SEDRIS Spatial Reference Model

1
SEDRISSpatial Reference Model
Dr. Ralph Toms SRI International ralph_toms_at_sri.co
m
Dr. Paul A. Birkel The MITRE Corporation pbirkel_at_m
itre.org

• STC 2000, Snowbird, Utah
• August 22-25, 2000

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SRM Tutorial
DESCRIPTION Accurately locating objects is key
to the operation of any system which contains
information about real-world entities.
Consistency in description, nomenclature, and the
treatment of models of the earth and related
spatial reference frames and coordinate systems
is critical to achieving effective data
interchange and system interoperability. The SRM
provides the means to define a unified approach
to representing spatial location information and
precisely relating different descriptions of
spatial location. The tutorial provides a
detailed review of the SRM requirements
significant discussion of earth reference models,
map projections, coordinate systems, and spatial
reference frames how and why modelers use
coordinate systems (and the many issues which
must be addressed in order to insure
interoperability) issues in implementing precise
and efficient coordinate transformations and an
overview of the SRM reference implementation
software. WHO SHOULD ATTEND Anyone interested
in gaining a more complete understanding of
spatial referencing and coordinate systems, or
intending to use the SRM reference implementation
whether as a stand-alone component or as part
of a SEDRIS-based application development
project. Prior knowledge of other SEDRIS
technologies is not required, however, some
knowledge of the Data Representation Model (DRM)
may be helpful. WHAT TO EXPECT At completion,
the attendee should have gained an appreciation
for the complexities involved with the proper
use, and accurate description, of spatial
locations, object/earth reference models,
coordinate systems, and the spatial reference
frames commonly used in the environmental
representation / modeling community. Some
mathematical formulations will be presented,
however a strong mathematics background is not
required in order to appreciate the results, and
their implications for modelers. Illustrations
and visual imagery will be used as much as
practical in order to make complex topics
accessible to the non-expert.
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Outline

• I. Simulation Interoperability and a Spatial
  Reference Model
• II. Introduction of Basic Terminology
• III. Map Projections
• IV. Augmented Map Projections and Geometric
  Distortions
• V. Selection of a SRF for Models and Simulations
• VI. ERM-based Geometry and Trigonometry
• VII. Computational Considerations
• VIII. Implementation of SRFs
• IX. Implementation of SRF-related Operations

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Section I
Simulation Interoperability and aSpatial
Reference Model (SRM)
5
In the Beginning ...

• Defining and using a consistent spatial reference
  framework is critical for simulation
  interoperability
• System models (men, equipment, material, )
• Environmental data, models, phenomena
• At all levels of simulation detail/resolution
• Aggregated / Entity Constructive / Virtual / Live

SNE Synthetic Natural Environment
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Simulation Interfaces

• The interfaces between each class of simulations
  makes achieving interoperability difficult.
• The traditional hierarchy of models does not work
  very well.
• Complex, labor intensive interface processing is
  required.
• Although there are software based linkages for
  connecting dissimilar models this does not
  guarantee that it is meaningful to do so.
• A major problem is the lack of commonality of
  interfaces due to the use of different earth
  reference models and coordinate systems.
• This leads to inconsistent positions and
  environmental representations.
• The lack of commonality is intensified by
  traditional aggregation policies.
• It is much easier, but certainly not pro forma,
  to interface between entity level simulation
  classes.
• The interface between aggregated constructive
  simulations and entity level simulations has been
  referred to as the Grand Canyon.

Blumenthal, Bridging the Grand Canyon, 1997
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A Spectrum of Constructive, Live and Virtual
Capabilities Support Training, Planning
Analysis
Aggregated Constructive
Entity Level
Ranges Live Exercises
Tactical Simulators
Constructive
War
Closed for analysis Interactive for
training Units normally are Battalions or
higher Environmental effects aggregated
Usually deterministic, not stochastic Attrition
often based on Lanchester differential
equations Entity states not maintained
Large
Closed for analysis Interactive for training
planning Player inputs tactics or uses SAF
Generally stochastic More detailed
environment Acquisition is modeled 2D
Graphics with 3D Display
Virtual, always interactive Principally for
training 3D graphics with 2D display
Acquisition by humans Principally for
distributed play OPFOR is primarily
constructive Protocol standards enforced
Real Platforms Emulated weapons delivery
Simulated BDA Acquisition by HITL in real
environment Training and procurement support
Simulations used for fire control solutions,
sensor pointing, guidance and control
S c o p e
Low
Low
High
Level of Detail
Lots of detail does not necessarily imply
accuracy, fidelity or functional completeness
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A Spectrum of Constructive, Live and Virtual
Capabilities Support Training, Planning
Analysis
Aggregated Closed Interactive Constructive
Entity Level
War
Tactical Simulators
Ranges Live Exercises
Constructive
CEM,THUNDER, RSAS, METRIC ENWGS, JTLS TACWAR, ITEM
Large
Seminar War Games
JTF/Theater
Corps/ CVBG/ARV
CBS TACSIM
S c o p e
NTC
VIC EAGLE
Div
JTCTS
SPECTRUM (MOOTW)
Bde/ CVW
JTS
JCATS
BBS MTWAS
EADSIM
Janus
Operational Planners
Bn/Wng
MODSAF/ CCTT SAF
DFIRST
Co/Sqdn
CCTT
SIMNET
Embedded Fire Control, Guidance, etc.
Low
Plt
SOFNET
Sqd
Low
High
Engineering Simulations
Level of Detail
Lots of detail does not necessarily imply
accuracy, fidelity or functional completeness
R. Toms 2/18/97
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The Interface Canyons
A notional view of the scope of the problem
Aggregated Constructive
Entity Level
Tactical Simulators
Ranges Live Exercises
Constructive
War
Low
High
Level of Detail
A consistent SNE is criticalto addressing the
interface problem
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Why is a Spatial Reference Model(SRM) Needed?

