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  ![](media/image1.emf){width="0.4305555555555556in"
  height="0.4583333333333333in"}**Fermilab**

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> **Doc. No. :** Beams-doc-1148
>
> **Old Doc. No.** **:** MI-0209

#  Version : 4.0

> **FERMILAB COORDINATE SYSTEMS**
>
> **Babatunde \"O\'Sheg\" Oshinowo**
>
> **Survey Alignment and Geodesy Group**
>
> **Fermi National Accelerator Laboratory**
>
> **P.O. Box 500, MS 334**
>
> **Batavia, Illinois 60510**
>
> **05/15/97**
>
> **ABSTRACT**
>
> The objectives of this document are to relate Fermilab to a global
> coordinate system, to define a mapping projection and establish all
> the coordinate systems that will be used at Fermilab and for the
> Fermilab Main Injector (FMI) project, and to describe all the
> parameters that are necessary for the several coordinate systems. The
> main objective is to define a new **assimilated** DUSAF coordinate
> system for the Fermilab site. This implies that the new system will
> have the same origin and coordinate axes definitions as the DUSAF
> coordinate system. The Survey Alignment and Geodesy (SAG) group, has
> defined a **Fermilab Site Coordinate System** (**FSCS**) for the site
> and a **Local Tunnel Coordinate System** (**LTCS**) for the FMI
> project. A **Double Stereographic Projection** has been adopted to
> define both coordinate systems.
>
> **Table of Contents**

**1. Introduction**

> 1.1 DUSAF Coordinate System
>
> 1.2 Map Projection

**2. Geodetic Coordinate System (GCS)**

2.1 Orthometric Height

**3. Geodetic Cartesian Coordinate System (GCCS)**

**4. Fermilab Site Coordinate System (FSCS)**

4.1 FSCS:XYZ

4.2 FSCS:XYH

**5. Local Tunnel Coordinate System (LTCS)**

5.1 LTCS:XYZ

5.2 DSP:XYH

5.3 LTCS:XYH

6**. Local Geodetic System (LGS)**

**7. Oblique Mercator Projection System (OMPS)**

**8. Illinois State Plane System (ISPS)**

**9. Acknowledgment**

**References**

**Appendix A -** Spatial Definition of the FMI Plane

**Appendix B -** Notes on Scale Compatibilities

**Appendix C -** Notes on Precision and Accuracy

**Appendix D -** Double Stereographic Projection

**Appendix E -** Lattice Program Documentation and Source Codes

**FERMILAB COORDINATE SYSTEMS**

**1. Introduction**

Several years ago a DUSAF coordinate system was established at Fermilab
and the original documentation could not be traced. It is a Cartesian
coordinate system with no known map projection associated with it.
Therefore, it is not possible to directly convert these DUSAF plane
coordinates to the global geodetic (or geographic) coordinate system
using the appropriate Earth parameters. The need for a map projection
and a tie to a global coordinate system also stems from the future
global experiments being planned at Fermilab. These global experiments
use the Global Positioning Systems (GPS) technology in which data is
collected in the geodetic coordinate system.

The objectives of this document are to relate Fermilab to a global
coordinate system, to define a mapping projection and establish all the
coordinate systems that will be used for the Fermilab site and for the
Fermilab Main Injector (FMI) project, and to describe all the parameters
that are necessary for the several coordinate systems.

Since all lattice coordinates for all Fermilab and FMI projects are
provided in the DUSAF coordinate system, the main objective is then to
define a new **assimilated** DUSAF coordinate system for the Fermilab
site. This implies that the new system will have the same origin and
coordinate axes definitions as the DUSAF coordinate system.

**1.1 DUSAF Coordinate System**

The DUSAF coordinate system is a right-handed Cartesian coordinate

system defined as follows:

Origin - A0

Y-axis - NORTH axis. Positive along the extraction line towards the

Neutrino Area

X-axis - EAST axis. Positive to the right and perpendicular to the

y-axis

Z-axis - ELEVATION axis. Positive up at A0 and perpendicular to

both X- and Y-axes.

