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Map Projections
And State Plane Coordinate Systems
By
Thomas Maxell Beer, PLS
Map Projections and State Plane Coordinate Systems
A Sun Cam online continuing education course
Introduction
The origin for the modern State Plane Coordinate System in the United States began with two
highway engineers from North Carolina. In 1932, George Syme and O.B. Bester contacted Dr.
Oscar. S. Adams of the United States Coast and Geodetic Survey, about an idea they had about
initiating a plane grid system for use in North Carolina. Being highway engineers, they were
well aware of the problems of using 2D plane surveying techniques over large areas of the
curved earth’s surface. Dr. Adams and his assistance, Charles Claire, thus began what is called
the State Plane Coordinate System (SPCS).
This lesson will focus on the history and rational of the use of the various Mapping Projections
and origin of SPCS systems in the United States. The lesson is not intended to be used as a
guide to performing all the necessary computations for use of any particular SPCS.
Cartesian Coordinates
The system of mathematics we call Cartesian coordinate system was derived from the mind of
the French Philosopher and Mathematician Rene' Descartes (Latinized form Renatus Cartesius).
Descartes lived from 1596 to 1650. He is considered to be the father of modern analytical
geometry that incorporates the study of geometry with the principles of algebra. This is widely
used in engineering today. Cartesian coordinates allow any point on a plane 2D surface to be
located by two numbers an X or Y coordinate. They allow any point in 3D space to be located
by three numbers being N X, Y and Z coordinate as illustrated below.
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Illustration of 3D Cartesian Coordinates
Notice that the origin point of the three number lines X, Y, and C have a common point of origin
with an intersection point of 0,0,0. The Y-axis represents a north-south direction in civil
engineering and surveying, the X-axis represents an east-west direction; the Z-axis represents an
elevation above a particular datum.
In a 2D Cartesian system a plane is divided into four sections or quadrants. By custom, they are
numbered clockwise with quadrant I being the northeast quadrant, II for the southeast quadrant,
III for the southwest and IV for the northwest quadrant.
In a 3D Cartesian system space is divided into eight octants. These coordinates depend on
where they are positioned in relation to the point of origin otherwise 0,0,0 as follows:
First octant (+, +, +), top-back-right (−, +, +), top-back-left (−, −, +), top-front-left (+, −, +),
bottom-front-left (+, −, −), bottom-back-left (−, −, −), bottom-back-right (−, +, −), bottom-frontright (+, +, −)
In regards to establishing point locations with SPCS, we are concerned with the northeast
quadrant only so the coordinates for both X and Y are positive numbers.
Euclidean distances (a straight line distance between two points) can be calculated between a set
of coordinates using formulas based on the Pythagorean Theorem: The sum of the areas of the
two squares on the legs (a and b) equals the area of the square on the hypotenuse (c).
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Distance for a two-dimensional coordinates
Distance for a three-dimensional coordinates
Angles from a 2 dimensional plane
Angles can be calculated between a set of coordinates using trigonometry rations for sin, tangent
and cosine.
The angle A of the hypotenuse then derived from the process of inversing using the arc functions
of arc-tan, arc-sin, or arc-cos.
Plane and Geodetic Surveying
Plane surveying is defined as a surveying system that assumes that the surface of the earth is a
perfectly flat plane. It is not concerned about making corrections for the curvature of the earth.
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Measurements are made in two dimensions. The theory assumes that in small areas any
distortion due to curvature is insignificant.
The Plane Surveying system relies on simple plane trigonometry. This is defined as the study of
triangles in the Euclidean plane including the use of functions defined by the ratios of sides of
right triangles.
In contrast, geodetic surveying is highly concerned with the aspect that the earth is curved.
Trigonometry deals with triangles on a sphere. Mathematically, it is a far more complex system
than the simpler mathematics used in ordinary plane surveying.