• Traditionally the MS community has not been
  consistent in the treatment of models of the
  earth and related coordinate systems.
• Consistency is required for joint distributed
  simulation in order to
• Achieve a reasonably level playing field,
• To support meaningful VVA.
• A number of different earth reference models
  (ERMs) are currently employed and this affects
• Representation of the environment in simulations
  authoritative data bases.
• Dynamics formulations, both kinematics and
  kinetics (movement).
• Acquisition modeling and processing
  (inter-visibility).
• Approximations in coordinate transformation
  algorithms made to reduce processing time may
  introduce additional inconsistencies.
• An SRM is needed to promote lossless and accurate
  transformations.
• A nomenclature inconsistency evolves when there
  is no SRM
• For example, how do these variables relate?
• Altitude, elevation, height, geodetic height,
  ellipsoidal height, orthometric height, height
  above sea level, height above mean sea level,
  terrain height, pressure altitude, temperature
  altitude, nap of the earth, ...

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SRM Requirements

• Completeness
• Must include coordinate frameworks in common
  usage.
• Must tie those systems together into a common
  framework.
• Must educate the system developer.
• E.g., Whats a horizontal datum? A vertical
  datum?
• Accuracy
• Generally higher than required for C4ISR systems.
• Typically better than 1 cm. up past
  geosynchronous orbit.
• Performance
• Never fast enough!
• Many environmental data sets dominated by
  location data
• Therefore efficient interconversion key to
  meeting 72 hour ready-to-run mandate.
• Federate costs for distributed simulation using
  heterogeneous coordinate systems can be
  substantial (e.g., 20 or more).

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Location is Not Enough

• The key in MS is not so much where you are, but
  who you can interact with
• Remembering that who includes both systems and
  things in the environment itself
• A complete SRM must address
• Direction (azimuth and elevation angle)
• Range
• (support for) Geometric intervisibility
• Gravity (and implied slope pitch/roll)
• Location interconversionsbetween common
  spatialreference frames do notnecessarily
  preserve these

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A Shared Solution is Required!
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Section II
Introduction of Basic Terminology
15
Its a Loooong Way Up
S E A M L E S S E N V I R O N M E N T
IONOSPHERE 25 Satellites
DMSP
SHUTTLE
MESOSPHERE
Space Weather
STRATOSPHERE
U-2
Terrestrial Weather
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Heres an Interesting Perspective
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Not All Projections are Geodetic!
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More Familiar?
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Spatial, the Final Frontier
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Coordinates

• Coordinates are linear or angular quantities that
  designate the position of a point in a reference
  frame.
• In the SEDRIS Data Representation Model (DRM),
  they are represented as ltLocationgts

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DRM ltLocationgt (Page 15)
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Spatial Reference Frames

• Spatial Reference Frames serve to locate
  coordinates in a multi-dimensional space
  (generally either two- or three-dimensional).
  They are specified in two parts
• A geometric description (model) of a reference
  object embedded in (and serving to orient) that
  frame referred to as an Object Reference Model
  (ORM)
• An Earth Reference Model (ERM) is a special case
  of an ORM
• A Coordinate System specifying how a tuple of
  values uniquely determine a location with respect
  to the origin of that frame. By extension, that
  tuple also specifies a location with respect to
  the reference object.
• SRF ORM CS

Naked coordinate systems have limited utility!
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SRF Example

• A common spatial reference frame is a combination
  of a
• Specification of the Earth Reference Model (ERM)
  (e.g., an ellipsoid with specific parameters),
  and a
• Right-handed Cartesian coordinate system in a
  specific relationship to the center or origin,
  and rotational plane (equator) and axis of the
  ERM.

One specific binding of (1) a right-handed
Cartesian coordinate system to (2) the rotational
plane (equator), prime meridian, and rotational
axis (via the north pole) of an ERM. Point P is
then a location defined with respect to that
coordinate system.
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Named SRFs in SEDRIS
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Named SRFs in SEDRIS
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Coordinates Coordinate Systems

• Coordinates are linear or angular quantities that
  designate the position of a point within a
  coordinate system. By extension, they also
  designate the position of a point within a
  spatial reference frame.
• Coordinate Systems are a collection of rules by
  which a tuple of values may be used to spatially
  relate a location to a unique (coordinate system)
  origin location.
• Cartesian Coordinate Systems are based on an
  ordered set of mutually perpendicular axes formed
  by straight lines. The point of intersection of
  the axes is termed the origin. The directions of
  successive axes are normally related to each
  other by a right hand rule.
• Cartesian Coordinates (two-dimensional) uniquely
  locate points on a plane using a doublet of
  values, e.g., (x, y).
• Cartesian Coordinates (three-dimensional)
  uniquely locate points within a volume using a
  triplet of values, e.g., (x, y, z).

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ERMs

• Topographic Surface is the interface between the
  solid and liquid/gas portions of the Earth. It
  corresponds to the surface of the land and the
  floor of the ocean.

• Earth Reference Model (ERM)is a specification of
  the math-ematical shape of the Earth, usually in
  terms of a com-bination of ellipsoidal and
  equipotential (geoidal) surfaces. It excludes
  the topographic surface, and therefore generally
  corresponds with mean sea level.

In the SRM, there is no Earth there are only
models of the Earth.
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Standard Ellipsoids
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Standard Spheres
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Standard Horizontal Datums
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Standard Vertical Datums
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ORMs

• Since the Earth is an important reference
  object in our spatial environment, many Spatial
  Reference Frames will consist of an Earth
  Reference Model plus a Coordinate System.
• There are spatial reference frames which
  incorporate models of alternative reference
  objects
• E.g., a spacecraft body-centric reference as
  often used with the Space Shuttle, or a non-Earth
  planetary body, or even a celestial body
• Local Space Rectangular (LSR) uses a minimalist
  ORM which does not locate the CS origin, but
  orients the axes through specification of up
  and forward and a right-hand rule
• In general SRF ORM CS
• Commonly SRF ERM CS

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Many SRFs

• There are a potentially infinite set of spatial
  reference frames based on the same Cartesian
  coordinate system with the same relationships to
  an ERM, but with differing ERMs.
• E.g., as defined by different ellipsoidal models
• Or which may share a common ERM, but which have
  different relationships between the Cartesian
  coordinate system and the ERM.
• E.g., Geocentric (GC) and Local Tangent Plane
  (LTP)
• Each of these spatial reference frames is unique,
  unambiguous, and can be exactly related to the
  others.
• And each has a customer-base for whom it is the
  preferred (only?) choice ...