The parameters that define the DUSAF coordinate system origin are:

X = 100000.000 ft (30480.06096 m) ; False Easting at A0

Y = 100000.000 ft (30480.06096 m) ; False Northing at A0

Z = 720.000 ft ( 219.45644 m) ; Elevation

The DUSAF elevation Z is referenced to the DUSAF (Vertical) Datum which
is an arbitrary datum. At A0, Z = 720.000 ft implies that the elevation
of A0 is 720 ft above the DUSAF Datum.

**1.2 Map Projection**

The Survey Alignment and Geodesy group has adopted a **Double
Stereographic Projection** (see Appendix D) to define a **Fermilab Site
Coordinate System (FSCS)** for the Fermilab site. The **FSCS** is
referenced to the North American Datum of 1983 (NAD83) and its reference
ellipsoid of 1980 (GRS80). The Double Stereographic Projection was
chosen to map the GRS80 geodetic coordinates onto the conformal mapping
plane which defines the Fermilab Site Coordinate System. The Double
Stereographic Projection was also adopted to define a **Local Tunnel
Coordinate System (LTCS)** for the FMI project.

Before these new coordinate systems and other projection systems are
defined in greater details, the **Geodetic Coordinate System (GCS)**
will be described. The geodetic coordinate system is the fundamental
coordinate system with respect to which all other coordinate systems are
defined.

**2. Geodetic Coordinate System (GCS)**

**GCS** is a curvilinear coordinate system, based upon a geocentric,
bi-axial reference ellipsoid (GRS80), called the North American Datum of
1983 (NAD83), with axes coincident with those of the Geodetic Cartesian
Coordinate System, axis of rotation along the Z-axis (see section 3
below). It is the fundamental coordinate system with respect to which
all other coordinate systems are defined. The geometric parameters of
the ellipsoid are:

*a* = 6378137.000 m (exactly, by definition)

*1/f* = 298.257 222 101 (to 12 significant digits)

where *a* is the semi-major axis of the ellipsoid and *f* is the
flattening of the ellipsoid.

The coordinates of a given point, P~i~, within this system are defined
as:

φ~i~ = Geodetic Latitude

λ~i~ = Geodetic Longitude

h~i~ = Ellipsoidal Height (height above the ellipsoid, normal

to the ellipsoidal surface)

Since the ellipsoidal height is directly related to the orthometric
height (elevation), this relationship will be described in the following
section.

**2.1 Orthometric Height**

The orthometric height, H, of a point, P, is defined as the geometric
distance between the geoid and the point, measured along the plumb line
through the point. The orthometric height is based on the North American
Vertical Datum of 1988 (NAVD88). The orthometric height above the NAVD88
datum is determined by

H~NAVD88~ = h - N

where

h = Ellipsoidal Height, as defined in the Geodetic Coordinate

System

N = Geoid Height, the separation between the Reference

Ellipsoid and the Geoid at point, P.

The geoid height values are interpolated from a grid of regularly-spaced

estimates using the GEOID93 Model based on GRS80 (or WGS84, which is
equivalent to NAD83, as defined below). The actual computation is
performed as an interpolation from a regularly-spaced grid of points.
The interpolation is accomplished by a locally fit biquadratic function.
The polynomial surface is fit to nine data points defining the area
surrounding the point where the interpolation is to take place.

The NAVD88 orthometric heights are derived from heights above the DUSAF
datum using the expression:

H~NAVD88~ = H~DUSAF~ + dH~NAVD88~

where dH~NAVD88~ = - 0.17308 m is the correction to tie the DUSAF datum
to the NAVD88 datum.

The relationship between the ellipsoidal height in the **GCS** system
and the orthometric height above the DUSAF datum is given by:

h = H~DUSAF~ + dH~NAVD88~ + N

**3. Geodetic Cartesian Coordinate System (GCCS)**

**GCCS** is a geocentric, right handed Cartesian coordinate system, with
axes defined to be coincident with those of the Reference Ellipsoid
GRS80, used for NAD83 datum. It is used as an intermediate system by

which other systems are converted to and from the geodetic coordinate
system **GCS**. The orientation is defined as follows:

Z-axis - Parallel to the direction of the BTS-84 Terrestrial Pole

BTS-84 = Bureau International de l'Heure (BIH) Terrestrial

System of 1984

X-axis - Parallel to the intersection of the BTS-84 Reference

Meridian and the plane of the equator of the BTS-84 pole

Y-axis - Perpendicular to both X- and Z-axes, to complete the right

handed coordinate system.

