This document discusses different types of coordinate systems used in surveying and mapping. It begins with an overview of why coordinate systems are important when working with geospatial data. It then describes several types of coordinate systems including geocentric, geodetic, plane, and local systems like State Plane coordinates. Key concepts covered include ellipsoids, datums, projections, and how transforming between systems can distort data. The document aims to explain the fundamentals of different coordinate systems for surveying applications.

1. FUNDAMENTALS
O F
COORDINATE SYSTEMS
Denver Water Training John Hunter, PLS
Coordinate Systems

2. • Why We Need To Talk About Coordinate Systems
• Types of Coordinate Systems
• Geocentric
• Geodetic
• Plane
• Ellipsoids
• Horizontal Datums
• Projections
• Heights
• Scale Factors & Grid to Ground
• Modified Projections & LDP’s
OVERVIEW
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3. 10/4/2017
WHY DO WE NEED TO TALK ABOUT
COORDINATE SYSTEMS?
I like to think of a coordinate system as the language in which
data has been written.
In language, sometimes things get “lost” in the translation, or
there is no direct translation and often the next closes
alternative word is used. The native meaning has then become
distorted or may not even make sense after it is translated.
Like language translation, when we transform data from one
coordinate system to the other, we risk distorting that data to
make it fit into any other given coordinate system.

4. 10/4/2017
WHY DO WE NEED TO TALK ABOUT
COORDINATE SYSTEMS?
The preservation of data is fundamental to all mathematical and
science based analytics.
We know what happens to data when we transform it.
Once it is transformed, or distorted, any reporting or analytic
that is produced thereafter has been distorted as well and a
statement to the degree of which distortion has occurred should
be made in the reporting.

5. TYPES OF COORDINATE
SYSTEMS
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6. Types of Coordinates
and Their Coordinate Systems
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Coordinate Type Coordinate System Note
Earth Centered, Earth Fixed
P(x,y,z)
Geocentric GPS/GNSS Points
Traditional Cartesian

7. Traditional Cartesian Coordinates
The X axis:
From the origin extending, horizontally, through the
intersection of the Prime Meridian, and the Equator
forming a 90° angle with the Z axis in the vertical
plane.
The Y axis:
From the origin extending, horizontally, through the
Equator forming a 90° angle with the X axis in the
horizontal plane, and a 90° angle with the Z axis in the
vertical plane.
The Z axis:
From the origin (center of the ellipsoid, or center of the
earth) extending straight through the north pole is the
Z axis (not the same as elevation).
Derived within Geocentric Reference Frames such as
ITRF and the soon to be NATRF system which will
come with the modernization of the NSRS in 2022.
Not great for being used in relatively localized
engineering settings. Very large coordinate values, in
meters, and true 3D points.
These are true ground points. However, because we
like to view points on a flat plane such as paper maps,
the points must be transformed through the use of
ellipsoids, datums, and projections…
Geocentric Coordinate Systems
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ge·o·cen·tric
adjective: geocentric
Having or representing the earth as the center, as in former
astronomical systems.
Astronomy
Measured from or considered in relation to the center of the earth.

8. Types of Coordinates
and Their Coordinate Systems
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Coordinate Type Coordinate System Note
Latitude & Longitude
DD°MM’SS.SSSS”N, DDD°MMM’SS.SSSS”W
Geodetic Differs with Ellipsoid
GRS80 Vs. Clarks

9. Traditional World Coordinate Systems
Considering the same Cartesian system with X, Y, and
Z axis, the Geodetic coordinates utilize a measure of
angles to compute positions.
Latitude (geodetic) is the angle of measurement from
the equator to the point being observed, THAT IS
PERPENDICULAR TO THE ELLIPSOID. This line
does not extend through the origin except at the poles
and along the equatorial plane. As in not geocentric.
Because the constraining factor in latitude is that it is
perpendicular to the size and shape of the earth,
makes it geodetic.
Longitude is the angle measured AROUND the Z axis,
from the Prime Meridian to the point of observation.
This angle is measured counter clock-wise
The common misconception among GNSS/GPS
surveyors are that these are the coordinates of
GNSS/GPS and have become synonymous with
WGS84 coordinates. GRS80 and WGS84 are
essentially the same.
The truth is that geodetic coordinates such as Lat Long
are calculated by the GNSS/GPS controller from ECEF
coordinates.
Because these points are on the ELLIPSOIDAL
surface, the issue of heights, and elevations must be
accounted for.
Geodetic Coordinate Systems
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ge·o·det·ic
Of or relating to geodesy, especially as applied to land
surveying.
ge·od·e·sy
The science dealing with the shape and size of the earth
or large portions of it.

