WGS84 and Projections: The Mathematics of Global to Local Mapping

Overview

Every GPS measurement is initially calculated in a global coordinate system called WGS84 (World Geodetic System 1984). However, engineering designs are typically created on flat, rectangular site grids. Mastering the gap between the curved ellipsoidal world of WGS84 and the flat projected world of the construction site is fundamental to modern geomatics 14, 15.

Why This Matters

If you treat WGS84 coordinates as if they were on a flat plane, the Earth's curvature will introduce massive distortions over just a few kilometres. Furthermore, WGS84 coordinates refer to a mathematical surface (the ellipsoid) that can be up to $100 \text{ m}$ away from the physical "sea level" surface (the geoid) 16, 17. Understanding this relationship is critical for accurate heighting and distance measurement 13.

Background

The WGS84 system is a geocentric datum, meaning its origin is at the Earth's centre of mass 15, 18. It was established using observations from roughly $1600$ sites globally and is the standard for all satellite positioning 18. For all practical engineering purposes, it is identical to the GRS80 ellipsoid 19.

Theory

1. Reference Ellipsoid: Since the physical Earth is too irregular to define mathematically, we use a smooth, symmetrical ellipsoid that best fits the Earth's shape 14.
2. Geodetic Coordinates: Position is defined by Latitude ($\phi$), Longitude ($\lambda$), and height above the ellipsoid ($h$) 20.
3. Projections: To create a map or site plan, we mathematically project these coordinates onto a flat surface (like a cylinder in the Transverse Mercator projection) 21.

Mathematical Principles

1. WGS84 Ellipsoid Parameters

The size and shape of the WGS84/GRS80 ellipsoid are defined by:

• Semi-major axis ($a$): $6,378,137.0 \text{ m}$ 19, 22
• Inverse Flattening ($1/f$): $298.257223563$ 19, 22

2. Cartesian $(X, Y, Z)$ to Geodetic $(\phi, \lambda, h)$

To convert satellite Cartesian coordinates to mapping coordinates: $\tan \lambda = \frac{Y}{X}$ 23$\tan \phi = \frac{Z + e^2 \nu \sin \phi}{(X^2 + Y^2)^{1/2}}$ 23$h = \frac{X}{\cos \phi \cos \lambda} - \nu$ 23(Note: $\nu$ is the radius of curvature in the prime vertical, and $e^2$ is the eccentricity squared) 23.

Field Workflow: WGS84 to Site Grid

Step-by-Step Example: Ellipsoid Elements

Problem: Given the semi-major axis ($a$) and flattening ($f$), find the semi-minor axis ($b$) and eccentricity squared ($e^2$).

1. Semi-minor axis ($b$):$f = \frac{a - b}{a} \implies b = a(1 - f)$ 16
2. Eccentricity squared ($e^2$):$e^2 = \frac{a^2 - b^2}{a^2}$ 17

Formula Breakdown

The Height Problem:$h = H + N$ 12 Where:

• $h$ = Ellipsoidal Height (from GPS) 30.
• $H$ = Orthometric Height (Elevation above Mean Sea Level) 30.
• $N$ = Geoid-Ellipsoid Separation 12, 30.
  In the UK, the WGS84 ellipsoid lies about $50 \text{ m}$below the geoid. Failing to account for this $N$ value will result in elevation errors of $50 \text{ m}$, even if your horizontal position is perfect 18.

Practical Tips

• Uniform Scale: For small engineering sites ($< 10 \text{ km}$), you can often use a single scale factor for the whole site 31.
• Coordinate Order: In mapping, the $X$-axis is Easting and the $Y$-axis is Northing, which is the reverse of pure mathematical convention 32.
• Check the Grid Convergence ($\gamma$): Grid North and True North are only the same on the Central Meridian. Ensure your Total Station orientation accounts for this 33, 34.

Common Mistakes

• Assuming Flatness: Neglecting the Earth's radius ($R \approx 6,380 \text{ km}$) in distance reductions 35.
• Sign Errors in Transformations: Helmert transformation parameters often have specific sign conventions for rotations. Always test with a known point 26.

FAQ

Conclusion

The WGS84 system provides the global foundation for all modern surveying. By rigorously applying geodetic formulas and understanding the transition from ellipsoid to plane, the surveyor ensures that satellite-derived data remains precise and useful for local engineering.

References

Schofield, W. (2001). Engineering Surveying. 5th ed. Butterworth-Heinemann.