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Spatial Reference Model

• Spatial Reference Model is a well-defined set of
• spatial reference frames,
• object reference models, and
• coordinate systems,
• that allows coordinates to be specified
  succinctly, and
• converted accurately between different spatial
  reference frames.

The degree of accuracy (and performance)
required is generally application-dependent.
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Operations on Coordinates

• Coordinate Conversion is the process of
  determining the equivalent spatial location of a
  point in a SRF which is based on the same object
  reference model (e.g., ERM), but a different
  coordinate system.
• Coordinate Transformation is the process of
  determining the equivalent spatial location of a
  point in a SRF which is based on the same
  coordinate system, but a different object
  reference model (e.g., ERM).
• Converting coordinates between two arbitrary
  Spatial Reference Frames may require both
  Coordinate Conversion and Coordinate
  Transformation.
• Other types of operations are also required,
  e.g.
• Direction (azimuth and elevation angle)

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Types of Operation Errors

• Formulation (Algorithmic) Errors Inherent in
  optimized algorithms for coordinate operations
  where accuracy may be traded for performance.
• Implementation Errors Includes errors due to
  finite precision arithmetic and software
  implementation.
• Usage Errors Includes errors due to extension of
  projection-based SRFs beyond reasonable limits.
• Where coordinate operations with respect to an
  alternative SRF are required.

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Section IIB
A Few More Facts and Diagrams
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Ellipsoidal Earth Reference Model(ERM) Geometry
Notation
Ellipsoids are standard in current geodesy
practice. For SNE data modeling, spheres are
often used to simplify dynamics equations.
Geocentric coordinates (GCC) are defined by the
point P(X, Y, Z). Geodetic coordinates (GDC)
are defined by the point P(Ø, ?, h).
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Latitude, Longitude and Heightfor Ellipsoids
Spheres
P(X, Y, Z) or P(Ø, ?, h)


Z
Z
h
hos
Pe
Ps
ø
Y
Y
?
?
X
X
For ellipsoids Latitude, longitude and geodetic
height are defined as per this diagram. The line
through P is perpendicular to the ellipsoid.
Longitude ? is generally referenced to the Prime
Meridian.
For spheres Longitude is the same as for the
ellipsoidal case, ? is the geocentric latitude,
and hos is height above the sphere. The line
through P is per-pendicular to the sphere. In
mapping, charting and geodesy, spherical ERMs are
almost never used.
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Cross-Section of theGeoid, Ellipsoid and Earth
Surface
Earth's Physical Surface
H
Geoid
h
Geoid
Ellipsoid
Geoid Separation N
Ellipsoid
The geoid is a gravity equipotential surface
selected to match mean sea level as well as
possible.
Geoid Separation - N
h is the geodetic height H is the orthometric
height N is the separation of the geoid
For more on this see NIMAs Geodesy for the
Layman on http//164.214.2.59/geospatial/product
s/GandG/geolay/toc.htm
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Gravitational Field and the Geoid,Ellipsoid
Earth Surface
Gravity vector depends on latitude,
longitude, and H (or h)
Gravity potential results in a gravity field
Earth's Physical Surface

P
H
Geoid
h
Geoid
Geoid Separation N
Ellipsoid
h is the geodetic height H is the
orthometric height N is the separation of the
geoid
Geoid Separation - N
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SRM Refresher
Earth Referenced, Projection-based 2D and 3D SRFs
Earth Referenced 3D (and 2D) SRFs
GCS
Geocentric (ECEF)
Geodetic (3D and 2D)
With and without Augmentation
Geomagnetic Geocentric Equatorial
Inertial Geocentric Solar Ecliptic Geocentric
Solar Magnetospheric Solar Magnetic
43
Section III
Map Projections Map projections were inventedto
support paper map development a long time ago
44
Development of Surfacesto Generate Maps
Developable Surfaces
A cone or cylinder can be cut and laid out flat.
Non-developable Surfaces
The surface of an ellipsoid cannot be cut so it
will lie flat without tearing or stretching.
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Map Projections

• Associate points on the surface of an ERM with
  points on an X-Y plane or more formally ...
• A map projection is a mathematical transformation
  from a three dimensional ellipsoidal or spherical
  ERM surface onto a two dimensional plane.

Y
X
Since spheres and ellipsoids are not
developable, distortions must occur. Note
that the transformation is from three to two
dimensions, and that there is no vertical axis in
the plane.
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Projecting from 2D to 1D
Projection from the point Nof all points on the
circle onto a line.
N
Note that the red points do not map!

• Note the stretching of the length of the arc s
  after the projection.
• The concept of a projection can be extended to
  projecting the points on the surface of an ERM
  onto a plane.

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Cylindrical Projections
48
Planar Projections
49
A Stereographic Projection
50
Conic Projections
51
Mercator Projection
A Mercator projection is a cylindrical projection.
From N. Bowditch, American Practical
Navigator, U.S. Navy Hydrographic Office, 1966 Ed.
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Oblique Mercator Projection
From N. Bowditch, American Practical
Navigator, U.S. Navy Hydrographic Office, 1966 Ed.
A Transverse Mercator (TM) Projection is defined
when the cylinder is parallel to the equator.
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Transverse Mercator Map of theWestern Hemisphere
In geodetic coordinates the origin is at (0,
-p/2, 0) The longitude of the origin is
shown as -90º W.
From N. Bowditch, American Practical
Navigator, U.S. Navy Hydrographic Office, 1966
Ed.
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Universal Transverse Mercator (60)
Widely used for paper maps by the U. S. Army.
Defined on six degree wide regions with 60
origins on the equator. A grid numbering scheme
is used to define the Military Grid Reference
System
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Lines of Constant Heading
Mercator projection commonly used for maritime
navigation. Line of constant heading is called
a rhumb line or loxodrome.
Mercator Projection
From N. Bowditch, American Practical
Navigator, U.S. Navy Hydrographic Office, 1966 Ed.
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Great Circle Arc between Moscowand Washington
D.C.
Mercator Projection
Oblique Mercator map for a sphere with the
central meridian on the great circle arc between
cities.
From N. Bowditch, American Practical
Navigator, U.S. Navy Hydrographic Office, 1966 Ed.
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Many Map Projections are Conformal

• Conformal means that the mathematical
  trans-formation preserves angles for example
• The curves between A B and B C are on the
  surface of an ERM.
• The projected curves (not necessarily the same
  shape) are on the plane.
• For a conformal transformation the angles ABC and
  abc are the same.

