10.3 Coordinate Systems, Grid vs. Ground, and Plan Orientation

10.3 Coordinate Systems, Grid vs. Ground, and Plan Orientation

Land surveying bridges the physical curved surface of the Earth (the ellipsoid/geoid) and flat two-dimensional map representations (CAD drawings and engineering plans). To depict large land areas accurately without noticeable distortion, surveyors utilize mathematically defined coordinate projections, most notably the State Plane Coordinate System (SPCS). Understanding the mathematical relationship between flat grid coordinates and actual ground measurements is a core requirement for CST survey technicians.

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State Plane Coordinate System (SPCS) Basics

The State Plane Coordinate System (SPCS) was developed in the 1930s by the U.S. Coast and Geodetic Survey (now NGS) to provide a localized, high-accuracy rectangular coordinate system (Northings and Eastings). By dividing states into specific zones, projection distortion is limited to less than 1 part in 10,000.

Conformal Map Projections in SPCS

SPCS uses two main conformal map projections based on zone geometry:

1. Lambert Conformal Conic Projection: Used for states or zones that have a primary geographic extent in the East-West direction (e.g., Tennessee, Pennsylvania, Washington, North Carolina). The projection surface is a secant cone intersecting the ellipsoid along two standard parallels of latitude.
2. Transverse Mercator Projection: Used for states or zones extending primarily North-South direction (e.g., Indiana, Illinois, Florida, New Jersey). The projection surface is a secant cylinder intersecting the ellipsoid along two lines parallel to the central meridian.
3. Hotine Oblique Mercator Projection: Used for zones aligned obliquely, such as the Alaska panhandle.

Geodetic Datums & Realizations

• NAD27 (North American Datum of 1927): Based on the Clarke 1866 ellipsoid with origin at Meades Ranch, Kansas. Units in U.S. Survey Feet.
• NAD83 (North American Datum of 1983): Geocentric datum based on the GRS80 ellipsoid. Modern realizations include NAD83(2011) Epoch 2010.00. Units in Meters or Feet.
• NATRF2022 / Modernized NSRS: Terrestrial reference frames introduced by NGS to replace NAD83 using GNSS-based origin.

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Grid North vs. True North vs. Magnetic North

A common source of error on survey plans is confusing the three distinct definitions of "North":

1. True North (Geodetic North): The direction along a geographic meridian toward the Earth's geographic North Pole (rotational axis).
2. Grid North: The direction of grid lines running parallel to the central meridian of a specific State Plane zone. Grid North matches True North only along the zone's central meridian.
3. Magnetic North: The direction indicated by a magnetic compass needle pointing toward the Earth's magnetic north pole. Magnetic north changes over time (secular variation) and fluctuates daily.

                    Grid North   True North
                        │          / 
                        │         /  
                        │  γ     /   
                        │ angle /    
                        │      /     
                        │     /      
                        └────/─────── (Central Meridian)

Mapping Concepts

• Mapping Convergence Angle ($\gamma$): The angular difference between True North and Grid North at a given site point. If a site lies East of the central meridian, Grid North lies East of True North.
• Magnetic Declination: The horizontal angle between True North and Magnetic North at a specific date and geographic location.

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Combined Scale Factor (CSF) & Grid vs. Ground Conversions

Field measurements (total station EDM distances or tape measurements) occur on the physical surface of the Earth (Ground Distances). However, CAD drawings constructed in State Plane coordinates exist on a mathematical projection surface (Grid Distances). Converting between Grid and Ground requires the Combined Scale Factor (CSF).

$\text{Combined Scale Factor (CSF)} = \text{Grid Scale Factor } (k) \times \text{Elevation Factor } (EF)$

1. Grid Scale Factor ($k$)

Accounts for distortion when projecting the curved ellipsoid onto the flat map plane. $k = 1.000000$ along standard parallels/meridians, $k < 1.000000$ between secant lines, and $k > 1.000000$ outside secant lines.

2. Elevation Factor ($EF$)

Reduces physical ground distances measured at an elevation ($h$) above sea level down to the mean ellipsoid surface:

$EF = \frac{R_m}{R_m + h}$

Where $R_m$ is the mean radius of the Earth (approximately $20,906,000 \text{ feet}$ or $6,371,000 \text{ meters}$), and $h$ is the ellipsoidal height (or orthometric elevation).

Grid / Ground Conversion Formulas

To convert a physical Ground Distance measured in the field to a flat Grid Distance in CAD:

$\text{Grid Distance} = \text{Ground Distance} \times \text{CSF}$

To convert a CAD Grid Distance back to an actual physical Ground Distance for staking in the field:

$\text{Ground Distance} = \frac{\text{Grid Distance}}{\text{CSF}}$

Numerical Example Calculation

Given:

• Measured Ground Distance $= 1,500.00 \text{ feet}$
• Grid Scale Factor $k = 0.999920$
• Elevation Factor $EF = 0.999940$

Step 1: Compute Combined Scale Factor (CSF) $\text{CSF} = 0.999920 \times 0.999940 = 0.999860$

Step 2: Compute Grid Distance for CAD $\text{Grid Distance} = 1,500.00 \times 0.999860 = 1,499.79 \text{ feet}$

Step 3: Convert Grid Distance of $800.00 \text{ feet}$ back to Ground Distance $\text{Ground Distance} = \frac{800.00}{0.999860} = 800.11 \text{ feet}$

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Basis of Bearings Statement on Survey Plans

Every professional boundary plat must include an explicit Basis of Bearings statement in the General Notes. Bearings are relative angular directions; without specifying the reference meridian, a boundary survey cannot be legally reconstructed.

Standard Types of Basis of Bearings:

1. State Plane Grid North: Tied directly to an established geodetic datum and zone (e.g., "Bearings shown hereon are referenced to Grid North of the Washington State Plane Coordinate System, South Zone, NAD83 (2011) Epoch 2010.00, derived from GNSS observations").
2. True Astronomical North: Established by solar or Polaris observations.
3. Deed / Record Bearing: Based on a historical recorded plat or prior deed survey line (e.g., "Bearings are based on the East line of Lot 4 holding a record bearing of N 01° 15' 00" E per Deed Book 150, Page 45").
4. Assumed North: An arbitrary meridian assigned to a baseline (e.g., "Bearings are assumed based on the centerline of Main Street assigned a direction of Due North"). Note: Assumed bearings are generally restricted in modern professional boundary practice.

Elements of a Legal Basis of Bearings Note:

• Reference Meridian System (Grid, True, Record, or Assumed).
• Geodetic Datum and Epoch (e.g., NAD83 2011 Epoch 2010.00).
• SPCS State and Zone Name.
• Identified physical baseline monuments (e.g., "between found control monuments #101 and #102").