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.
2. Grid Directions and Angles
a. Convergence Angle
The angle from Geodetic North to Grid North is the Convergence, γ. In NAD 27 systems, it was called the mapping angle. Along the CM, Geodetic and Grid North coincide so there is no convergence. Convergence magnitude increases with distance east or west from the CM.
On a conic projection, Figure H7, geodetic meridians (red) converge to the North Pole.
Figure H7 Conic Convergence 
The geodetic meridians on a transverse cylinder are more complex. On the grid they are curved lines converging to the North Pole, Figure H8. The closer to the pole, the greater their curvature.
Figure H8 Cylindric Convergence 
Convergence is positive (+) east of the CM, negative () west, Figure H9.
Figure H9 Math Sign 
b.Projected lines
A geodetic line on the ellipsoid doesn't project as a perfectly straight line on the grid but is slightly curved. The curvature amount depends on the line's orientation. On a conic projection, Figure H10(a), it is most pronounced for eastwest lines and is concave toward the Central Parallel. On a transverse cylindric projection, the curvature is largest for northsouth lines, Figure H10(b), and is concave toward the Central Meridian.
(a) Conic Projection 
(b) Transverse Cylindric Projection 
Figure H10 Projected Lines 
Geodetic meridians, being geodetic lines, also project slightly curved Figure H11 shows a highly exaggerated depiction of the angle and direction relationships between straight and projected lines.
Figure H11 Projected Line Relationships 
Geodetic north (solid red) on the grid is tangent to the projected meridian (dashed red) at the user's position, point C. When the observer sights point D, his line of sight (dashed green) is tangent to the projected line CD (dashed black). The angle between the line of sight and the straight chord CD is the arctochord correction, δ.
Other elements in Figure H11 are
T  grid azimuth of the projected line of sight
t  grid azimuth of the chord
α  geodetic azimuth of the projected line of sight
Equations H7 and H8 are the relationships between the various angles and azimuths.
Equation H7  
Equation H8 
The arctochord correction is also called the second term or (tT) correction.
Equations H9 and H10 are arctochord correction equations for conic projections.
Equation H9  
Equation H10 
Either can be used although Equation H9 is set up for metric units.
Equations H11 and H12 are for transverse cylindric projections.
Equation H11  
Equation H12 
Either can be used although Equation H11 is set up for metric units.
In the equations:
δ_{12}  Correction for line 1 to 2; seconds 
N_{1}, E_{1}  Coordinates of from point 1 
N_{2}, E_{2}  Coordinates of to point 2 
N_{o}  Northing at intersection of CM and Central Parallel 
r_{o}  Mean radius at origin latitude 
E_{o}  CM Easting 
The Central Parallel of a conic projection is not the same as the Origin Parallel nor is it midway between North and South Standard Parallels. Its northing, N_{o}, is a constant but it must be computed from a zone's parameters.
Approximate coordinates can be used for both points in all four equations.
The mathematical sign on the correction indicates its direction: positive (+) is an angle to the right, negative () is an angle to the left.
If arctochord corrections are significant they should be applied to the backsight and foresight lines of measured horizontal angles, Figure H12, using Equation H13,
Figure H12 Corrected Angle 
Equation H13 
where:
β_{H}  measured horizontal angle 
β_{G}  grid angle 
δ_{BS}  backsight arctochord correction 
δ_{FS}  foresight arctochord correction 
For most surveys,the corrections are generally smaller than angle measurement accuracy. To determine if they should be applied, compute the angle correction for a survey's the worstcase scenario; if it isn't significant, they can be ignored for all the angles. If that's the case, Equation H7 becomes:
Equation H14 
c. Forward and Back Directions
Because grid meridians are parallel, the forward and back grid azimuth of a line, Figure H13, differ by exactly 180°, Equation H15.
Figure H13 Forward and Back Grid Azimuths 
Equation H15 
The line's forward and back geodetic azimuths, Figure H14 , must account for the convergence difference at each end, Equation H16.