Triangle on Sphere
Unlike the earlier example of an ordinary simple straight line calculations in Euclidean
geometry, lines on a sphere are the shortest path from point A to point B are the shorter arc of a
great circle passing through points on A and points B.
A great circle is defined as a circle that cuts the earth into two equal halves. Any circles that are
not cutting the sphere into equal halves are called a small circle. These are not the shortest
distance between points. On a sphere, the sum of the angles of a triangle does not equal 180°. A
sphere is not a Euclidean space. It lies within a curved non Euclidean space.
Therefore, geodetic surveys are typically carried with great precision and require highly trained
personnel. They take into account the true shape of the Earth; they are used to determine the
size, shape, and gravity fields of the earth, and are necessary to develop engineering plans for
projects that cover great lengths and large areas.
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True Shape of the Earth
Sometime around 600 BC, ancient Greek philosophers belonging to the Pythagorean School of
Thought, followers began to postulate that the Earth was a sphere. It is a sphere, but it is not a
perfect sphere.
The Earth is flatter at the poles that at the equator. The reason for this is that the earth consists
of a molten core and it spins on its axis. Do to this rotation, the forces of angular momentum
created a bulge at the equatorial plane.
These forces of angular momentum cause the polar axis to be a shorter diameter than the
diameter of circle that bisects the equatorial plane. The Earth is thus referred as being an oblate
spheroid.
The earth has a radius of 6378.137 km (3,963 statute miles) along the equatorial major axis and a
radius of 6356.752 km (3,949 statute miles) length along the polar minor axis.
Earth Dimensions
Radius of sphere of equal area
6,370,997 meters
3,958 statute miles
Area of earth
510,900,000 sq. km
197,260,000 sq. miles
Equatorial Circumference
40,075 kilometers
24,899 miles
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Mass
The earth has no weight because it is a body in space that is in free fall due to its orbiting. This
24
is just like any other object in space. It does have a mass of 5.9736×10 kg.
Surface Elevations
Lowest point on dry land 422 meters (1385 ft) below sea level (Dead Sea).
Highest point 8,850 meters (29035 ft.) above sea level (Mt. Everest).
Lowest point is at bottom of the ocean at a depth of 10,916 meters (35,814ft) (Mariana Trench).
Datum For Earth Models
In order to create a mathematical model of a sphere, it requires a set of parameters that define a
coordinate system and includes control points we call a datum. The datum is an attempt to
approximate the true shape of the Earth.
Horizontal Datum NAD 27 and NAD 83
The two foremost map datums used in the United States and the rest of North America
are the NAD 27 Datum of 1927 and the more modern NAD 83 Datum of 1983.
The older NAD 27 is based on the Clark Spheroid of 1866. Alexander Clark developed it an
English geodetic scientist. It was one of the first model datums of the Earth that took into
accounting of the flattening of the polar region and the equatorial bulging previously discussed.
NAD 27 bases its geodetic center at a point in place in Kansas called Meaes Ranch. This point is
located at the approximate geographic center of the contiguous United States and served its
purpose as long as the system was only used in North America.
With the advent of the space age, there was a need to shift to a system that could be used
worldwide. NAD 27 would not support the GPS that we depend on so much today.
NAD 1983 is based on a newer defined spheroid that geocentric datum having no initial point or
initial direction.
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In 1980 The General Assembly of the International Union Of Geodesy and Geophysics (IUGG)
adopted the geodetic reference system (GRS 80) that was based on a 6,378,137-meter semimajor axis and a 6,356,752.31414-meter semi minor polar axis.
When comparing NAD 27 coordinates to NAD 83 a difference of up to nearly 100
Meters occur in the contiguous United States. There is no simple formula to calculate these
shifts since they vary widely from place to place.
The Nation Geodetic Survey offers a free program (NADCON) that computes the
difference in height between the two datum with reasonable results.
http://www.ngs.noaa.gov/PC_PROD/NADCON/
Vertical Datum NGVD 29 and NAVD 88
A vertical datum, in regards to surveying and geodesy, is a reference surface from which
measurement of elevations and depths are taken. The first attempts of true geodetic leveling in
the United States was based on sea level and occurred along the eastern seaboard in 1856.
