variances (more correction to weaker observations).
- Plane triangles: required sum = 180°00'00″. Spherical triangles: required sum = 180° + ε.
- Horizon (station) condition: angles around a single point = 360°00'00″.
- Loop condition: elevation or coordinate misclosure in a closed level/traverse loop must close to zero.
- Correction sign is OPPOSITE to misclosure sign — this is the #1 exam mistake.
- After applying all corrections, the adjusted observations must satisfy the condition equation exactly.
- In a closed traverse (loop): ΣΔx = 0, ΣΔy = 0; distribute linear misclosure (closure error) proportionally.
Key definitions
Condition Equation
Example
∠A + ∠B + ∠C = 180° (plane triangle); ∠A + ∠B + ∠C = 180° + ε (spherical triangle with excess ε).
A mathematical statement enforcing a known geometric relationship that adjusted observations must satisfy exactly (e.g., triangle angles = 180°).
Misclosure
Example
Triangle angles sum to 180°00'12″ instead of 180°00'00″ → misclosure = +12″.
The amount by which raw observations violate the condition equation; the error to be distributed.
Figure Adjustment
Example
Adjusting three triangle angles so their sum = exactly 180°00'00″.
Process of correcting observations to satisfy geometric conditions by distributing the misclosure equally or weighted.
Spherical Excess (ε)
Example
Large survey triangle with ε = 5″ → required angle sum = 180°00'05″, not 180°00'00″.
Additional angle sum in a spherical triangle due to curvature; target condition becomes 180° + ε, not 180°.
Diagrams to know
- Condition equation workflow: measure → calculate misclosure → distribute correction → verify sum
- Triangle with three angle measurements and misclosure arrows showing distribution
- Horizon condition diagram: four or more angles around a central point summing to 360°
- Closed traverse (loop) with closure error vector and residual adjustment arrows
Reactions or equations
∠A_adj + ∠B_adj + ∠C_adj = 180°00'00″
Note
This is the verification step — if adjusted angles do NOT sum to exactly 180°, there is an arithmetic error.
Conditions
Plane triangle; all adjustments applied; no residual misclosure
∠A_adj + ∠B_adj + ∠C_adj = 180° + ε (spherical)
Note
Failing to include ε is the #2 common exam error after misclosure sign.
Conditions
Large triangle on Earth's surface; ε calculated from area or using spherical excess formula
w_i × c_i = constant (weighted distribution)
Note
Heavier (more reliable) observations get smaller corrections; lighter observations get larger corrections.
Conditions
Unequal observation weights w_i; corrections c_i are inversely proportional to weight
Equal-Weight Adjustment (Most Common in Exams)
Formulas
c = −(Misclosure) / n
Meaning
c = correction per observation; n = total number of observations; negative sign reverses misclosure
Watch out
Do NOT forget the negative sign. If misclosure is +8″, each of 4 angles gets correction −8″/4 = −2″. If you write +2″, the answer is wrong.
When to use
All observations have equal precision/reliability; is the DEFAULT in most exam problems unless stated otherwise
Adjusted value = Original value + c
Meaning
Apply the same correction to every observation
Watch out
Arithmetic sign errors are fatal. Use a simple table: [Original] + [c] = [Adjusted].
When to use
After computing c from misclosure/n, apply to each original measurement
Verification: Σ(Adjusted values) = Required total
Meaning
Sum of all corrected observations must equal the geometric requirement
Watch out
If verification fails, do NOT submit the answer; find the arithmetic error first.
When to use
ALWAYS verify at the end; if this fails, recheck c calculation or sign
Important facts
- Equal-weight method is used when problem does NOT specify weights or standard deviations.
- Simple formula: correction = −misclosure / count.
- Each observation receives IDENTICAL correction (same value, same sign).
- The sum of all corrections must equal −(misclosure) (verification of distribution).
- Equal-weight is faster than weighted; examiners expect it unless weights are given.
Key definitions
Equal-Weight Adjustment
Example
Three angles each get −4″ correction to eliminate a +12″ triangle misclosure.
Distribution of misclosure equally (divided by n) to all observations; assumes equal measurement precision.
