Survey Calculations

Overview#

Survey calculations form the mathematical backbone of every survey deliverable. Whether computing the position of a boundary corner, adjusting a traverse, calculating an area for a deed description, or laying out a highway curve, the surveyor must command a toolkit of computational methods and understand when each applies.

These calculations bridge the gap between raw field observations and the coordinates, areas, and geometries that appear on maps and in legal documents. A surveyor who understands the underlying mathematics -- not just which button to push in the software -- can verify results, catch errors, and solve problems that software alone cannot.

The surveyor who understands the math behind the software is the one who catches the errors the software cannot. Blind trust in software output is not professional practice.

Coordinate Geometry (COGO)#

Coordinate geometry -- universally called COGO -- is the collection of mathematical operations used to compute positions, directions, and distances from coordinate data. COGO is the daily language of the survey office.

Inverse Computation

The inverse computes the azimuth and horizontal distance between two known coordinate pairs.

Given two points $P_1(N_1, E_1)$ and $P_2(N_2, E_2)$:

$\Delta N = N_2 - N_1$ $\Delta E = E_2 - E_1$

$\text{Distance} = \sqrt{(\Delta N)^2 + (\Delta E)^2}$

$\text{Azimuth} = \arctan\left(\frac{\Delta E}{\Delta N}\right)$

The arctangent must be adjusted to the correct quadrant:

The inverse is arguably the most frequently used COGO function. It is essential for computing bearings and distances for legal descriptions, checking field measurements against computed values, and verifying map data.

Traverse Computation

A traverse computes successive coordinates from a starting point using measured angles and distances. Given a starting point $P_1(N_1, E_1)$ and a series of azimuths and distances:

$N_{i+1} = N_i + D_i \times \cos(\text{Az}_i)$ $E_{i+1} = E_i + D_i \times \sin(\text{Az}_i)$

where $D_i$ is the horizontal distance and $\text{Az}_i$ is the azimuth of the $i$-th course.

Intersection Computations

Intersections compute the position of an unknown point from known data. The three primary types:

Direction-Direction Intersection -- Given two known points and the directions from each toward the unknown point, the intersection computes the unknown position. Requires two azimuths from two known points.

Distance-Distance Intersection -- Given two known points and the distances from each to the unknown point. This produces two possible solutions (one on each side of the baseline); the surveyor must select the correct one based on field conditions.

Direction-Distance Intersection -- A hybrid: one direction from a known point and one distance from another. Also produces two possible solutions in some configurations.

Resection

A resection computes the position of an unknown occupied point by observing angles to three or more known points. This is a powerful field technique for establishing instrument positions without occupying control points. The mathematical solution involves solving a system of equations based on the observed angles and the known coordinates.

Beware the "danger circle." If the occupied point and the three known points all lie on (or near) the same circle, the resection solution becomes indeterminate. Always check the geometry before relying on a resection.

Perpendicular and Offset Computations

Common office calculations that support fieldwork and mapping:

• Perpendicular offset -- Distance from a point to a line, plus the station along the line where the perpendicular meets
• Point on line at distance -- Given a line and a distance along it, compute the coordinate
• Parallel offset -- Compute a line parallel to a given line at a specified offset distance

Traverse Adjustment#

Error of Closure

After computing a closed traverse (one that returns to its starting point or connects between two known points), the closure error reveals the accumulated errors in the measurements:

$\text{Linear error of closure} = \sqrt{(\Delta N_{\text{error}})^2 + (\Delta E_{\text{error}})^2}$

$\text{Relative error of closure} = \frac{\text{Linear error}}{\text{Total traverse length}} = \frac{1}{n}$

where $n$ is the denominator of the relative closure ratio. For example, a linear error of 0.05 ft over a 5,000 ft traverse gives a relative closure of 1:100,000.

Minimum closure standards depend on the survey purpose:

Compass Rule (Bowditch)

The compass rule distributes the closure error proportionally to each course based on its length. It assumes that errors in angles and distances are equally probable and is the most commonly used adjustment method for property surveys:

$\text{Correction}_N = -\Delta N_{\text{error}} \times \frac{D_i}{D_{\text{total}}}$

$\text{Correction}_E = -\Delta E_{\text{error}} \times \frac{D_i}{D_{\text{total}}}$

where $D_i$ is the length of the $i$-th course and $D_{\text{total}}$ is the total traverse length.

Transit Rule

The transit rule distributes the closure error proportionally to the absolute values of the latitude and departure of each course:

$\text{Correction}_N = -\Delta N_{\text{error}} \times \frac{|L_i|}{\sum |L_i|}$

$\text{Correction}_E = -\Delta E_{\text{error}} \times \frac{|D_i|}{\sum |D_i|}$

where $L_i$ and $D_i$ are the latitude (northing component) and departure (easting component) of the $i$-th course. The transit rule is less commonly used than the compass rule but is sometimes applied when distances are measured with higher precision than angles.