The two most widely used vertical datum used in the United States are the NCVD 29 and NAVD
88 Datum.
The National Geodetic Vertical Datum 29 (NGVD29) standard was adopted in 1929. It is also
referred to as Sea Level Datum of 1929 or simply “mean sea level” on early quad maps.
The NGVD29 datum established an independent mean sea level elevation observation at 26 base
monuments, 21 in the United States and five along the Canadian Coastlines. By decree, it was
declared that mean sea level would be 0.00 feet at each of these local stations pending tidal
observations over a long period of time. That period is 19 years. The sun, earth, and moon,
come back into the same relative position every 19 years. This period reflects all of the
influences of gravity of the moon and the sun alignment that subsequently influence tides.
The problem with a system based on “mean sea level” is that no such thing exists. The sea is not
level; it varies from place to place. Besides gravity wind, barometric pressure, currents, the
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salinity of the water, and seabed topography are a few contributing factors that contribute to sea
levels.
The North American Vertical Datum (NAVD 88) was adopted in 1991 as a means of correcting
the inadequacies of the NGVD 29 system. It adopted a single primary benchmark reference as
its basis. This basis is known as the Great Lakes Datum of 1985. Its local mean sea level value
at a station at Father Point in located Rimouske, Quebec, Canada.
It is no simple matter to convert NGVD 29 elevations to NAVD 88 elevations. The conversion
factor varies from place to place and can be as much as three vertical feet.
The Nation Geodetic Survey offers a free program (VERTCON) that computes the difference in
height between the two datums. The program uses the Latitude and Longitude of a particular
point to compute reasonable results.
http://www.ngs.noaa.gov/PC_PROD/VERTCON/
Geographic Coordinates
Latitude is based on the equator being an equal distance between the North and South Poles. It
was given a value of 0 degrees. The poles are given a value of 90 degrees. Lines of the same
degrees are said to be parallel to the equator. For example, the dividing line between North and
South Korea is an arbitrary east west line along the 38th Parallel.
Lines in the Northern Hemisphere are given positive numbers and those in the Southern
Hemisphere are given negative numbers. Points are described in degrees, minutes and seconds.
Instead of a positive or negative numbers the letters N or S are most often used. Sometimes the
Greek letter (φ) phi is used.
Longitude is based on an arbitrary line running from pole to pole. Unlike latitude, there is no
point that can be naturally being assumed as a starting point. This starting point is referred to as
being the prime meridian. It is only an assumed line and is arbitrary in its location.
By international convention in 1884, the line used today runs through the Royal Observatory in
Greenwich England. All longitudinal lines or meridians are great circles. The Prime Meridian
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was given the value of 0 degrees. From the Prime Meridian, lines are designate as being +180°
eastward and −180° westward. Again the letters E and W are most often used instead of the
Greek letter λ (lambda).
Royal Observatory located at the Prime Meridian
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Accurate positioning on longitude is based on determining the difference from local time to that
of Greenwich time. This could be accomplished using a quadrant or sextant to determine local
noon based on an observation of the sun at its highest point and compare that time with
Greenwich time. These instruments were used to accurately determine latitude from observing
the sun or particular stars.
A Quadrant Circa 14th Century
The invention that made determining longitudes possible was the marine chronometer. Prior to
this navigators, depended on dead reckoning to keep track of longitude. This system was
inaccurate, to say the least. It depended on keeping track of a ship’s speed, time and direction,
and plotting position on a chart.
In 1714 the Longitude Act was passed in England by Parliament. The Act provided a monetary
prize for anyone who invented a simple means to accurately measure longitude. The prize was
eventually awarded to John Harrison, an English self-taught clockmaker, in 1773. His devise
incorporated a balance wheel attached to a spring that compensated for motion.