Equal Reliability
Example
Three theodolite angle measurements from the same instrument and operator → assume equal weights.
All observations have the same measurement uncertainty; each should carry equal weight in adjustment.
Diagrams to know
- Three angles in a triangle with identical −4″ correction arrows pointing to each
- Bar chart showing original values, correction bars, and adjusted values
Reactions or equations
c_1 = c_2 = c_3 = ... = c_n = −M/n (where M = misclosure)
Note
All corrections are identical. This is what makes equal-weight so straightforward.
Conditions
Equal weights; all observations measured with same precision
Weighted Adjustment (Unequal Reliability)
Formulas
c_i = −(Misclosure) × (σ_i² / Σσ_j²)
Meaning
c_i = correction for observation i; σ_i² = variance (inverse of weight) for observation i; sum in denominator is total variance
Watch out
This is an INVERSE relationship: LARGER variance → LARGER correction. Weaker measurements corrected more. Do NOT mix up weight and variance signs.
When to use
When observations have unequal precision or explicit weights are given; more uncertain observations get larger corrections
Weight w_i = 1 / σ_i² or w_i = constant / σ_i²
Meaning
Weight is inversely proportional to variance; heavier (lower variance) observations have higher weight
Watch out
If σ values are given, calculate w = 1/σ². If weights are given directly, use them. Do NOT confuse weight with variance.
When to use
Converting standard deviation or uncertainty to weight for weighted adjustment
c_i = −(Misclosure) × (w_i / Σw_j)
Meaning
Alternative form using weights directly: correction proportional to weight fraction
Watch out
Verify that Σw_i = n (approximately) or is normalized; adjust the formula if weights are not normalized.
When to use
When weights are given instead of standard deviations
Important facts
- Weighted adjustment requires knowledge of observation precision (standard deviations or weights).
- Weaker (higher variance, lower weight) observations receive LARGER corrections.
- Stronger (lower variance, higher weight) observations receive SMALLER corrections.
- If only one observation has different precision, it still follows the weighted formula.
- Verification: Σ(w_i × c_i) = −Misclosure × Σ(w_i) / Σ(w_j) = −Misclosure (check sums match proportionally).
Key definitions
Weighted Adjustment
Example
One angle measured with ±1″ precision, another with ±2″; the ±2″ observation gets 4× larger correction (variance ratio 4:1).
Distribution of misclosure proportional to observation variances (or inverse of weights); accounts for unequal measurement reliability.
Variance (σ²)
Example
Observation A: σ² = 1″², Observation B: σ² = 4″² → B is 4× less reliable; gets 4× larger correction.
Measure of measurement uncertainty; larger variance = less reliable observation.
Weight (w)
Example
If w_A = 4, w_B = 1, then A is 4× more reliable and gets 1/4 the correction of B.
Inverse of variance; w = 1/σ². Higher weight = more reliable observation.
Diagrams to know
- Three observations with different variances (bars of different heights) and corresponding correction magnitudes
- Weight vs. correction relationship graph (inverse linear: higher weight → lower correction magnitude)
Reactions or equations
c_i / c_j = σ_i² / σ_j² (ratio of corrections equals ratio of variances)
Note
Quick check: if one observation is twice as uncertain, its correction should be 4× larger (variance is squared).
Conditions
Same misclosure, different observation precisions
Σ(c_i × w_i) = −Misclosure (verification for weighted adjustment)
Note
This verifies the correction distribution; must sum to exactly the negative misclosure.
Conditions
All weights normalized or proportional
Triangle Angle Adjustment (Plane Triangles)
Formulas
Required sum = 180°00'00″ (plane) or 180° + ε (spherical)
Meaning
For plane triangles, exactly 180°; for spherical, add the spherical excess ε
Watch out
Forgetting spherical excess (ε) for large triangles is a common exam error. Always check the problem statement for 'spherical' or large area.
When to use
SET this as the reference before calculating misclosure
Misclosure = (∠A_obs + ∠B_obs + ∠C_obs) − (180° + ε)
Meaning
Sum of three measured angles minus the required total (including excess if spherical)
Watch out
Pay attention to sign. If observed sum is 180°00'15″ and required is 180°00'00″, misclosure = +15″ (positive). Correction will be −15″/3 = −5″ per angle.
