H. Ground to Grid Conversion
1. Grid Distances
a. General
In the Horizontal Datum topic, we defined three Earth surfaces and their height relationships, Figure H1.
Figure H1 Surfaces and Heights

Adding a grid introduces another surface, Figure H2.
Figure H2 Inserting a Grid Surface 
To convert a ground distance to grid, Figure H3, requires going from the horizontal measurement on the physical earth, D_{H}, to the ellipsoid, D_{E}, then to the grid, D_{G}.
Figure H3 Ground Distance to Grid Distance 
D_{E} is a geodetic distance since it is on the ellipsoid's surface.
b. Ground to Ellipsoid
Going from ground to the ellipsoid, Figure H4, is done using a proportion, Equation H1.
Figure H4 Ground to Ellipsoid 
Equation H1 
R_{E} is the ellipsoid radius. The ellipsoid has different radii depending on a line's direction and location. Because a survey line covers such a small part of the ellipsoid, its radius can be assumed constant. 6,372,200 m or 20,609,000 ft is typically used for R_{E}. Section 3 discusses discusses different radii and their effect on distance reduction.
The ground distance, D_{H}, is a straight line so D_{E} computed from Equation H1 is a chord. The geodetic distance should be along the ellipsoid's surface, D_{E'}. The difference between the two isn't significant: the chord distance for a 20,000.0000 ft geodetic distance is 19,999.9992 ft. The distance from Equation H1 can be used directly as the geodetic distance.
If there is a significant elevation difference between ground points and they are far enough apart, the horizontal distance between them differs based on which point it is referenced to. In Figure H5, the distance at point B's elevation is longer than it is at point A's.
Figure H5 Average Elevation 
To reduce the distance to the ellipsoid, the line's average elevation is used so Equation H1 is modified to Equation H2.
Equation H2 
The term inside the brackets is called the Elevation Factor, EF, since it accounts for the orthometric and geoid heights.
The geoid doesn't vary much over a line's length so the geoid height at either end, or an average, can be used. Geoid heights over larger areas can be determined using NGS's GEOID18 Computation Tool.
The geodetic distance is independent of the grid system projection. The projection only matters when going from the ellipsoid to the grid.
c. Ellipsoid to Grid
The geodetic distance is multiplied by the scale factor to obtain the grid distance, Equation H3.
Equation H3 
k  scale factor (aka, grid scale)
Scale is a function of the grid system location, as discussed in Chapter D. It is either computed using projection equations or obtained from the grid attributes at a control station. The latter is covered in Chapter I.
For short lines, the grid scale at either end can be used. Where the scales different significantly, their average, or weighted average (Simpson's Rule), Equation H4, can be used.
Equation H4 
Because the ellipsoid isn't spherical, midpoint scale, k_{m}, is not the average of the scales at both its ends.
d. Combined Factor
The Elevation Factor and scale can be multiplied to create the Combined Factor, CF, Equation H5. Each ground distance can be multiplied by the CF to go directly to grid, Equation H6.
Equation H5  
Equation H6 
Because they use average values, Equations H5 and H6 work best over smaller project areas where elevation changes are not excessive.