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Harrison’s Extremely Accurate Marine Chronometer H4
Prior to this invention, clocks with pendulums were useless at sea because of a ship’s movement.
Complex astronomical observations of heavenly bodies were the only means of accurately
determining longitude. Chronometers used during this time were set at precisely at 1:00 pm with
a ball dropping at the Naval Observatory located at 0 degrees longitude.
From combining both latitude and longitude there is a system that is called geographic
coordinates. These coordinates designate any point on the globe as illustrated below. For
example Washington DC has the coordinates of Latitude: 38°53' North, Longitude: 77°02' West.
The NADCON software also allows conversion of Geographic to NAD 27 and 83 systems.
Locating Geographic Coordinates on the Surface of the Globe.
Map Projections
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The ideal model of a map is a globe. It truly represents a map that is three-dimensional. When
trying to cast a 3D image on to a 2D plane, a distortion occurs like the case as plane surveying
vs. geodetic surveying. One can envision the problem by peeling an orange and placing the
remains on a flat surface. There would be no end to the differences number of outcomes based
how the orange was peeled. Therefore, there are many ways to project a 3D surface on a plane
surface. Each projection has particular advantages and disadvantages. There is no perfect way
to project the Earth’s surface other than a globe.
To various degrees, map projections are selected on the following four parameters:
1)
Area: If your main concern is that areas on the map are uniform and proportional in
scale, one should select a projection that would produce an equal area map.
Equal Area Projection
Examples of equal map areas are relatively proportional. There is no perfect solution other than
a globe to produce a perfect equal area map on a flat surface.
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2) Direction: A map that is used in navigation is mostly concerned with using azimuth to
correctly plot a course. These maps are accurate for this purpose only. They make a poor choice
for other purposes. The most famous example is the Mercator projection that originated in the
16th century and was used by early navigators.
Mercator Projection
Above is an example of a directional projected based on a Mercator projection? Note the gross
distortion of areas of Greenland and South America. They appear to be the same size, but in
reality Greenland is about a one eighth the size of South America. Mercator never intended his
map to be used for any other purpose that navigation. Compare the size ratio of Greenland and
South America with the equal area map shown above.
Map projections have even been used for propaganda purposes. For example, the upper latitude
countries of the former Soviet Union and Red China are grossly misrepresented as their true
areas in a Mercator projection. By using this projection, in addition using a vivid color of red, it
made these areas seem much more menacing during the Cold War. If the author of this map
wanted to adhere to the true, correct projection would be equal area projections with neutral
colors.
3) Distance/Scale: A map that preserves distance from the center of the projection to any other
point on map is refereed to being an equal-distant projection.
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4) Shape/Conformity: This projection concerns itself with the preservation of the shape. It is
used for topographic mapping and navigation. Its advantages are that every parallel line
intersects with every meridian at right angles, and that for any point; scale must be the same in
all directions. This projection is used primarily on smaller scaled maps. This feature is also
referred to as conformity. A map cannot be both equal area and conformal.
Although though there are numerous projections available for use, the State Plane Coordinate
system predominately use two. They are the Transverse Mercator projection and the Lambert
Conformal Conic projection.
Mercator Projection
The Mercator projection is a cylindrical projection of Earth's surface created in the mid 1500’s
by Gerhardus Mercator, a Flemish mathematician and cartographer. Gerard Mercator's parents
were Hubert and Emerentia Kremer. The name Kremer means a merchant in German.
Gerhardus Mercator is the Latinized version of his name.
The Mercator projection is to be considered as one of the most important nautical navigation
tools of all times. It made possible many the great explorations of the past 500 years and is still
used today and will be in the future.
Mercator projected the world on a cylinder with the parallels and meridians intersecting each
other at right angles. He stretched the distances of the parallels with the same factor as the
distances of the meridians. As a result, angular measures are preserved with this projection.