10. Types of Coordinates
and Their Coordinate Systems
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Coordinate Type Coordinate System Note
State Plane Coordinates
N:1697814.29 E: 3157197.06
Plane Round Surface Projected to
Flat Surface.
Coordinates ≠ Ground
Distances

11. Types of Coordinates
and Their Coordinate Systems
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Coordinate Type Coordinate System Note
Modified State Plane
N:697814.29 E: 157197.06 CSF: 1.00026
Sometimes more modifications are made
Plane Truncate Coordinates &
Apply a Scale Factor.
Coordinates = Ground
Distances

12. Types of Coordinates
and Their Coordinate Systems
10/4/201712
Coordinate Type Coordinate System Note
Earth Centered, Earth Fixed
P(x,y,z)
Geocentric GPS/GNSS Points
Traditional Cartesian
Latitude & Longitude
DD°MM’SS.SSSS”N, DDD°MMM’SS.SSSS”W
Geodetic Differs with Ellipsoid
GRS80 Vs. Clarks
State Plane Coordinates
N:1697814.29 E: 3157197.06
Plane Round Surface Projected to
Flat Surface.
Coordinates ≠ Ground
Distances
Modified State Plane
N:697814.29 E: 157197.06 CSF: 1.00026
Sometimes more modifications are made
Plane Truncate Coordinates &
Apply a Scale Factor.
Coordinates = Ground
Distances

13. Geodetic Coordinate Systems
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Differences Between Geocentric, and Geodetic

14. ELLIPSOIDS
The Horizontal Ellipse
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15. The Ellipsoid
Definitions
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• Biaxial ellipsoid has dimensions a,
and b
• Mathematical approximation of the
shape and size of the earth.
• Defined by two dimensions; a Semi-
Major axis, (a), and a Semi-minor
axis, (b).
• The flattening ratio, (ƒ) of an
ellipsoid.
• ƒ = 1-(b/a)
• The eccentricity, (e) is how much
the ellipsoid deviates from a circle.
• e = √(2 ƒ - ƒ 2).
• Typical notation: A precise
calculation of the ellipsoid can be
performed with the Semi-major
dimension, (a), and the flattening
ratio, ƒ.

16. The Ellipsoid
Orientation & Initial Point
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• Attaches the ellipsoid to a physical
point on the earth’s surface, known
as an initial point.
• Points chosen were best suited for
a particular region.
• Historically observatories were
used.
• Five Parameters:
1. Semi-Major (a)
2. Semi-Minor (b)
3. Latitude
4. Longitude
5. Azimuth to reference point
• NAD 1927 fixed at Meade’s Ranch
in Kansas and uses Clarke 1866
Spheroid.

17. The Ellipsoid
Geocentric Datums
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• Attaches the ellipsoid to a theoretical point at the
center of the earth, known as a geocentric
system.
• NAD 1983 used new technologies to take
measurements.
• Satellite Laser Ranging (SLR)
• Lunar Laser Ranging (LLR)
• Very Long Baseline Interferometry (VLBI)
• Doppler Orbitography (DORIS)
• Some (relatively few) GPS
• Realizations are needed to become useful in
Land Surveying.
• Realization can be thought of as a snapshot in
time. The center of the earth is a fixed point,
whereas the surface is constantly moving and
changing.
• Readjusted to incorporate additional “survey
measurements & observations”.
• Classical survey measurements
• EDM-measured baselines
• A lot of GPS observations
• Other
• NAD 1983 is fixed at, and attached to the center
of the earth, uses the GRS80 Spheroid, and
requires realizations.

18. HORIZONTAL DATUM
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19. Horizontal Datum
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A horizontal datum forms the
basis for computations of
horizontal positions which may be
defined at an origin point of an
ellipsoid such that the center of the
ellipsoid coincides with the earth’s
center, or actual fixed points on the
earth’s surface. A proper
horizontal datum will define:
• The dimensions of the
reference ellipsoid.
• The orientation of the
coordinate system.
• The location of the origin of the
coordinate system.
So really… A datum essentially
defines the coordinate system.
o NAD27
o NAD83
o NAD83 (2011)
o NAD83 (CORS96)
o 2022 Modernization

20. Projections
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A projection, is a process to represent a
portion of the actual earth on a plane.
All projections will result in some sort of
distortion.
The Secant Lambert Conformal Conic
(SLCC) projection is one of many types of
projection methodologies. This conformal
conic projection preserves the integrity of
angles but suffers a varying distortion in
distances, dependent upon several
factors. The SLCC is the projection
utilized in the Colorado State Plane
Coordinate System, therefore the only one
that will be discussed.
However, there are many types of
projections, and some of the popular
methodologies are: Planar, Conical,
Cylindrical

21. Projections
Secant & Conformal
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Secant: In geometry, a secant line
of a curve is a line that (locally)
intersects two points on the curve.
A chord is an interval of a secant
line, the portion of the line that lies
within the curve. The word secant
comes from the Latin word secare,
meaning to cut.
In the SLCC, the cone intersects the ellipsoid at
two points, hence the name secant. But
because the cone and the ellipsoid are 3
dimensional, these points actually create lines
of latitude, also known as standard parallels.
In a conformal projection, the angles are
preserved but the distances are distorted.
Conformal: (map projection or a
mathematical mapping) Preserving
the correct angles between
directions within small areas,
though distorting distances are
likely to occur.