When to use
Calculate misclosure FIRST; this determines the magnitude and direction of correction
c = −(Misclosure) / 3
Meaning
For three angles with equal weight, divide misclosure by 3 and reverse sign
Watch out
This is the most direct formula in triangle problems. Three angles always → divide by 3.
When to use
Equal-weight adjustment of a plane or spherical triangle
Spherical excess ε (seconds) = A / (2R²) × (180 × 3600 / π)
Meaning
A = triangle area in m²; R = Earth radius (≈ 6,371 km); ε in arc-seconds
Watch out
This formula requires consistent units (area in m², radius in m). The result is very small for typical survey triangles (few arc-seconds).
When to use
When area is given and excess must be calculated; rarely required for adjustment itself, but good to know
Common values
Value
180°00'00″ or 180°
Symbol
Target for condition equation
Quantity
Plane triangle angle sum
Value
6,371.0 km (mean); 6,378.137 km (equatorial)
Symbol
R
Quantity
Earth radius (WGS84)
Value
0.5″ to 2″
Symbol
ε
Quantity
Spherical excess (typical survey triangle, ~1 km sides, area ~0.5 km²)
Important facts
- Plane triangle: always use 180°00'00″ as target unless problem says otherwise.
- Spherical triangle: ALWAYS add excess ε to the 180° target; ε is positive and typically 1″–20″ for surveying.
- Misclosure is independent of whether plane or spherical; the difference is in the required total.
- Three-angle adjustment with equal weights is THE most common exam problem.
- Verification: sum of adjusted angles must equal required sum (180° or 180° + ε) within rounding.
- If two angles are weighted differently, apply weighted formula; otherwise, use equal-weight (divide by 3).
Key definitions
Plane Triangle
Example
Local building survey triangle with sides ~500 m.
Triangle on a flat surface; angle sum = 180°00'00″; used for small surveys (< 100 km typical sides).
Spherical Triangle
Example
Regional geodetic network triangle with sides 10 km or larger; ε ≈ 1″ to 10″.
Triangle on Earth's curved surface; angle sum = 180° + ε (spherical excess); used for large surveys (sides > 1 km).
Spherical Excess (ε)
Example
Triangle with area 100 km² → ε ≈ 3″ (rough approximation).
Additional angle sum in a spherical triangle; proportional to area and inversely to Earth radius squared.
Diagrams to know
- Equilateral triangle with three angle values, misclosure calculation, and three equal correction arrows
- Large triangle on globe showing spherical excess ε conceptually
Reactions or equations
∠A_adj + ∠B_adj + ∠C_adj = 180°00'00″
Note
This is the MANDATORY verification step. Must hold exactly (within rounding to nearest 0.1″).
Conditions
Plane triangle; all equal-weight corrections applied
∠A_adj + ∠B_adj + ∠C_adj = 180° + ε
Note
Same verification, but target is shifted by ε. Both sides must balance.
Conditions
Spherical triangle; excess ε pre-calculated
Horizon (Station) Angle Adjustment
Formulas
Required sum = 360°00'00″
Meaning
Angles around a single point always sum to 360° by definition
Watch out
This is analogous to a closed polygon. Do NOT confuse with triangle (180°).
When to use
Whenever correcting multiple angles (3 or more) observed from one station
Misclosure = Σ(observed angles) − 360°
Meaning
How much the sum of all station angles exceeds or falls short of 360°
Watch out
Common exam trick: 4 angles might sum to 360°00'08″ → misclosure = +8″. Students sometimes forget to subtract from 360°.
When to use
Calculate misclosure after summing all angles from the station
c = −(Misclosure) / n, where n = number of angles at station
Meaning
Equal correction to each angle; n is the count of angles (could be 3, 4, 5, etc.)
Watch out
Use the COUNT of angles, not the number of sides (they differ by 1 in closed polygons). If you measure 4 angles, n = 4.
When to use
Equal-weight distribution among station angles
Common values
Value
360°00'00″ or 360°
Symbol
Reference total
Quantity
Horizon angle sum (required)
Important facts
- Horizon (station) condition is simpler than triangle: always 360°, no excess term.