Least Squares Adjustment

Least squares is the most rigorous adjustment method. It uses all redundant observations simultaneously to find the most probable values of the unknown coordinates by minimizing the sum of weighted squared residuals:

$\mathbf{v}^T \mathbf{W} \mathbf{v} = \text{minimum}$

where $\mathbf{v}$ is the vector of residuals and $\mathbf{W}$ is the weight matrix (derived from the estimated precision of each observation).

Advantages of least squares over compass/transit rule:

• Incorporates the precision of each measurement through weighting
• Provides statistical measures of the quality of the adjusted coordinates (error ellipses, confidence intervals)
• Can combine different observation types (angles, distances, GPS baselines, bearings) in a single adjustment
• Detects blunders through residual analysis

Least squares is the standard of practice for any survey of consequence. The compass rule remains useful for quick checks and simple traverses, but it does not provide the statistical rigor or blunder detection that least squares offers.

Area Calculations#

Coordinate Method

The coordinate method (also called the shoelace formula) computes the area of a polygon from the coordinates of its vertices. For a polygon with vertices $(N_1, E_1), (N_2, E_2), \ldots, (N_n, E_n)$ listed in order:

$2A = \sum_{i=1}^{n} (E_i \times N_{i+1} - E_{i+1} \times N_i)$

where the subscript $n+1$ wraps back to subscript 1. The result is positive for counterclockwise vertex ordering and negative for clockwise. The absolute value gives the area.

This method is exact for straight-sided polygons and is the standard computational approach in survey software.

Double Meridian Distance (DMD)

The DMD method is a traditional hand-computation technique that produces the same result as the coordinate method. The DMD of a course is defined as the sum of the departures of the preceding course, the current course's own departure, and the departure used to compute the previous DMD. The area is:

$2A = \sum_{i=1}^{n} \text{DMD}_i \times L_i$

where $L_i$ is the latitude (northing component) of the $i$-th course.

Area with Curves

When a boundary includes circular arcs, the area of the circular segment must be added to or subtracted from the straight-line (chord) polygon area:

$A_{\text{segment}} = \frac{R^2}{2}(\Delta - \sin\Delta)$

where $R$ is the radius and $\Delta$ is the central angle in radians. The segment area is added when the arc is on the exterior of the chord polygon and subtracted when it is on the interior.

Unit Conversions

Curve Calculations#

Circular Curve Elements

Circular curves are the most common curves in surveying, used for roadways, boundaries along rights-of-way, and property lines adjacent to curves. The key elements are:

Degree of Curve

Two definitions are in common use:

Arc definition (used by highway agencies): The degree of curve $D$ is the central angle subtended by a 100-foot arc:

$R = \frac{5729.578}{D}$

Chord definition (used by railroads): The degree of curve $D$ is the central angle subtended by a 100-foot chord:

$R = \frac{50}{\sin(D/2)}$

Always confirm which definition is being used before performing curve calculations. The results differ, and using the wrong definition introduces systematic error.

Curve Layout

To lay out points along a circular curve, deflection angles from the tangent are computed:

$\delta_i = \frac{l_i}{2R} \times \frac{180}{\pi}$

where $l_i$ is the arc length from the PC (Point of Curvature) to the $i$-th point, and $\delta_i$ is the deflection angle in degrees.

Spiral Curves

Spiral (transition) curves provide a gradual change from a tangent to a circular curve. The most common spiral is the clothoid (Euler spiral), where the radius decreases linearly with distance along the spiral:

$R \times L_s = A^2$

where $L_s$ is the length along the spiral and $A$ is the spiral parameter. Key spiral elements include:

Vertical Curve Calculations#

Vertical curves connect grades (slopes) along a roadway profile. The standard vertical curve is a parabola, which provides a constant rate of change of grade.

Key Elements

Given two grades $g_1$ (incoming) and $g_2$ (outgoing), expressed as percentages, and the curve length $L$:

$A = g_2 - g_1 \quad \text{(algebraic difference of grades)}$

The elevation at any station along the curve:

$y = y_{\text{BVC}} + g_1 \times x + \frac{A}{200L} \times x^2$

where $x$ is the horizontal distance from the BVC (Beginning of Vertical Curve) and $y$ is the elevation.

High/Low Point

The station of the high or low point on a vertical curve:

$x_{\text{turn}} = \frac{-g_1 \times L}{A}$

This point is critical for drainage design and sight distance calculations.