Y
A
a
B
b
C
c
X
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Section IV
Augmented Map ProjectionsandGeometric
Distortions
These are used, used and used but, they are
distorted, distorted and doubly distorted.
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Augmented Projection-Based SRFs

• Simulations usually require three dimensions.
• Some SRFs are three dimensional by definition.
• Map projections (two-dimensional) are commonly
  augmented with a vertical axis to create a three
  dimensional system.
• Various vertical measures are used for these
  augmentations, such as
• Mean sea level height, orthometric height,
  geodetic height, pressure altitude, and others.
• This practice adds additional geometric
  distortions.

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3D Projection-based SRFs
These result in augmented projection-based
spatial reference frames.
61
Coordinate Operations affectGeometrical
Relationships

• The taxonomy for classifying mathematical
  trans-formations is complex and there are many
  types
• isometric, linear, bi-linear, conformal,
  orthogonal, affine, isomorphic, ...
• For SEDRIS, two classifications are sufficient
  for operations associated with earth referenced
  spatial reference frames
• Geometry Preserving Transformations (GPT) That
  class of conversions/transformations between
  spatial reference frames that do not distort
  geometrical relationships.
• Geometry Distorting Transformations (GDT) That
  class of conversions/transformations between
  spatial reference frames that distort some
  geometrical relationships.

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SRF Operation Relationships
Operations between map projections and these are
GDT.
Operations among earth referenced 3D systems are
GPT.
Projection-based
GCS
Local Tangent Plane
Geocentric (ECEF)
Geodetic
With and without Augmentation
Geomagnetic Geocentric Equatorial
Inertial Geocentric Solar Ecliptic Geocentric
Solar Magnetospheric Solar Magnetic
63
Geometric Distortions

• Distortion caused by Projection Inherent in the
  use of 2D projection-based SRFs, where the
  distortion varies non-linearly with distance from
  the line(s) or point of projection.
• Distance (on the surface of the ERM)
• Azimuth
• Distortion caused by Augmentation Inherent in
  the extension of projection-based 2D SRFs to 3D
  SRFs, where the distortion varies non-linearly
  with both elevation and distance from the line(s)
  or point of projection.
• Distance (with respect to altitude)
• Elevation Angle

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Distance and Elevation Angle
N
Note that the red points do not map
s
Distance is distorted by the projection
Z
X
An augmented projection produces another 2D
system. Note that there are now two distortions
with respect to the original rectangular system.
This process can be extended to the 3D case but
even if the projection is conformal, elevation
angles are distorted by the augmentation.
65
Distance DistortionCan Be Mitigated, Somewhat
N
Note that the red points do not map
s
Scale here 1
Tangent
Secant
Scale here lt 1
On the green dotted line the average distortion
is reduced
66
Azimuth

• Transverse Mercator SRF
• In geodetic coordinates the origin is at (0,
  -p/2, 0)
• The longitude of the origin is shown as 90º W
• And were all familiar with Rhumb lines
  (loxodromes), and great circle distances ...

From N. Bowditch, American Practical
Navigator, U.S. Navy Hydrographic Office, 1966
Ed.
67
How Much Azimuth Distortion?
Convergence of the Meridian
P is a reference location in the
projectedsystem. The central meridian appears
as a straightline (the y axis). Meridians may
or may not curve(one exception is the Mercator
projection). T is the tangent line to the
meridian at P. ? is the Convergence of the
Meridian. The convergence of the meridian is
theangle between SRF North and true Northon
the source ellipsoid (positive clockwise). The
angle ? is the rotation angle needed torotate a
vector in the Projection-based SRFto true North
when siting a reference vectoronto a canonical
LTP.
The formula for computing the convergence of the
meridianranges from very simple (0 for the
Mercator-based SRFs)to very complex for
Transverse Mercator.
68
Flattening the ERMDistance and Geometry

• Augmented UTM (AUTM) example
• Several distortions are introduced, especially at
  the higher latitudes.

The results of such elevation angle and range
distortions may not be so apparent when all
simulations involved use AUTM. However, in a
federation involving real world coordinate
systems the distortions may become evident. Use
of AUTM increases visibility, causes interactions
to prosecute too fast, leads to an uneven playing
field and is not recommended for use in joint
simulations.
69
Section V
Selection of a SRFfor Models and Simulations
70
Modelers Often PreferCartesian Coordinate
Frameworks

• Dynamics equations can be simplifiedso that
  they are cheaper computationally.
• Velocity and acceleration components generally do
  not contain trigonometric functions.
• In Cartesian real world systems straight lines
  are linear functions.
• Shortest distance paths are straight lines.
• The Euclidean metric requires only a square root
  operation.
• In other coordinate systems minimum distance
  paths may be non-trivial to compute.
• Segments of ellipses lead to elliptic integrals.
• Shortest path on the surface of an ellipsoid is a
  geodesic (not an arc segment of an ellipse).
• The Earth and its natural environment are modeled
  with an ERM and a SNE.
• In this model shortest distance (or time) paths
  are not unique, and
• almost always are not geodesics or straight lines.