In these respects, NAD83 is similar to the other modern global reference
system, such as the World Geodetic System of 1984 (WGS84). The Global
Positioning Systems (GPS) data are given in the WGS84 coordinate system.
In principle, the three dimensional coordinates of a single physical
point should be the same in both systems; in practice, small differences
are sometimes found. For these coordinate system definitions and
transformations, it can be assumed for all practical purposes that the
NAD83 and WGS84 are entirely coincident. There are standard formulas
used for conversion between the **GCCS** coordinates (XYZ) and the
geodetic coordinates (φ, λ,h) of the **GCS**. Transformation between the
**GCCS** coordinates (XYZ) and other Cartesian coordinate systems are
also possible.

**4. Fermilab Site Coordinate System (FSCS)**

The Fermilab Site Coordinate System (FSCS) is an **assimilated** DUSAF
coordinate system. Its origin and rotation axes are located at A0,
preserving as nearly as possible the DUSAF coordinate system.

**4.1 FSCS:XYZ**

**FSCS:XYZ** coordinate system corresponds to the lattice version of the

**FSCS**. This coordinate system was developed for the beamlines
extracting from the Main Ring and the Tevatron at A0. It is a
right-handed Cartesian coordinate system defined as follows:

Origin - A0

Y-axis - NORTH axis rotated by Geodetic Azimuth α from the

Geodetic North. Positive along the extraction line towards

the Neutrino Area

X-axis - EAST axis. Positive to the right and perpendicular to the

Y-axis

Z-axis - Positive up and perpendicular to both X- and Y-axis. It is

aligned with the direction of the normal to the ellipsoid at

A0.

The parameters that define the **FSCS:XYZ** coordinate system origin at
A0 are:

X = 100000.000 ft (30480.06096 m)

Y = 100000.000 ft (30480.06096 m)

Z = 720.000 ft ( 219.45644 m)

To relate the **FSCS:XYZ** system to the **GCS** system, the ellipsoidal
height of A0 is given as:

h~0~ = 186.49880 m

The Geodetic Azimuth α defining the alignment of the Y-axis at A0 is
given as:

α = 38° 16′ 48.01429″

**4.2 FSCS:XYH**

**FSCS:XYH** is a right-handed Cartesian coordinate system based on a

Double Stereographic Projection and heights above the DUSAF datum. At
the origin A0, the (XYH) coordinates in this system are identical to the
(XYZ) coordinates in the **FSCS:XYZ** system. Figure 1 shows the
relationship between the **FSCS:XYZ** and **FSCS:XYH** systems.

**FSCS:XYH** is a two dimensional mapping plane whose coordinates (X,Y)
or (E, N; Easting, Northing) are generated by a Double Stereographic
Projection of geodetic coordinates (φ,λ), with the origin defined as the
point on the ellipsoid corresponding to A0. The Double Stereographic
Projection is performed in two steps: namely, the projection from the
reference ellipsoid to a conformal sphere and from the sphere to a
plane.

The basic parameters for performing the Double Stereographic Projection
are as follows:

φ~0~ = Geodetic Latitude of the origin

λ~0~ = Geodetic Longitude of the origin

h~0~ = Ellipsoidal Height of the origin

X~0~ = False Easting at the origin

Y~0~ = False Northing at the origin

F~0~ = Scale factor at the origin

α = Geodetic Azimuth of the Y-axis (North) at the origin

The geodetic coordinates (φ,λ) are projected into E and N using the
double Stereographic projection, with the origin defined as the point on
the ellipsoid corresponding to A0. The E and N coordinates are rotated
about the Z-axis by the angle α (geodetic azimuth) to obtain the X and Y
coordinates. The X and Y coordinates are then re-scaled by the scale
factor F~0~ to give true scale at the origin at 720 ft above the DUSAF
datum. False coordinates are applied at the origin.