- Any number of angles (3, 4, 5, ...) can be corrected using the same formula.
- Misclosure is typically small (< 30″ for reasonably careful work).
- More angles → correction per angle is smaller (misclosure divided by larger n).
- If only two angles and misclosure is significant, check for measurement error (it's rare to measure just 2 angles at a station).
Key definitions
Horizon Condition
Example
Observer at station P measures 4 angles to surrounding points A, B, C, D; ∠APB + ∠BPC + ∠CPD + ∠DPA = 360°.
All angles measured around a single point sum to exactly 360°.
Station Misclosure
Example
Four angles sum to 360°00'06″ → station misclosure = +6″.
The angular closure error at a single observation point; usually a few arc-seconds.
Diagrams to know
- Station P with 4 or 5 rays to surrounding points, labeled angles, misclosure, and equal correction annotations
- Circle divided into sectors (angles) showing 360° closure
Reactions or equations
∠_1_adj + ∠_2_adj + ... + ∠_n_adj = 360°00'00″
Note
Verification must hold exactly. If it does not, recheck arithmetic.
Conditions
n angles at a station; all equal-weight corrections applied
Closed Loop (Traverse / Level) Adjustment
Formulas
Linear misclosure (traverse) = √[(ΣΔx_obs − 0)² + (ΣΔy_obs − 0)²]
Meaning
Magnitude of closure error in a closed traverse; ΔX and ΔY are easting and northing increments
Watch out
This is a VECTOR misclosure (2D). Do NOT just add ΣΔx + ΣΔy; calculate the magnitude (hypotenuse).
When to use
After summing all course increments in a closed traverse; should equal zero but rarely does
Correction c_i = −(Misclosure) × (Length_i / Σ(Lengths))
Meaning
Distribute linear misclosure in proportion to course lengths (longer courses get more correction)
Watch out
Misclosure is negative in sign (reverse direction of closure error). Multiply by the length fraction to get per-course correction.
When to use
Equal-length-weight method: each course corrected proportional to its measured length
Elevation misclosure (level loop) = ΣΔh_obs − 0
Meaning
In a closed level loop, sum of elevation differences should return to starting elevation; misclosure is the failure
Watch out
Sign: if final elevation is above start, misclosure is positive (upward). Correction reverses this.
When to use
Leveling loop closure check
c_i = −(Elevation misclosure) × (L_i / Σ(L))
Meaning
Distribute elevation misclosure proportional to sight length (longer sights = more correction)
Watch out
In leveling, longer sight distances have larger potential errors; proportional distribution is standard.
When to use
Leveling loop adjustment with equal-weight assumption
Common values
Value
1:1000 to 1:10000 (better surveys are 1:5000 or tighter)
Symbol
R_closure
Quantity
Relative closure spec (traverse)
Value
0.05–0.20 m (if closure spec 1:1000 → expect ~0.10 m or less)
Symbol
M_closure
Quantity
Typical linear misclosure (100 m traverse)
Important facts
- Traverse misclosure has both magnitude and direction (it is a vector); use ΔX² + ΔY² = R².
- Distribute proportional to length: longer courses absorb more error.
- After correction, all courses are adjusted; the loop closes perfectly (ΣΔx_adj = 0, ΣΔy_adj = 0).
- Leveling loop misclosure is 1D (just elevation); distribute proportional to sight distance.
- Relative closure error = (closure magnitude) / (perimeter); typical spec is 1:1000 to 1:10000.
- If closure error is too large, suspect measurement or systematic error (not just random variation).
Key definitions
Closed Traverse
Example
Polygon survey with 5 courses: A→B→C→D→E→A; if all courses are measured, the loop must close.
A sequence of measured courses that return to the starting point; ΣΔx = 0, ΣΔy = 0, ΣΔz = 0 exactly.
Closure Error (Misclosure)
Example
Traverse closes 0.15 m short in easting, 0.08 m over in northing → linear closure error = √(0.15² + 0.08²) ≈ 0.17 m.
The linear or angular deviation from exact closure in a traverse or level loop.
Loop Misclosure Distribution
Example
A 100 m course in a 500 m loop gets 20% of the total misclosure correction.