71
Non-real World 3-D Systemsare often Used. Why
?
Newtons second law of motion for a fixed or
inertial reference frame is For a rotating
system Va can be written In which case the
relative acceleration becomes Transforming these
to any system of earth related coordinate systems
will result in very complex equations. However,
under certain assumptions they can take on a
relatively simple form, particularly in
rectangular coordinates for a spherical ERM. An
example would be using Augmented Lambert
Conformal Conic projected off of a sphere as a
framework for an atmospheric model. The
assumptions involved must be understood when
using such data to create an SNE data set in a
simulation coordinate system.
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Relationships Between Coordinate Systems and
Simulations
73
MCGI and Dynamics Modeling
74
MCGI
Mapping, Charting, Geodesy Imagery
75
Dynamics Modeling
76
Section VI
ERM-based Geometry Trigonometry
Distance measures may have multiple
meanings Angular measures direction may have
multiple meanings True north is referenced to the
north pole of the ERM which is different than
the north pole in the real world and different
from the magnetic north pole, too.
77
Euclidean Distancein a Rectangular Framework
The Euclidean distance in a rectangular space
is given by D ? (xa - xb)2 (ya- yb) 2
(za - zb) 2. A straight line is the
minimum distance path between a and b. The
variables are linearly related by the parametric
equations of a line. The equation of the
line segment from a to b is given by x xa
(x - xb) µ y ya (y - yb) µ
z za (z - zb) µ where the parameter µ
is in 0 , 1.
78
Defining Geometrical Concepts

• ERMs are useful for defining geometrical concepts
  such as
• Three dimensional location with respect to the
  ERM.
• Distance between two points on the surface of the
  ERM.
• Angles between three points on the ERM surface.
• Normals and elevation angles.
• Trigonometric relationships.

79
The Traditional Surveying Process

• ERM position determination is closely related to
  surveying.
• Surveying is a process used to estimate locations
  on the earth.
• The ERM is needed to define a measurement
  reference baseline (distance, angle, elevation,
  surface normals, ...).
• Due to the terrain surface on the earth these
  parameters cannot be measured directly.
• Indirect measurements are used with the
  mathematical ERM model to estimate the reference
  parameters.

80
Surveying is a Complex Process

• Involves many different types of instrumentation.
• Geometrical concepts of triangularization,
  traverses and chains.
• Measurement corrections are needed to account for
  several environmental effects.
• Surveying approaches depend on whether they are
  for small areas, medium areas or global.

81
Traditional Distance Determination in Surveys

• Distance determination requires a leveling
  transit (theodolite) and a distance measuring
  device called a chain (the process is called
  chaining).
• Historically a metal chain was used but evolved
  to metal cables, metal rods and then steel tapes.
• Euclidian distances between points on the terrain
  surface are measured with the chain. The
  theodolite is used to keep the points on the same
  ground line and for measuring angles.
  Corrections for catenary, thermal and optical
  effects are required.
• Today the chain/theodolite combination uses a
  laser for distance measuring.

82
Surveying on a Flat ERM
By definition there is no terrain surface.
y
z
north

?
B
north

A
x
The points A and B are on the ERM surface.
The reference distance AB on the ERM is also the
Euclidean distance from A to B in the
coordinate frame. The normals at A and B are
co-planar. The azimuth ? (or any angle) is
readily defined. Note that z is not involved
and chaining would be easy.
83
Surveying on a Flat ERM with a Terrain Model
Added
Represents a surveyorstarget (pole).

• Chaining is depicted in the AB plane view.
• The transit provides elevation angles.
• The distances aiai1 are computed by trigonometry
  and summed to yield AB.
• Angular measurements are not affected.
• Intervisibility is required.
• Gravity and thermal models could be added.

84
Surveying on a Spherical ERM
The blue dotted curve is a small circle arc
generated by the intersection of a plane, that
does not contain O, with the sphere. The red
curve is a great circle arc generated by the
intersection of a plane, containing O, with the
sphere. The great circle arc is the minimum
distance path on the surface of a sphere.
The length of the great circle arc can be
computed using spherical trigonometry which
involves trigonometric functions. The normals
at A B are coplanar. When an environmental
model is added, e.g., terrain, minimum distance
paths on a sphere are almost never great circle
arcs.
85
Chaining on a Spherical ERMwith a Terrain Model
Added
Represents a surveyorstarget (pole).

• Chaining is depicted in the AB plane view.
• The transit provides elevation angles.
• The distances aiai1 are computed by spherical
  trigonometry and summed to yield AB.
• Angular measurements are not affected.
• Intervisibility is required.
• Gravity and thermal models could be added.

86
Surveying on an Ellipsoidal ERM

• Minimum distance paths are geodesics.
• Surface normals on an ellipsoidal ERM may not be
  coplanar.
• Non-coplanar normals greatly complicate the
  definition of a bearing angle.
• Ellipsoidal trigonometry is required which is
  mathematically and computationally complex.

87
Normal Section of an Ellipsoid
The gray plane is tangent to the ellipsoid at
A. The normal NA at A is orthogonal to the
tangent plane. The red plane, containing NA is
the normal plane at A. The curve from A to
B, the intersection of the ellipsoidal surface
and the the normal plane, is called the
normal section. The normals at A B are
generally not coplanar. Construction of a
normal plane at B which passes through A will
generate a different normal section. Normal
planes may not contain the origin.
88
Distance on an Ellipsoidal ERM
The red curve A to B is the normal section at
A. The red curve B to A is the normal section
at B. The green dotted curve, the
minimum distance path on the surface of
an ellipsoid, is called a geodesic. There is
no plane that contains the geodesic. The
length of a geodesic is an incomplete elliptic
integral. The curvature in the figure is
exaggerated for the purpose of exposition.
For short distances all three curves have
nearly the same length. When an environmental
model is added, e.g., terrain, minimum distance
paths on an ellipsoid are almost never geodesics.
89
Bearing Angles (Ellipsoidal ERM)
From the previous slide there are three
possible definitions for defining the bearing
angle ß1, ß2 and ß3.
The angle ß2 is preferred because of its unique
definition but it is computationally complex to
compute. For distances under a hundred
kilometers all three angles are nearly the
same. The normal section from A to B is
generally used for bearing computations in
practice. Note that A and B are on the surface
of the ERM.
90
Bearing when a Point P is above the ERM Surface
(Ellipsoid)
In this case the normal plane is rotated to
the point P. A and B are on the ERM
surface (B is below the horizon). The
green and blue arrows represent the normals NA
and NB. This does not define the same
bearing angle. Mathematical adjustments need
to be made to compute bearing. This process
is called reduction.
This is an edge view of the tangent plane
(purple dotted) and the normal plane (red) at
A. A and B are on the ERM surface (B is
below the horizon). The green and blue
arrows represent the normals NA and
NB, respectively. This is the
defining geometry for bearing angle.
NA
NA
P