![](media/image2.emf){width="6.388888888888889in"
height="9.079166666666667in"}

The defined parameters of the Double Stereographic Projection for the
**FSCS:XYH** are given below:

Geodetic coordinates at the origin A0

φ~0~ = N 41° 50′ 14.312704″

λ~0~ = W 88° 15′ 41.143123″

h~0~ = 186.49880 m

The relationship between the ellipsoidal height h~0~ in the **GCS**
system and the orthometric height above the DUSAF datum is given by:

h~0~ = H~720DUSAF~ + dH~NAVD88~ + N~0~

where H~720DUSAF~ corresponds to 720 feet above the DUSAF datum and

dH~NAVD88~ = -0.17308 m is the correction to tie the DUSAF datum to the
NAVD88 datum. N~0~ is the geoid height (geoid93 model) at A0 and is
equal to -32.78456 m. At any other point the ellipsoidal height is given
as:

h = H~DUSAF~ + dH~NAVD88~ + N

The orthometric height at A0 is given by:

H~720NAVD88~ = h~0~ - N~0~

The orthometric height at A0 in the **FSCS** = 219.45644 - 0.17308 =
219.28336 m. This relationship must be considered when transforming
between the **FSCS** and other coordinate systems.

The orthometric heights are converted to heights above the DUSAF datum
using the expression:

H~DUSAF~ = H~NAVD88~ - dH~NAVD88.~

Geodetic Azimuth of the Y-axis of **FSCS** at the origin A0 is given by:

α = 38° 16′ 48.01429″

False coordinates applied at the origin are given by:

X~0~ = 100000.000 ft (30480.06096 m) ; False Easting at A0

Y~0~ = 100000.000 ft (30480.06096 m) ; False Northing at A0

At A0, the scale factor corresponding to the height of 720 ft above the
DUSAF datum (H~720DUSAF~) is given by:

F~0~ = 1.000029251309483

**5. Local Tunnel Coordinate System (LTCS)**

The Local Tunnel Coordinate System (LTCS) was established to meet the
stringent accuracy requirements of the FMI project. Its origin and
rotation axes are located at a point, CFMI, which lies at the centroid
of the FMI Plane and whose (X,Y) coordinates are in the **FSCS**. The
coordinates in the LTCS are defined in the FMI Plane. The FMI Plane is
that plane defined by the nominal designed orthometric heights of the
cell boundaries 308, 522, and 620. The coordinates of the cell
boundaries 308, 522, and 620 in the FMI Plane (and other coordinate
systems) are given in Table A1 in Appendix A. To convert the LTCS
coordinates to global geodetic coordinates the FMI Plane must be tilted
by a small angle to a projection plane

**5.1 LTCS:XYZ**

The **LTCS:XYZ** coordinate system corresponds to the lattice version of
the **LTCS**. The X-Y plane of the **LTCS** is coincident with the FMI
plane. It is a right-handed Cartesian coordinate system defined as
follows:

Origin - CFMI (Centroid of the FMI Plane)

Y-axis - NORTH axis rotated by Geodetic Azimuth α from the

Geodetic North.

X-axis - EAST axis. Positive to the right and perpendicular to the

Y-axis.

Z-axis - Positive up at CFMI and perpendicular to both X- and

Y-axes.

The parameters that define the **LTCS:XYZ** coordinate system origin at
CFMI are:

X = 100,661.49800 ft (30681.68595 m)

Y = 95,856.98802 ft (29217.26838 m)

Z = 715.664 ft ( 218.13491 m)

To relate the **LTCS:XYZ** system to the **GCS** system, the ellipsoidal
height of CFMI is given as:

h~0~ = 185.19035 m

The Geodetic Azimuth α defining the alignment of the Y-axis at CFMI is
given as:

α = 38° 16′ 29.97831″

**5.2 DSP:XYH**

**DSP:XYH** is a right-handed Cartesian coordinate system based on

Double Stereographic Projection and heights above the DUSAF datum.

The (XYH) coordinates in this system are identical to the (XYZ)

coordinates in the **LTCS:XYZ** system at the origin CFMI. The
relationship between the **LTCS:XYZ** and **DSP:XYH** systems is shown
in Figure 2.

**DSP:XYH** is a two dimensional mapping projection plane whose
coordinates (X,Y) or (E, N; Easting, Northing) are generated by a Double
Stereographic Projection of geodetic coordinates (φ,λ), with the origin
defined as the point on the ellipsoid corresponding to CFMI. This
projection plane is tilted from the **LTCS:XYH** FMI plane by an angle
see Figure 2 The basic parameters of the projection are the same as
those of **FSCS:XYH**. The parameters for FMI have been selected to
minimize

![](media/image3.emf){width="6.933333333333334in"
height="9.036805555555556in"}

the effect of the point scale factor for all points around the FMI
tunnel. The parameters of the projection are given below.