Method of correcting course increments to satisfy closure; typically proportional to course length.
Diagrams to know
- Closed 5-course traverse polygon with course lengths, closure error vector, and proportional correction arrows
- Leveling loop: series of benchmark sights showing elevation increments and cumulative closure error
Reactions or equations
ΣΔx_adj = 0, ΣΔy_adj = 0 (verification for closed traverse)
Note
Must hold exactly within rounding. If not, recheck correction calculations.
Conditions
All course increments adjusted; loop closed
ΣΔh_adj = 0 (verification for leveling loop)
Note
Same verification principle applied to leveling.
Conditions
All elevation differences adjusted; loop closed vertically
Quick Problem-Solving Workflow
Important facts
- Step 1: Identify the condition (triangle angle sum, horizon sum, loop closure, etc.).
- Step 2: Calculate the required total (180° for plane triangle, 180° + ε for spherical, 360° for horizon, 0 for loop).
- Step 3: Sum the observed values.
- Step 4: Calculate misclosure = (observed sum) − (required total).
- Step 5: Determine if weights are equal or unequal from problem statement.
- Step 6: Apply equal-weight formula (misclosure / n with sign reversal) OR weighted formula (variance/weight-based).
- Step 7: Apply corrections to each observation.
- Step 8: Verify that adjusted observations sum to the required total (±0.1″ rounding tolerance).
- Step 9: Check that corrections sum to −(misclosure), confirming distribution is complete.
Must remember
- Correction sign is OPPOSITE the misclosure sign. This is the #1 exam mistake. If misclosure is +12″, correction is −12″/n.
- For equal-weight adjustment, divide misclosure by the count of observations: c = −Misclosure / n. The same correction applies to every observation.
- Plane triangle required sum = 180°00'00″. Spherical triangle required sum = 180° + ε (where ε is spherical excess in arc-seconds).
- Horizon (station) condition = 360°00'00″ always. Angles around a single point sum to 360° by definition.
- Always verify after applying corrections: adjusted observations must sum to the required total. If not, find the arithmetic error.
- For weighted adjustment, weaker observations (higher variance σ², lower weight w) receive LARGER corrections. Stronger observations get smaller corrections.
- Closed traverse misclosure is a VECTOR: use √((ΣΔx)² + (ΣΔy)²), not addition. Distribute proportional to course length.
- If the problem does NOT mention weights or standard deviations, assume equal-weight adjustment (divide by n).
- Spherical excess ε is typically 1″–10″ for typical survey triangles (area 0.5–100 km²). Forgetting ε for a spherical triangle is a common exam error.
- The sum of all corrections must equal −(misclosure). Use this as a second verification step to confirm the distribution is complete and correct.
Last minute tips
- Before any calculation, identify the condition type (triangle angle, horizon, traverse, level loop). This determines the required sum and formula. Write it down explicitly.
- Always calculate misclosure first and record its sign. The correction is NEGATIVE of this. Double-check the sign before applying to observations.
- For equal-weight problems (the most common type), use the simple formula c = −M/n. This is straightforward and requires no variance/weight data. If the problem gives weights, you will need the weighted formula instead.
- After applying corrections, verify by summing the adjusted observations and confirming they equal the required total. This takes 10 seconds and catches ~80% of arithmetic errors. Always do this verification step.
- If a problem mentions 'spherical triangle' or gives a large area (km²), you must include spherical excess ε in the required angle sum. Defaulting to 180° is wrong. Check the problem statement carefully.