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Mercator wrote on his map “spread on a plane the surface of a sphere in such a way, that the
positions of all places shall correspond on all sides with each other, both in so far as true
direction and distance are concerned and as concerns true longitudes and latitudes.”
Straight lines on the Mercator projection are called rhomb or loxodrome line. They represent
lines of constant compass bearings that make this projection so useful for navigation and
surveying.
Mercator Map of the World 1569
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Transverse Mercator
The standard Mercator projection illustrated above has a cylindrical axis that is the same as the
Earth's axis. A variation of this projection has a cylindrical axis that is perpendicular to the
Earth’s axis. This projection is called transverse.
In this projection the point of tangency between cylinder and globe is a meridian, or line of
longitude. This meridian is called the central meridian. This projection is often used to map
areas with larger north-south extensions than east west. For example the state of Florida has
most of its landmass running from north to south.
Universal Transverse Mercator Zones in the United States
In the 1940’s the United States Army Corps of Engineers adopted a standard universal grid for
military use throughout the world called the Universal Transverse Mercator (UTM). The sphere
is divided into 60 zones running along meridians. Each zone covers a 6° area. They are
numbered from zone 1, located between West longitude 180°, and 174°. From here the zones
run eastward around the globe to zone 60. The zones for the contiguous United States are
numbered from zone 10 in the west to zone 19 in the east.
Lambert Conformal Conic Projection
Johann Heinrich Lambert was born in 1728 in an enclave of Switzerland. He was a
mathematician and cartographer. One of his achievements formulated hyperbolic functions into
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the field of trigonometry concerning triangle on concave surfaces. Lambert developed the
Lambert Conformal Conic projection in 1772.
This projection superimposes a cone over the sphere. The intersection line produces tworeference parallels secant to the globe and intersecting it. Instead of the references being
longitudinal, as in the case of the Transverse Mercator projection, they are associated with
latitudinal lines. This projection is often used to map areas with larger east-west extent than
north-south extent. For example, the state of Tennessee has most of its landmass running from
east to west.
The Lambert Conformal Conic map projection is one of the most widely used map projections in
the United States. The USGS uses this projection for many 7.5 and 15-minute topographic maps.
Conic Projection
State Plane Coordinate System (SPCS)
State Plane Coordinate systems were developed for specifying positions of geodetic stations
using plane rectangular coordinates for surveying and mapping. Zones were created based on
the premise that distortion less than 1 in 10,000 would occur. In order to accomplish this ratio,
the width of zone must be limited to 158 miles.
There are 124 separate number zones in the Unites States. Each zone has a code number.
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The main advantages of SPCS are that all surveys and mapping are permanently tied to a single
system so that no point can be lost. Simpler methods of plane geometry can be applied over large
statewide areas.
All SPCS systems use either the Transverse Mercator or the Lambert Conformal polyconic
projections with the exception of one unique zone in Alaska that uses an oblique Mercator
projection. This projection was chosen for the panhandle area because it lies at an angle.
State Plane Zones United States
Note SPCS zone boundaries running in from east to west like the five zones Texas are Lambert
Conformal Polyconic projections. Those running north to south in extense like the two zones in
Illinois are Transverse Mercator.
Florida has two zones that are Transverse Mercator and one zone covering it east to west extense
of the panhandle as being a Lambert Conformal. Some states like North Carolina and
Tennessee have only one zone. Alaska has the most with as many as ten zones.
SPCS systems employ three basic units of measurement for distances. These are the
International Foot, the US Survey Foot, and the meter. It depends in which state the SPCS is
located, as to whether the international foot or US Survey foot is used. For example by state
statute in North Carolina the unit is the US Survey Foot. In South Carolina it is the International
Foot and in Virginia it’s the US Survey foot. Even counties within Virginia use use the
International Foot!