22. Projections
Lambert Conformal Conic (SLCC) projection
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Standard Parallels are the only points (lines) in this projection
with a scale of 1. Standard parallels are important in a projection
because they essentially function as the boundaries.
The Origin is the anchor point for the projection. It is the point
from which Northing and Eastings are derived.
The Central Meridian. (in a coordinate systems) A line of
longitude that defines the center of a projected coordinate
system. In planar rectangular coordinate systems of limited
extent, such as state plane, grid north coincides with true north
at the central meridian.
The False Northing (map projections) The linear value added to
all y-coordinates of a map projection so that none of the values
in the geographic region being mapped are negative.
The False Easting (map projections) The linear value added to
all x-coordinates of a map projection so that none of the values
in the geographic region being mapped are negative.
The false northing and easting is used to ensure that there are no negative coordinate values
within the projection. These are the parameters that are crucial to ensuring that
transformations are performed correctly.

23. HEIGHTS (……AND GEOIDS)
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24. Heights
Ellipsoidal Height
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The Ellipsoidal Heights is the distance from the ellipsoid to a corresponding
point on the surface of the Earth. The height is measured along a line
perpendicular to the ellipsoid. This distance is known by more than one name. It
is called the ellipsoidal height and is also called the geodetic height, and it is
usually symbolized by h.
Note: Not all h values are the same, and the horizontal datum CAN impact h.

25. Heights
What is a Geoid and what is a Geoid Height
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The Geoid is an equipotential surface that best fits mean sea level. Meaning
that, across the geoid, the potential of gravity is always the same. The geoid and
mean sea level could be the same if the oceans of the world could be utterly still,
completely free of currents, tides, friction, variations in temperature, and all
other physical forces, except gravity.
A Geoid Height is the ellipsoidal height
from an ellipsoidal datum to a geoid.
That means that geoid height models are
directly tied to the geoid and ellipsoid that
define them. In other words, geoid height
models are not interchangeable.
Note that the geoid is a vertical datum surface.

26. Heights
Putting it together
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Orthometric Height (H) = h-N
Ellipsoidal Height (h) = H+N
Geoid Height (N) = h-H
Keep this in mind: An elevation that was established by differential leveling
should not be used to determine an Ellipsoidal Height because the level is
based upon gravity. However, this is negligible in most cases but reinforces the
need to ensure geoids are properly paired with the correct horizontal datum.

27. SCALE FACTOR
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28. Scale Factor
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An Elevation Factor (EF) is determined by the ratio of the ground distance (DGND) to the corresponding geodetic
distance (DGDC): EF = DGND / DGDC.
The Scale Factor (SF) is determined by the ratio of the geodetic distance (DGDC) to the corresponding grid distance
(DGRD): SF = DGDC / DGRD.
A Combined Scale Factor (CSF) is perhaps the most useful, and is determined by the ratio of the ground distance
(DGND) and the grid distance (DGRD): CSF = DGND / DGRD.

29. Scale Factor
Grid to Ground and Ground to Grid
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Combined Scale Factor (CSF) = 1.00046
Grid to Ground
Ground Distance = Grid Distance x CSF
Ground to Grid
Grid Distance = Grid Distance/CSF
The horizontal distance between two section corners is measured on the surface
with a GPS unit, and has a ground distance of 5,280’. To convert the distance to
grid we take the ground distance (5280’) and divide it by the CSF (1.00046), which
gives us 5277.57’.
The horizontal distance between two section corners is measured on the surface
with a GPS unit, and has a grid distance of 5,280’. To convert the distance to grid
we take the ground distance (5280) and multiply it by the CSF (1.00046), which
gives us 5282.43’.

30. 10/4/2017
WHY DO WE NEED TO TALK ABOUT
COORDINATE SYSTEMS?
Because Far too Often
 We know the lingo but are far less comfortable with the details
Usual Response: What do you mean my acreages are not correct on my State
Plane Map.
 We do not really care as long as our datasets matchup
Usual Action taken: Click on OK instead of Transformation when the
Transformation Dialog Box comes up.
 The outcome is given more importance than the foundation
Meaning: We have a tendency to focus on the actual analysis more than the
quality of the data.
 There is a misunderstanding between collection and transformation
What I usually hear: You mean I have to go back out and collect this in state
plane?!