P

NB
A
A


NB

B

B
91
Once the Environment is Included,There are Many
Feasible Paths
Minimum distance (or time) paths are much
harder to determine and are not unique. This
is true for land, maritime and airborne assets.
Sometimes paths are constrained by roads,
trafficability, political boundaries, hostile
sites, underwater structure and many others.
92
Placing a Solid Cube on an ERM
Excavation is not allowed because there is no
environmental model. The ERM is a mathematical
concept it is not the terrain! The cube
can only contact the ERM at one common
point. Once the terrain is modeled there
are many ways to site a cube by using
excavation.
93
Doing the Math ...
Transforming a Cube from an Augmented Conformal
Projection-based SRF to GD
Every point on the base (red) is on the
plane. Interior angles of the base are 90.
All other interior angles are 90. All sides
are of the same length. The vertical sides are
parallel planes. The cube is a convex hull.
Only points in the red region are
transformed by the map projection. Every point
on the base (red) is on the ERM. Since the
projection is conformal interior angles of
the base are 90. All other angles are
generally not 90. In general none of the sides
are equal. The vertical lines are not parallel
and are not even coplanar. The 3D volume is
no longer convex.
94
Placing a Cube on theTerrain Skin Model
These might not be very acceptable.
95
Long Linear Structures
Transformation from a Projection Based System To
GDC
Z
Side view in geodetic coordinates.
X
Y
In map projection coordinates.
Y
Top view in geodetic coordinates.
X
Top view.
96
Vectors inAugmented Projection-based SRFs

• Augmented projected coordinate systems are
  rectangular systems in which linear vector spaces
  can be embedded.
• A method is needed for transforming vectors in
  such a space to a geodetic coordinate system
  associated with an ellipsoidal (or spherical) ERM.

97
Vectors in a Curvilinear System
The notion of a surface vector is well known.
Np

p
A surface vector is well defined at p by
constructing a normal vector at p and
prescribing a unit length. A set of surface
vectors defined on a subset of the surface for
an ellipsoidal ERM do not form a linear vector
space. Common vector operations, such as scalar
products, cross products, and transformation
matrices are not defined.
98
Defining a CanonicalLocal Tangent Plane SRF
A local tangent plane is defined at the point
p.
Orthogonal x-y axes are defined at p and the
normal to the local surface is used to define
the z axis. The resulting rectangular system is
called a Local Tangent Plane (LTP) coordinate
system. If the surface is an ERM and the y axis
is defined to point to true north (the pole)
the resulting system is a Canonical LTP (CLTP)
spatial reference frame. A linear vector
space can be associated with a CLTP.
99
A LTPEmbedded in a GD SRF
Top Views
North Pole
W
Azimuth angle
Y
Azimuth
Z

North Pole
Y

X

PO
PO
X
Rotated to North Pole
Meridian of Origin
North Pole
V
Y

X
PO
Equatorial Plane

PO and alignment with Northdefine a Canonical
LTP (CLTP)
U
Central Meridian
U
100
Defining a CLTPEmbedded in a GD SRF
Top View
W
North Pole
Y
Z

North Pole
Y
X


PO
X
PO
PO and alignment with Northdefine a Canonical
LTP (CLTP)
Meridian of Origin
V
Equatorial Plane

U
Central Meridian
U
101
Reference Vectors

• In SEDRIS, reference vectors are unit vectors
  associated with a reference position (point).
• In a Augmented Projection-based Coordinate System
  (PCS) this is some point (x,y) in the plane of
  the projected system.
• In a CLTP, the reference position (point (x,y))
  ison the x-y plane of the CLTP.

102
Reference Vector Transformations

A reference vector is a unit vector associated
with a point P in a spatial reference frame. If
the point P and the reference vector are in a PCS
the inverse transformation of the PCS is used to
find the origin of the canonical LTP system on
the surface of the ERM. That is Locate the LTP
origin at T-1(P(x,y)), where T is the projection
transformation. The z component of a reference
vector in a CLTP is always perpendicular to the
LTP plane. A reference vector in a PCS or APCS
is referenced to grid north (the y axis) and
generally not to true north. By definition the
CLTP has its y axis pointing to true north. This
means that the vector must be rotated with
respect to the CLTP y axis. The required
rotation angle is the convergence of the meridian
(COM). This is an angle whose value depends on P.
103
SRF North may not be True North
Dashed grid lines are alloriented to map (or
grid)north. Solid curves are meridianthrough
the pi. Tangents to the meridian at pi point
to true north. The ?i are angles betweengrid
north and true northand vary with x-y
position. A reference vector at pi willnot
point north
y
?1
?2
?3