Comparison tables
Condition Types and Required Sums
| Condition Type | Number of Observations | Required Sum | Common Misclosure Range | Adjustment Method |
|---|---|---|---|---|
| Plane Triangle | 3 angles | 180°00'00″ | ±10″ (typical) | Equal-weight (most common) or weighted by angle precision |
| Spherical Triangle | 3 angles | 180° + ε | ±10″ (after subtracting ε) | Equal-weight or weighted; ε must be pre-calculated |
| Horizon (Station) | n ≥ 3 angles | 360°00'00″ | ±15″ (typical) | Equal-weight: divide by n angles |
| Closed Traverse | Multiple courses | ΣΔx = 0, ΣΔy = 0 | 0.05–0.20 m (depends on perimeter) | Proportional to course length (linear distribution) |
| Level Loop | Multiple elevation diffs | ΣΔh = 0 | 0.01–0.05 m (depends on distance) | Proportional to sight distance or number of setups |
Equal Weight vs. Weighted Adjustment
| Aspect | Equal Weight | Weighted (Unequal Precision) |
|---|---|---|
| When to use | All observations have same precision; no weights given | Observations have different standard deviations or explicit weights; precision varies |
| Formula for correction | c = −(Misclosure) / n | c_i = −(Misclosure) × (σ_i² / Σσ_j²) or c_i = −(Misclosure) × (w_i / Σw_j) |
| Correction magnitude | Identical for all observations | Varies: weaker observations (higher variance) get larger corrections |
| Exam difficulty | Straightforward; most common problem type | Requires variance or weight values; more calculation |
| Physical interpretation | All measurements equally reliable; distribute error equally | Some measurements more trustworthy; less correction to reliable observations |
Common Exam Mistakes and Fixes
| Mistake | Why It Happens | Correct Approach | Example Impact |
|---|---|---|---|
| Wrong sign for correction | Forgetting to negate the misclosure | Always: c = −(Misclosure) / n. If misclosure +12″, correction is −12″/3 = −4″ per angle. | If original angle is 60°00'10″ + (−4″) = 60°00'06″. With wrong sign: 60°00'14″ (makes angle larger, wrong direction). |
| Forgetting spherical excess ε | Confusing plane and spherical triangles; ε is small and easy to overlook | For large triangles (sides > 1 km), ALWAYS check if excess must be included. Required sum is 180° + ε, not 180°. | ε = 5″; observed sum = 180°00'18″. Misclosure = 180°00'18″ − (180°00'05″) = 13″, not 18″. |
| Dividing by wrong number in equal-weight | Confusing number of observations with something else (perimeter, angle count) | Count the observations being corrected. For 3 triangle angles: divide by 3. For 4 station angles: divide by 4. | Three angles, misclosure +9″ → correction per angle is −9″/3 = −3″, NOT −9″/4 or other value. |
| Not verifying the result | Rushing; assuming calculation is correct without checking | ALWAYS verify: Sum of adjusted values must equal the required total (180°, 360°, ε, 0, etc.). | If adjusted angles sum to 180°00'02″ instead of 180°00'00″, there is an error. Go back and recheck. |
| Confusing horizon (360°) with triangle (180°) | Both are common conditions; easy to mix up under exam stress | Horizon = angles around ONE POINT = 360°. Triangle = three angles of a polygon = 180°. Read problem carefully. | Four angles at a station sum to 360°00'08″ → misclosure +8″. (NOT 180° condition.) |
| Mixing variance and weight | Forgetting that w = 1/σ², so larger σ → smaller w, but correction c ∝ σ² | Higher variance = less reliable = LARGER correction. If σ_B = 2σ_A, then σ_B² = 4σ_A² → c_B = 4 c_A. | Observation A (σ = 1″) and B (σ = 2″) with misclosure +12″ over 2 obs. c_A = −12″ × (1/5) = −2.4″, c_B = −12″ × (4/5) = −9.6″. |
| Linear misclosure confusion in traverse | Adding ΣΔx + ΣΔy instead of using Pythagorean theorem; treating as 1D instead of 2D | Traverse misclosure is a VECTOR. M = √((ΣΔx)² + (ΣΔy)²). Always use distance formula, not sum. | ΣΔx = +0.12 m, ΣΔy = +0.09 m → M = √(0.12² + 0.09²) = √(0.0225) = 0.15 m. (NOT 0.21 m from adding.) |
Diagrams
Condition Equation Adjustment Workflow
Complete step-by-step flowchart for solving condition equation and figure adjustment problems. Covers condition identification, misclosure calculation, correction method selection (equal vs. weighted), and verification.
Condition Equations: Concept Map
Hierarchical mind map showing the relationship between condition types, misclosure calculation, adjustment methods (equal vs. weighted), and verification steps. Useful for understanding the overall structure of the topic.