Coordinate systems based on the more modern NAD 83 use the meter as the primary unit of
measurement. The International Foot was defined in 1959 as being 0.3048 meters. The US
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Survey Foot is defined as being 1200/3937 meters. Although the differences between the twofoot units are small, they can amount to a significant amount over a large statewide area.
It is also important to round off numbers to at least nine significant figures to avoid large errors.
For example one meter should equal 3.280833333 feet. It is also important not to mix
calculations from two different SPCS since they are not based on a common framework.
The following is taken from a letter written in January 1991 by the National Geographic Survey
regarding policy:
“In preparation for the adjustment of the North American Datum of 1983, 31 states enacted
legislation for the State Plane Coordinate System of 1983 (SPCS 83). All states defined SPCS 83
with metric parameters. Within the legislation, the U.S. Survey Foot was specified in 11 states
and the International Foot was specified in six states. In all other states the meter is the only
referenced unit of measure according to the SPCS 83 legislation. The remaining 19 states do not
yet have any legislation concerning SPCS 83”.
State Plane Coordinate System for North Carolina
As mentioned in the beginning of this lesson, the first state responsible for the initiation a SPCS
was North Carolina. It would be impracticable to examine all of the various state systems in this
short lesson, so let us begin with the first state to become involved.
North Carolina has most of its landmass in an east west direction. It is considerably narrow in its
north south direction. Therefore, the State of North Carolina best fits using the Lambert
Polyconic Conformal projection, because it is so narrow in the north south direction it only
requires one zone.
Referring to the map on the next page the two standard parallels or the lines where the conic
projection intersects the globe are along 36° 10' 00” in the north and 34° 20' 00”
In the south. Along these parallel lines the scale is exact. This is because they intersect with the
globe along these lines.
The area between the two parallels are said to have a scale projection that is too small. These
area covers 2/3 the distance of the north-south projection limits. The reason is that the grid plane
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is below the curved surface of the ellipsoid. The two areas above and below the parallels each
covering 1/6 of the area of the projections limits are said to be too large in scale. This is because
the curved surface of the ellipsoid is below the grid plane.
The base meridian for the North Carolina SPCS is 79 degrees. This system is depicted on the
map on the next page. The map is from a state publication entitled “DEVELOPMENT &
APPLICATION OF THE STATE COORDINATE SYSTEM” dated rev. 1996.
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Converting Ground Distances to Grid
Below is an illustration that explains terms used in converting ground distances to grid distances.
Note that ground distances will be longer than grid or ellipsoid distance. The Combined Factor
(CF) consists of the elevation factor and grid factor both of which are used to reduce ground
distances to grid distances.
Note from the illustration that the grid scale is only exact where the north latitude intersects the
ellipsoid. Between these intersections, where the ellipsoid surface and ground is above the grid
plane, the scale projection is too small. Outside the scale projection it is too large because the
ellipsoid surface is below the grid plane.
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To obtain the combined factor, the ground horizontal distances must first be reduced to the
ellipsoid.
The formula for this reduction is as follows:
Ellipsoid Distance = Horizontal Distance
X
-
=R
H + Geoid Height + R
-
-
TERMS
H = elevation (NGVD29/NAVD)
Geoid height = Use the average geoid height for the particular area, in this case for North
Carolina -33 meters is sufficient for most surveys.
R = Mean radius of the Earth in (North Carolina uses 6370944 meters.)
Often tables are compiled for this data using the above formula in the following format:
(This is for a -33 meter geode height.)
Elevation H meters
25
50
100
200
300
400
500
600
700
800
900
1000
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Ellipsoid factor
1.00000126
0.99999733
0.99998948
0.99997379
0.99995809
0.99994239
0.99992670
0.99991100
0.99989531
0.99987961
0.99986391
0.99984822
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A more precise means would be to average the geoid heights from data sheets. Below is an
example for a particular monument control station in Montgomery County, North Carolina.
This data sheet example states that the GH is -30.32 meters. A survey tying in two such
monuments would take the mean average of the two GH factors. The next step is taking the
calculated ellipsoid lengths and further reducing the original ground distance to the length they
would be on the particular State Plane coordinate grid.