Central Meridian
P1
P2
P3
x
Map Equator
104
Convergence of the Meridian
P is a reference location in the
projectedsystem. The central meridian appears
as a straightline (the y axis). Meridians may
or may not curve(one exception is the Mercator
projection). T is the tangent line to the
meridian at P. ? is the Convergence of the
Meridian. The convergence of the meridian is
theangle between SRF North and true Northon
the source ellipsoid (positive clockwise). The
angle ? is the rotation angle needed torotate a
vector in the Projection-based SRFto true North
when siting a reference vectoronto a canonical
LTP.
The formula for computing the convergence of the
meridianranges from very simple (0 for the
Mercator-based SRFs)to very complex for
Transverse Mercator.
105
Representing a Projection-based SRF Vector
in Terms of a CLTP
x, y and z refer to the PCS systemwhile X,Y
and Z refer to the CLTP Counter-clockwise
rotations are takento be positive ? is the
COM (Convergence Of the Meridian)and determines
the true north direction Let V be a unit vector
in a PSC (xc,yc,zc) Rotations are about the z
axis so that it issufficient to consider only
the planar rotations The X-Y system is the
planar part of the CLTP The vector V can be
represented in the X-Ysystem by a rotation of
the components of V Since vectors are not
position dependent, therotation can be viewed as
occurring at theProjection-based SRF origin A
counter-clockwise rotation from x-y to
X-Ysuffices. That is XC xcCos(?)
ycCos(?)YC -xcSin(?) ycCos(?)ZC
zc The vector V (XC,YC,ZC) is the
representationof V in the CLTP.
106
TM series for X and Y(from P. D. Thomas)
For
and where is a
small parameter of the ERM, the series expansion
for X is given by,
where
and
where
and where Sø is the arc length along the meridian
through P from the origin. Sø is given in a
series and is a function of latitude (ø)
only. COM computations for TM and UTM involve
inversion of this series to get the footpoint
latitude.
107
Computing theConvergence of the Meridian (COM)
By elementary differential geometry and the
definition of COM
In the TM forward expansion most of the
coefficients do not depend on and so the
partials are readily computed. Specifically,
Using the canonical origin values for latitude
and longitude in these expressions to get
the partials allows COM to be determined.

108
Second Method forComputing the COM
This method comes from a modification of P. D.
Thomas equation (294) page 97 but has been
written in a better form for computation. This
method requires longitude, sine and cosine of
latitude of the origin of the CLTP.
where,
109
Computing the Sine and Cosineof the COM
The angle is not needed in the rotation
matrix for the reference vector. The
trigonometric functions are computed directly
using only a square root.
For method one.
For method two.
110
Observations
The procedure developed to compute COM for
TMcan be used for any map projection, canonical
or not. The procedure avoids having to compute
the footpoint latitude. For most PCSs the
formulas reduce to simple forms so themore
complex procedure is only needed for TM, UTM
andProjection-based SRFs for which COM is
complex. This general procedure provides a
means of verifying commonCOM formulas when
arbitrary origins have been introduced. COMs
formulas for common SRFs other than TM and UTM
are PCS COM Mercator LCC PS
in northern
hemisphere otherwise. EC
111
Section VII
Computational Considerations
Accuracy Errors Efficiency Testing
112
Why is Accuracy Neededfor Coordinate
Transformations?

• In real world, it has been difficult to measure
  positions on the surface of the earth to better
  than 1 meter.
• This situation is changing due to new technology
  developments.
• GPS now can achieve absolute accuracy of about 21
  cm (SEP 90) over large regions.
• Real-world weapons applications mostly use
  relative coordinate systems
• Dynamically correct location errors by using
  on-board sensors.
• In the simulation environment, relative
  coordinate systems must be accurately portrayed.
• Mixing of live synthetic environments has
  special accuracy requirements.
• Mission planning, rehearsal, conduct of real
  operations have situation-dependent accuracy
  requirements.

Lucha, G. V., On the Consequences of
Neglecting Measurement Accuracy Issues in
Live and Virtual interactions, SIW Spring 1997
113
Shooting at a Target in the Real
World(Relative Coordinate System)

• Set up paper target with aim-point approximately
  1000 meters away
• Shoot N rounds
• Measure miss distances using bullet holes on
  paper target compute CEP or some other measure
  of accuracy

Never used any precise location or environmental
data
114
Simulation of Shooting at a Target (Simulation
of a Relative Coordinate System)

• Select rectangular (LTP) coordinate system origin
  at shooter.
• Define position location of target shooter
• Both with target plane oriented perpendicular to
  LOS
• Develop an aiming model with random inputs.
• Define shoot time T.
• Integrate bullet trajectory in time from T until
  it pierces the plane of the target (need air
  temperature, density, speed of sound, wind,
  etc.).
• Will have to access geodetic system for each of
  these.
• Will need an iterative scheme to get the impact
  point.
• Compute radial miss at target plane impact.

Any errors made in any of the position-location
computations, including those needed to compute
the correct environmental parameters, can, and
will dilute the accuracy of the result.
115
Error Sources in Coordinate Transformation
Software

• There are many possible error sources in
  development of software for coordinate
  transformations.
• Did we mention distortions yet?
• Truncation errors are due to the use of a finite
  number of terms in an infinite series.
• Approximation error is due to approximating one
  function with another (simpler to compute)
  function.
• Iteration error is the due to the use of a finite
  number of iterations in an iterative process.
• Formulation errors are due to the analyst
  developing the incorrect equations or logic.
  This includes improper formulations near singular
  points, improper treatment of signs, incorrect
  treatment of units and others.
• Implementation errors are due to improper coding
  of the correct formulation.
• Round-off errors are those caused by finite word
  length computers.

116
Definition of Error
Position error - If (X,Y,Z) is the true
value of a point and (XA, YA, ZA) the approximate
value. - Use the Euclidean metric E2 (X-
XA)2 (Y- YA)2 (Z- ZA)2 to determine anerror
ball of radius E. For two dimensional systems,
set the Zs to 0. Angular error - There two
types of geodetic points (lat, lon, h) or for
the map projections (lat, lon, 0). - Except
for UTM, the forward transformations are exact.
General approach - Generate a known set of
points (lat, lon, h). - When the exact
transformation is available, generate the
corresponding exact set of points (X,Y,Z). -
E in terms of position errors can always be
calculated in two or three dimensions. UTM is a
special case - Because there is no exact
transformation in either direction - Angular
measures can be converted to distance measures
using s rø. - Again, start with a known set
of exact points (lat, lon). - Given the
approximate point (latA, lonA) compute e2 (lat
- latA)RM2 ( lon - lonA)RN2 - Where RN is
the radius of curvature in the prime vertical and
RM is the radius of curvature in the meridian.
- e is the (approximate) radius of the positional
error ball. - When the angular errors are
small, the error measure e is nearly E.
117
Numerical methods