Types

• Crest curve -- Connects a positive grade to a less positive (or negative) grade. $A < 0$.
• Sag curve -- Connects a negative grade to a less negative (or positive) grade. $A > 0$.

Volume Calculations#

Average End Area Method

The average end area method computes the volume of earthwork between two cross-sections:

$V = \frac{(A_1 + A_2)}{2} \times L$

where $A_1$ and $A_2$ are the cross-section areas at each end, and $L$ is the distance between sections. This method is simple and widely used, but it overestimates volume when the two end areas differ significantly.

Prismoidal Formula

The prismoidal formula provides a more accurate volume by incorporating a middle cross-section:

$V = \frac{L}{6}(A_1 + 4A_m + A_2)$

where $A_m$ is the area of the cross-section at the midpoint between the two end sections. This formula gives the exact volume for a prismoid (a solid with plane faces and parallel end planes).

Prismoidal Correction

The prismoidal correction converts the average end area volume to the prismoidal volume:

$C_p = \frac{L}{12}(D_1 - D_2)(W_1 - W_2)$

where $D$ and $W$ are representative dimensions (typically center height and width) of the two cross-sections. The corrected volume is:

$V_{\text{prismoidal}} = V_{\text{avg end area}} - C_p$

Borrow Pit (Grid) Method

For irregular excavations, the borrow pit method divides the site into a grid of squares or rectangles. The volume of each column is the average of its corner elevations multiplied by the column area. The total volume is the sum of all columns:

$V = A_{\text{cell}} \times \left(\frac{\sum h_1}{1} + \frac{\sum h_2}{2} + \frac{\sum h_3}{3} + \frac{\sum h_4}{4}\right)$

where $h_1$ are heights at corners shared by 1 cell, $h_2$ shared by 2 cells, etc., and $A_{\text{cell}}$ is the area of each grid cell.

Computation Best Practices#

Significant Figures and Rounding

• Coordinates -- Carry at least 4 decimal places during computation; round to 3 for final reporting (feet) or 4 (meters).
• Angles -- Carry to the nearest second during computation. Report to the nearest second unless project accuracy warrants otherwise.
• Areas -- Report to appropriate precision for the size of the parcel. Typically to the nearest square foot for small parcels, nearest 0.001 acre for moderate parcels, and nearest 0.01 acre for large parcels.
• Never round intermediate results. Carry full precision through all intermediate steps and round only the final answer.

Independent Checks

Every critical calculation should be verified by an independent method:

• Compute the area by both the coordinate method and DMD -- they must agree.
• Check traverse closures both before and after adjustment.
• Verify curve computations by computing from both the PC and PT and confirming they produce the same PI.
• For volumes, compare results from different methods to confirm reasonableness.

Documentation

• Record all input data, formulas used, intermediate results, and final answers.
• Identify the software and version used for calculations.
• Note the datum, units, and coordinate system.
• Keep computation sheets (paper or digital) as permanent project records.

Key Takeaways#

• COGO is the surveyor's essential toolkit. The inverse, traverse, intersection, and resection computations are used daily and must be thoroughly understood.
• Traverse adjustment distributes errors according to defined rules. The compass rule is standard for property surveys; least squares is the most rigorous method and provides statistical quality measures.
• Area calculations require precision. The coordinate method is standard; always account for curved boundaries separately.
• Curve computations underpin roadway design and right-of-way surveys. Know the elements, the formulas, and the difference between arc and chord definition of degree of curve.
• Volume calculations serve earthwork estimation. The average end area method is standard practice; the prismoidal formula is more accurate when end areas differ significantly.
• Never round intermediate results. Carry full precision through all steps.
• Always verify calculations by an independent method. Software errors, input errors, and blunders are caught only through independent checks.

References#

1. Ghilani, C.D. & Wolf, P.R. Elementary Surveying: An Introduction to Geomatics (13th Ed.). Pearson, 2012. Chapters 9--11, 15, 17, 24--26.
2. Kavanagh, B.F. & Mastin, T.B. Surveying: Principles and Applications (9th Ed.). Pearson, 2014. Chapters 6, 11, 14.
3. Meyer, T.H. Introduction to Geometrical and Physical Geodesy. ESRI Press, 2010.
4. Mikhail, E.M. & Ackermann, F. Observations and Least Squares. IEP, 1976.
5. Davis, R.E., Foote, F.S., Anderson, J.M. & Mikhail, E.M. Surveying: Theory and Practice (6th Ed.). McGraw-Hill, 1981.
6. AASHTO. A Policy on Geometric Design of Highways and Streets (7th Ed.). American Association of State Highway and Transportation Officials, 2018.