The geodetic coordinates (φ,λ) are projected into E and N using the
double Stereographic projection, with the origin defined as the point on
the ellipsoid corresponding to CFMI. The E and N coordinates are rotated
about the Z-axis by the angle α (geodetic azimuth) to obtain the X and Y
coordinates. The X and Y coordinates are then re-scaled by the scale
factor F~0~ to give true scale at the origin at 715.664 ft above DUSAF
datum. False coordinates are applied at the origin. The defined
parameters of the Double Stereographic Projection for the **DSP:XYH**
are given below:

Geodetic coordinates at the origin at CFMI (Centroid of the FMI Plane):

φ~0~ = N 41° 49′ 38.134927″

λ~0~ = W 88° 16′ 08.184535″

h~0~ = 185.19035 m

The ellipsoidal height in the **GCS** system, h, is converted to an
orthometric height above the DUSAF datum using the same definitions and
parameters as those given for the **FSCS:XYH** coordinate height.

Geodetic Azimuth of the Y-axis of **LTCS** at the origin CFMI is given
by:

α = 38° 16′ 29.97831″

False coordinates applied at the origin are given by:

X = 100,661.49800 ft (30681.68595 m) ; False Easting at CFMI

Y = 95,856.98802 ft (29217.26838 m) ; False Northing at CFMI

At CFMI , the scale factor corresponding to the height of CFMI above the
DUSAF datum (H~CFMIDUSAF~) is given by:

F~0~ = 1.000029046120306

**5.3 LTCS:XYH**

The actual working plane of reference for the FMI tunnel is the FMI
Plane. Therefore the coordinates in the **DSP:XYH** projection plane
must be related to the FMI Plane. The **LTCS:XYH** coordinates in the
Lattice Program (see Appendix E) refers to the **LTCS:XYH** coordinates
in the FMI Plane. The relationship between the **LTCS:XYZ, DSP:XYH** and
**FSCS:XYH** systems is shown in Figure 2.

To rotate the coordinates on the **DSP:XYH** projection plane to the FMI

Plane a seven-parameter transformation is performed using the following

expression:

**[X]{.underline}** ~LTCS:XYH~ = **[X]{.underline}**~Translation~ + S \*
**R**(~XYH~) \* **[X]{.underline}**~DSP:XYH~

where **[X]{.underline}**~LTCS:XYH~ is the vector containing the (XYH)
coordinates in the FMI Plane; **[X]{.underline}**~DSP:XYH~ is the vector
containing the (XYH) coordinates in the projection plane;
**[X]{.underline}**~Translation~ is the vector containing the
translation parameters in XYH; **R**(~XYH~) is the rotation matrix; and
S is the scale.

The transformation parameters from the **DSP:XYH** projection plane to
the **LTCS:XYH** FMI Plane are defined as follows:

Scale = 1.00000 Fixed

X Rotation, ~X~ = 2.07594 arcsec

Y Rotation, ~Y~ = -0.74326 arcsec

H Rotation, ~H~ = 0.00000 arcsec Fixed

X Translation = -0.00079 m

Y Translation = -0.00220 m

H Translation = 0.38636 m

**6. Local Geodetic System (LGS)**

A left handed Cartesian coordinate system, defined to be topocentric
about a point, P, i.e., a coordinate system with the origin at a point,
P, of known geodetic coordinates. The left-handed coordinate system is
defined as

follows:

h-axis - Outward ellipsoid normal in the **GCS** system passing through

point P.

n-axis - Perpendicular to the h-axis, and directed towards Geodetic

North (i.e., lies in the plane defined by the Geodetic

Cartesian Z-axis and point P).

e-axis - Perpendicular to both the n- and h-axes, forming a left

handed coordinate system with the positive direction to the

East.

It should be noted that there exists an infinite number of local
geodetic coordinate systems, dependent upon the choice of point P.

**7. Oblique Mercator Projection System (OMPS)**

The Global Positioning Systems (GPS) data collected at the Fermilab are
in the geodetic coordinate system. Present software available transform
these data into a conformal mapping plane using the Oblique Mercator
Projection and the same basic parameters as defined for the Double
Stereographic Projection.