This grid factor is a scale factor and is based on the previously mentioned scale factor can be
exact, in the case of where parallels intersect the ellipsoid, or small or large depending on the
latitude of the points surveyed.
For example if the point falls on Lat 34 Degrees 20 minutes in North Carolina the scale factor is
1.000000 or unity. Refer to Map on page 23.
Scale factors are obtained from data sheets, by software or by published tables.
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North Carolina publishes this table in the following format:
Latitude
R (Meters) Tab diff.
K scale factor
….............................................................................................................................................
34 d 18 m
9138771.722
30.81354
1.00000940
34 d 19 m
9136922.910
30.81348
1.00000466
34 d 20 m
9135074.101
30.81342
1.00000000 Unity
34 d 21 m
9133225.296
30.81337
0.99999543
34 d 22 m
9131376.494
30.81331
0.99999093
…...........................................................................................................................................
R in above table refers to mapping radius at grid origin
The above is only a portion of the table going from scale too great to unity on the parallel
described above to scale factor too small.
We then can determine the combine factor that allows us to project ground distances to grid
distances.
Combined Factor = Ellipsoid Factor X Scale Factor
Grid Distance = Horizontal Ground Distance X Combined Factor
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Map Projections and State Plane Coordinate Systems
A Sun Cam online continuing education course
Convergence Angles
The next concern is that true north only occurs along the central meridian for each state plane
coordinate system. Referring to the diagram on page 30, this meridian is 79 degrees in the case
of the North Carolina Lambert projection. Any other place except the central meridian will need
to address the difference between the grid and geodetic azimuths at any point.
Grid meridians are parallel, Geodetic meridians converge at the geographic poles.
This angle has been called the convergence angle, the mapping angle, or simply the theta (θ)
angle. The angle is a positive number to the east of the Central Meridian and a negative to the
west of the Central Meridian.
Grid Azimuth equals Geodetic True Azimuth (-) theta (θ) + the 2nd term.
The second term (T-t) is a minor correction that is applied to surveys of great accuracies and is
not normally used in ordinary surveying. They apply to directions more than five miles and in
east-west directions. It is needed for surveys of great precision because measured direction
between two points is a curved line and when projected on a plane appears as a curved line.
The mapping angle is used only when it is necessary to convert grid azimuth to geodetic azimuth
or geodetic azimuth to grid azimuth. Grid azimuth can be computed from an inverse between two
points with plane coordinates. Geodetic azimuth is usually provided by NGS or from GPS data.
Example: For North Carolina 0.577170255241 is the ration of theta to the longitude difference
from the central meridian. Caution- this ration is for NC only.
(Central meridian – longitude) x 0.577170255241 = theta (θ)
(79 degrees-80 degrees) x 0.577170255241 = - 0.577170255241
= - 34' 37.8143” = theta (θ)
Conclusion
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Copyright 2014 Thomas Maxell Beer
Page 26 of 27
Map Projections and State Plane Coordinate Systems
A Sun Cam online continuing education course
State plane Coordinate Systems have contributed greatly to Surveying and Engineering as
follows:
1. Large projects, like highway projects, can be an undertaking using the methods of plane
surveying. The projects need not have to resort to the more difficult methods and calculations
employed with geodetic surveying.
2. Plane surveying is less expensive that Geodetic Surveying.
3. Surveyors do not have to be trained as extensively using the simpler methods of plane
surveying.
4. All surveys within the SPCS have a single established framework.
5. Lost monuments can be reestablished when tied to state plane coordinates.
6. Projects surveyed in different areas can be linked together by a common framework.
For further information regarding each State’s Plane Coordinate Systems visit the following web
site.
http://home.comcast.net/~rickking04/gis/spc.htm
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Copyright 2014 Thomas Maxell Beer
Page 27 of 27