• Analytic (closed form) solutions
• Taylor series
• Iteration
• Approximation methods

118
Efficient Evaluation of Special Functions
Common to Coordinate Transformations
Developed a generic machine independent timing
capability for SEDRIS. Transcendental functions
are frequently occurring and expensive to
compute. Eliminate them using identities or
in-line approximations. Relative cost of
evaluating common functions on modern workstation
below.
Normalized floating multiply
119
Analytic (closed form) Solutions

• Often can not be found.
• When available, always used in mapping, charting
  and geodesy applications.
• Advantages
• They provide exact reference values,
• Useful for derivations,
• ERM parameters embedded as variables.
• Disadvantages
• Usually involve many transcendental functions,
• Generally least efficient,
• Usually too accurate (wasted computation time).

120
Taylor/Maclaurin Series Methods

• Taylor series always exist for the type of
  transformations considered.
• Advantages
• Very useful for derivations,
• Can be used to get theoretical error bounds,
• ERM parameters embedded as variables,
• Power series can be inverted to yield the inverse
  function.
• Disadvantages
• Successive terms get very complex and hard to
  derive,
• Truncation error tends to grow rapidly away from
  expansion point,
• Almost always not as efficient as curve fitting
  or direct approximation.

121
Iterative Methods

• Coordinate transformation calculations can almost
  always be formulated using an iterative method.
• Advantages
• ERM parameters embedded as variables,
• If properly formulated they are almost always
  more efficient for the same accuracy as a power
  series,
• Usually the expressions involved are compact.
• Disadvantages
• Convergence rate depends on formulation and
  quality of the initial value,
• Initial guess must be efficient to compute,
• Efficient to compute stopping criterion needed,
• Even when this approach works, the direct
  approximation is almost always more efficient.

122
Direct Approximation of a Functionor its Inverse
Advantages - by far the most efficient
approach for a fixed accuracy criterion, -
maximum flexibility in efficiency vs. accuracy
tradeoffs, - can use piecewise approximation
for increased efficiency. Disadvantages -
more difficult to include ERM parameters as
variables, - requires non-linear approximation
tools to get coefficients, - requires
considerable analyst experience and intuition.
For ssin(ø)
becomes
and once the ERM is fixed becomes where
the Ci are constants.
123
Error in Power Series Expansions
Typical plot of the truncation error in one
dimension.
Checking at only one point, e.g., near D1, can
result in unwarranted confidence. When the
series expansion is three dimensional (the usual
case for coordinate system applications) there
may be many more zeros of the error function.
The error must be evaluated at all, or nearly
all, points wherethe series is expected to be
used.
124
Authoritative Sources Sometimes Appear to
Disagree
These are truncated series expansions for
Transverse Mercator taken from three different
sources are shown below.
(from TEC SR-7)
(from P.D. Thomas)
125
Error Analysis and Resolution of Disagreements

• For power series expansion it is possible to
  compute an exact upper bound on the error and
  this should be done.
• Testing should be done over the entire region of
  application using a very large and dense set of
  test points. This will insure sampling away from
  zeros of the error function, and will also
  validate the analytical error analysis and help
  find possible coding errors.
• The three representations shown on the previous
  slide, while appearing to be different, may be
  equivalent in the following sense
• Suppose that the test region is bounded and
  closed,
• that is R min. lon. , max. lon. X 0 , max.
  lon. X min. e , max. e where e is the
  eccentricity of the set of meridian ellipses
  being considered,
• then generate a dense grid on the three
  dimensional region R,
• then compare the three alternatives at each grid
  point,
• if the maximum absolute difference between them
  is less than some acceptable value (e.g., one
  millimeter) then they have equivalent accuracy.

Proper error analysis requires a very dense set
of test points in R.
126
Effect of Small Errors
For small regions all the map projections are
the same.
Take such statements with a grain of salt. -
The original Bowditch book is very old, - In
navigation meters of error are small, - In
geodesy a meter is small to some, large to
others, - In GPS applications a meter is big,
- To a gunner who aims at the turret ring of an
enemy tank a meter is really big, - It only
takes a small curvature distortion to hide or
uncover a target and on some terrain
surfaces this is a frequent occurrence, - In
some real time embedded systems small errors may
accumulate. The application domain determines
what is small, in MS very accurate
representations of spatial reference frameworks
are required for VVA support and to promote a
level playing field.
From N. Bowditch, American Practical
Navigator, U.S. Navy Hydrographic Office,1966 Ed.
127
Fuzzy Creep/Coordinate Drift

• Digital computers have finite word lengths.
• The set of numbers that they deal with is finite.
• They cannot perform real number arithmetic
• This complicates the problem of determining
  whether or not a point is inside or outside an
  interval or region (polygon).
• Coordinate transformations can be exact only in
  rare cases. There is always some combination of
  round-off and algorithm error.
• After a sequence of computations these errors may
  increase (although they are generally very
  small). This is sometimes called coordinate
  drift or fuzzy creep.
• In a particular application domain the developer
  must be aware of this possibility and institute
  the appropriate actions.
• Examples
• Using a GUI, a player places a combatant just
  inside a building. Due to round-off, the
  internal model representation of the combatant
  may be outside the building and can be acquired
  and attrited. If polygonal regions are involved
  and the original point is in another coordinate
  system, coordinate drift can cause this kind of
  behavior.
• Sequences of coordinate transformations may not
  be reversible near (e.g., UTM) zone boundaries.

128
Bounds Checking

• Coordinate transformations may only be valid for
  prescribed regions.
• Both input and output should be checked for
  validity.
• For many coordinate frameworks the mathematical
  formulation may exist everywhere even though the
  results may be nonsensical.
• Determining acceptable bounds requires an
  examination of distortion effects and in some
  cases computational accuracy.
• This is an emerging issue for SRM standardization
  efforts.
• The problem is multi-faceted
• What are the