The Oblique Mercator Projection Coordinate System (**OMPS**) is a two
dimensional mapping plane whose coordinates (X,Y) or (E, N), are
generated by a Oblique Mercator Projection of Geodetic Latitude and
Longitude (φ,λ) in the **GCS** system.

The defined parameters of the Oblique Mercator Projection for the
Fermilab site are given below:

Geodetic Origin at A0

φ~0~ = N 41° 50′ 14.312704″

λ~0~ = W 88° 15′ 41.143123″

h~0~ = 186.49880 m

The ellipsoidal height in the **GCS** system, h, is converted to an
orthometric height above the DUSAF datum using the same definitions and
parameters as those given for the **FSCS:XYH** coordinate height.

Geodetic Azimuth of the Central Meridian (center line) of the projection
at the origin A0 is given by:

α = 38° 16′ 48.01429″

False coordinates applied at the origin are given by:

X~0~ = 100000.000 ft (30480.06096 m) ; False Easting at A0

Y~0~ = 100000.000 ft (30480.06096 m) ; False Northing at A0

At A0, the scale factor corresponding to the height of 720 ft above the
DUSAF datum (H~720DUSAF~) is given by:

F~0~ = 1.000029251309483

**8. Illinois State Plane System (ISPS)**

It might be necessary sometime to relate the FSCS to an outside monument
coordinates defined by the Illinois State Plane Coordinate System
(**ISPS**). The Illinois State Plane Coordinate System is a two
dimensional mapping plane whose coordinates (x, y) or (E, N), are
generated by a 1983 State Plane Transverse Mercator Projection of
Geodetic Latitude and Longitude (φ,λ) in the **GCS** system. The zone
used in the vicinity of Fermilab site is the Illinois East Zone #1201 as
given by the National Geodetic Survey (Stem, 1989).

The parameters of the projection are given by:

Grid Origin:

φ~0~ = N 36° 40′ 00.00000″

λ~0~ = W 88° 20′ 00.00000″

False coordinates applied at the origin are given by:

E~0~ = 300000.0000 m ; False Easting at the grid origin

N~0~ = 0.0000 m ; False Northing at the grid origin

Central meridian scale factor at the grid origin

F~0~ = 0.9999750000

**9. Acknowledgment**

Thanks to all those who made this paper possible. The geodetic committee
comprising of the author, Virgil Bocean, and George Wojcik defined and
ratified the coordinate systems and projection parameters. Virgil Bocean
computed the projection parameters which were also recomputed by the
author for verification. Virgil Bocean also wrote Appendix A. Terry
Sager, Stu Lakanen, John Greenwood, and the committee members reviewed
this documentation. Helpful discussions with all the above mentioned are
also gratefully acknowledged.

**References**

Leick, A., (1990). GPS Satellite Surveying, John Wiley & Sons, Inc., New
York.

Schwarz, C. R., (1989). North American Datum of 1983, NOAA Professional
Paper NOS 2, Ed. Charles R. Schwarz, National Geodetic Survey Rockville,
MD.

Snyder, J. P., (1987). Map Projections - A Working Manual. U.S.
Geological Survey Professional Paper 1395. U.S. Government Printing
Office, Washington D.C.

Stem, J. E., (1989) . State Plane Coordinates System of 1983. NOAA
Manual NOS NGS. National Geodetic Information Branch, NOAA, Rockville,
MD.

The Fermilab Main Injector, Technical Design Handbook. Fermilab Main
Injector Department, Fermi National Accelerator Laboratory, August 1994.

The FMI Collaboration Internal Communication. Fermi National Accelerator
Laboratory, December 12, 1995.

Thompson, D.B., Mepham, M.P. and R.R. Steves, (1977). The Stereographic
Double Projection. The Department of Surveying and Engineering,
University of New Brunswick, Fredericton, N.B., Canada. Technical Report
No. 46.

Vanicek, P., and Krakiwsky, E. J., (1982). Geodesy: The Concepts,
North-Holland Publishing Company.

**Appendix A**

**Spatial Definition of the FMI Plane**

The objective of this section is to summarize the definition of the
plane that contains the Fermilab Main Injector (FMI) with respect to the
existing Tevatron and the local Earth parameters. The details of the FMI
location are determined by the requirements for transfers of both
protons and antiprotons into the Tevatron.