Figure H14 Forward and Back Geodetic Azimuths 
Equation H16 
3. Ellipsoidal Radii
Equation H1 requires a radius to reduce ground distance to geodetic. Because the ellipsoid is not a sphere, it does not have a single uniform radius. At each point on the ellipsoid, there are (at least) three different radii. These are dependent on the point's ellipsoid location.
a. Normal, N
The normal is the line perpendicular to the ellipsoid surface at the observer's position, Figure H15.
Figure H15 Normal 
It is in the observer's meridian and extends to the semiminor (polar) axis. The angle from the semimajor axis to the normal is the geodetic latitude, Φ.
The length of the normal varies between a at 0° lat and b at ±90° lat, Equation H17
Equation H17 
b. Meridional Radius, M
The shorter an ellipsoidal arc segment, the closer it comes to being a circular arc. The meridional radius, M, is the radius of an infinitesimally short arc in the meridian at the observer's position, Figure H16.
Figure H16 Radius in Meridian 
Like the Normal, its length depends on geodetic latitude, Equation H18.
Equation H18 
c. Radius in an Azimuth, R_{α}
A meridian is an ellipse with the same semimajor and minor axes as the reference ellipsoid. Connecting two points A and B not in the same meridian will create a new ellipse. If the points are at
 the same latitude, the ellipse will be at a 90° azimuth to A's and B's meridians and form a circle of fixed radius.
 different latitudes, the ellipse will be at an azimuth other than 90° (or 270°) and have semimajor and minor axes different than the reference ellipsoid.
The radius in the direction of line AB is similar to a meridional radius except it is for a section along the new ellipse, Figure H17 and Equations H19 and H20.
Figure H17 Radius in an Azimuth 
Equation H19  
Equation H20 
d. Radius R_{E}?
R_{E} is typically 20,906,000 ft or 6,372,200 m which is a mean value and not dependent on latitude or direction. How does geodetic reduction using this radius compare to the others?
An example can be used to illustrate. Given:
 Latitude of point A is 41°30'00"
 Azimuth and distance to point B are 55°45'00" N and 10,000.00 ft.
 Elevations of points A and B are 900.0 ft and 800.0 ft
 Area geoid height is 32.0 m
 GRS 80 ellipsoid a = 6,378,137.0 m, e = 0.08181 919104, e' = 0.08209 44382
Compute the radii
Compare to R_{E} = 20,906,000 ft
Radius 
Value 
diff 
diff 
R_{E} 
20,906,000 ft       
N  20,956,425.4 ft  +50,425.4 ft  +0.24% 
M 
20,877,499.8 ft  28,500.2 ft.  0.14% 
R_{α} 
20,931,361.3 ft  +25,361.3 ft  +0.12% 
Those seem like substantial differences. How do they affect distance reduction to ellipsoid? Use each in Equation H1 to determine the geodetic length of line AB and compare them to using R_{E}.
Radius 
D_{E} 
diff 
R_{E} 
9,9999.6436 ft    
N  9,9999.6445 ft  +0.0009 ft 
M 
9,9999.6432 ft  0.0004 ft 
R_{α}  9,9999.6441 ft  +0.0005 ft 
Despite the large radii variation, there is no significant effect on distance reduction. Even with a ground distance of 20,000 ft, the maximum difference is only 0.002 ft.
This basic analysis shows that 20,906,000 ft can be used for grid to geodetic distance reduction without affecting accuracy.
4. Reduction Summary
If it seems like there are a lot of computations involved, well, there can be. Measurements are distorted going from ground to grid and these computations are used minimize their impact.
Depending on the project scope or accuracy needs, some generalizations can be made to reduce computations. For example, it may be possible to use a single Combined Factor for distance reductions or arctochord corrections may be small enough to ignore. On the other hand, a control quality survey may require each line and direction be individually reduced.
Computations can also flow in the other direction: grid to ground. These are largely a reversal of the of the procedures described in this chapter and might be generalized based on project needs.