The FMI is situated southwest of the Tevatron Ring. The MI-60 straight
section is parallel to the Tevatron F0 straight section, and has the
FSCS Azimuth = 301^o^ 13' 53.9". The two beamlines are separated by
11.823 m horizontally. The design location of the MI-60 reference point
is 13.222 m downstream from the F0 TeV point, and offset from the TeV
straight section by 11.823 m (FMI Technical Design Handbook, 1994).

The FMI is designed to be a planar machine. The final vertical
definition of the FMI plane begins by specifying that cell boundaries
522 and 620 are designed at the nominal orthometric height of 218.15314
m (DUSAF Datum), and allow the initial constraint on delta elevation
between the FMI and TeV to vary (FMI Collaboration Internal
Communication, 1995). Based on the latest (1995) as found elevation of
the Tevatron F0 straight section quads string (220.48886 m DUSAF Datum),
and the nominal FMI design elevation (218.15314 m DUSAF Datum), the FMI
plane is located 2.33572 m below the Tevatron beam.

Additionally, to account for the relative tilt of the FMI and the
Tevatron planes, and also for fully constraining the FMI plane in the
geodetic space with respect to the local Earth parameters, it is further
specified that the cell boundaries 522, 620, and 308 are placed at the
same nominal design orthometric height of 218.15314 m (DUSAF Datum). The
coordinates of the cell boundaries 308, 522, and 620 in the FMI Plane
(and other coordinate systems) are given in Table A1.

![](media/image4.emf)

**Appendix B**

**Notes on Scale Compatibilities**

S~c~ Distance between two points inverted from the coordinates.

S~a~ Actual distance between two points as measured in reality.

**GCCS** This is a Cartesian coordinate system, which does not involve
the use of a projection. S~c~ and S~a~ will be equal.

**FSCS:XYZ** This is a Cartesian coordinate system. S~c~ and S~a~ will
be equal.

**FSCS:XYH** This is a projection coordinate system that has been
re-scaled by the appropriate scale factor F to be compatible with the
720 ft above DUSAF datum. Therefore a two dimensional distance S~a~
measured at the 720 ft height is related to S~c~ by the expression:

S~c~ = k**^.^**S~a~

where the projection scale factor k is equal to 1 at the origin.

Away from the 720 ft above DUSAF datum an additional scaling is
required:

S~c~ = k**^.^**$\frac{F\text{720}}{Fh}$**^.^**S~a~

where F~h~ is the height scale factor at the measurement height, and
F~720~ is the standard height scale factor (= 1.000029251309483).

**LTCS:XYZ** This is a Cartesian coordinate system. S~c~ and S~a~ will
be equal.

**LTCS:XYH** This is a projection coordinate system that has been

re-scaled by the appropriate scale factor F to be compatible with the
715.664 ft above DUSAF datum. In a similar definition to that used for

the **FSCS:XYH**, the relationship between the two dimensional distance
S~c~ and S~a~ is:

S~c~ = k**^.^**$\frac{F\text{CFMI}}{Fh}$**^.^**S~a~

where k is the projection scale factor, equal to 1 at the origin CFMI.

F~h~ is the height scale factor at the measurement height, and F~CFMI~
is the standard height scale factor (= 1.000029046120306).

**LGS** This is a Cartesian coordinate system. S~c~ and S~a~ will be
equal.

**OMPS** This is a projection coordinate system that has been re-scaled
by the appropriate scale factor F~0~ (= 1.000029251309483) to be
compatible with the 720 ft above DUSAF datum.

In a similar definition to that used for the **FSCS:XYH**, the
relationship between the two dimensional distance S~c~ and S~a~ is:

S~c~ = k**^.^**$\frac{F\text{720}}{Fh}$**^.^**S~a~

where k is the projection scale factor, equal to 1 along the center line
of the projection (azimuth α); F~h~ is the height scale factor at the
measurement height; and F~720~ is the standard height scale factor at
the DUSAF datum.

**ISPS** This is a projection coordinate system that has not been
re-scaled by to any datum.

The relationship between the two dimensional distance S~c~ and S~a~ is:

S~c~ = k**^.^**$\frac{1}{Fh}$**^.^**S~a~

where k is the projection scale factor, equal to 0.999975 along the
central meridian line of the projection (azimuth α); F~h~ is the height
scale factor at the measurement height, equal to 1 on the ellipsoid.

**Appendix C**

**Notes on Precision and Accuracy**

The precision of the conversion from one coordinate system to another
has been specified by the SAG group to be 1x10^-7^ m. Therefore in
converting from one system to another and back again, the coordinates
will be compatible to this level.

In terms of accuracy, it should be noted that a coordinate conversion
cannot compensate for the inaccuracies in the original data. Therefore
in converting to the **ISPS**, for example, the compatibility with other
data can only be as good as the survey that defined the geodetic
coordinates of the point A0.

**Appendix D**

**Double Stereographic Projection**

Traditional geodetic computations are carried out on the surface of a
biaxial ellipsoid (ellipsoid of revolution), which is the mathematical
figure that is the most convenient representation of the size and shape
of the earth. To perform the same computations on a plane, it is
necessary to map the ellipsoidal information (points, angles, lines,
etc.) on a plane mapping surface. A convenient mapping for geodetic
purposes is a conformal mapping in which ellipsoidal angles are
preserved on the mapping plane.

The Stereographic projection of an ellipsoid of revolution can be

approached by a double projection (hence double Stereographic
projection), in which the biaxial ellipsoid is conformally mapped to a
sphere, which is then \"stereographically\" projected to a plane.

The Stereographic projection of a sphere to a plane has the following

properties (Thompson, et. al., 1977):

\(i\) It is a perspective projection, the features being projected from
the

sphere onto the plane from a common point of perspective. This
perspective center lays on the sphere, and is the antipodal point of the
point at which the plane is tangent to the sphere.

\(ii\) It is an azimuthal projection, characterized by the fact that the

direction, or azimuth, from the center of the projection to every other
point

on the map is shown correctly.

\(iii\) It is a conformal projection, the relative local angles about
every

point on the map are shown correctly (angles not distorted).

\(iv\) Scale increases away radially from the origin (center) of the
projection.

\(v\) Great circles are projected as circles.

The Stereographic projection of a sphere to a plane is the only true

perspective projection of any kind that is also conformal. There is no
mapping of an ellipsoid to a plane that possesses all the
characteristics of the Stereographic mapping of a sphere to a plane. For
the conformal mapping of the biaxial ellipsoid on a plane, the
ellipsoidal data is first mapped conformally on to a sphere. Then, a
second conformal mapping of the spherical data to the plane completes
the double Stereographic projection. The result is a conformal mapping
of ellipsoidal data on a plane, since the two mappings are conformal.

The Stereographic projection is the most widely used azimuthal
projection, mainly used for portraying large areas of similar extent in
all directions. The Stereographic projection is very suitable when areas
of interest are circularly shaped, like the FMI area. Since all the
surveying activities will occur along the path of the ring, choosing the
origin of the projection at the centroid of the FMI Plane has as a
result that the scale along the tunnel will be standardized to the same
scale factor. The differences created by the fact that the FMI shape is
not perfectly circular is very insignificant.

**Appendix E**

**Lattice Program Documentation and Source Codes**

**errata**

**LATTICE PROGRAM DOCUMENTATION**

**On page 3,**

Geodetice Cartesian Coordinate System

should read

Geodetic Cartesian Coordinate System

**On page 6,**

X~GC~ X~GC~

**X~GC\ =~** Y~GC~ should read **X~GC\ =\ ~**Y~GC~

Y~GC~ Z~GC~

X~LG~ X~LG~

**X~LG\ =~** Y~LG~ should read **X~LG\ =\ ~**Y~LG~

Y~LG~ Z~LG~

**On page 8,**

X~FXYZ~ X~FXYZ~

**X~FXYZ\ =~** Y~FXYZ~ should read **X~FXYZ\ =\ ~**Y~FXYZ~

Y~FXYZ~ Z~FXYZ~

X~LG~ X~LG~

**X~LG\ =~** Y~LG~ should read **X~LG\ =\ ~**Y~LG~

Y~LG~ Z~LG~

**On page 8,**

a = Geodetic Azimuth of the Fermilab (XYZ) Origin

should read

= Geodetic Azimuth of the Fermilab (XYZ) Origin

**On page 12,**

= Geodetic Azimuth of the Fermilab (XYZ) Origin