Penn State Surveying Program

COORDINATE SYSTEMS

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Table of Contents

  1. Introduction
  2. Three-dimensional Rotations
  3. Fundamentals of Geodetic Astronomy
  4. Commonly Used Coordinate Systems
  5. Transformation between Datums
    1. Converting between a Celestial Reference Frame and a Terrestrial Reference Frame
    2. Helmert Transformation
    3. Molondensky Transformation
  6. Time Systems
  7. Satellite Coordinate Systems

 

INTRODUCTION

The determination of the coordinates of a point with respect to some established coordinate system is called point positioning. This method of positioning is also known as absolute positioning. This distinguishes this method from relative methods such as traversing, differential leveling, or networks where coordinates of points are relative to the values of one or more pairs of coordinates.

Traditionally, most point positioning was performed using observations made on celestial objects such as the stars, planets, sun, etc. These methods are still frequently used today on our most precise stations. However with the introduction of artificial stars (a.k.a. satellites), it has been possible to perform point positioning using tools such as GPS.

Point positioning can be performed in using three different types coordinate systems. They are (1) terrestrial coordinate systems for positioning of earth-based points, (2) celestial coordinate systems for sighting and positioning of earth-based points using celestial objects, and (3) orbital coordinate systems for positioning of earth-based points using satellite systems.

The earth has two important motions to consider when developing these coordinate systems. That is, the earth (1) revolves around the sun, and it (2) spins on its instantaneous axis. Terrestrial coordinate systems are earth-fixed and thus both revolve and spin with the earth. Celestial coordinate systems are sun (helio)-centered and thus do not revolve with the earth, but may spin with the earth. Orbital coordinate systems do not spin with the earth, but revolve with it!

In this lesson we will define the various coordinate systems and the methods used to transfer from one coordinate system to another. In order to transform one set of coordinates to another, we need to define the three-dimensional rotation matrices.

 

Three-Dimensional Rotations

The rotation of a three-dimensional coordinate system involves three rotations. Each rotation is a two-dimensional coordinate rotation where one coordinate axis is held fixed while the other two are rotated about this fixed axis. The rotation is considered positive for counter-clockwise rotations as viewed from the positive end of the rotating (fixed) axis.

 

R1 Rotation

The first rotation, shown in to the right, is a rotation of the Y' and Z' axis about the X' axis by an amount of q1. It is known as R1. From the figure it can be seen that the coordinates for y1 and z1 can be computed as

x1 = x'
y1 = y' cos q + z' sin q
z1 = -y' sin q + z' cos q

This can be written in matrix form as

X1 = R1X'

where

R2 Rotation

The second rotation, shown in the figure to the right, is about once rotated Y1 axis rotating the X1 and Z1 axes by an amount of q2. From the figure it can be seen that

x2 = x1 cos q - z1 sin q
y2 = y1
z2 = x1 sin q + z1 cos q

Thus, the R2 rotation can be written in matrix notation as

X2 = R2 X1

where 

R3 Rotation

The third and final rotation, shown in figure to the right, is about the twice rotated Z2 axis rotating X2 and Y2 by an amount of q3. From the figure it can be seen that the rotations can be expressed mathematically as

x = x2 cos q + y2 sin q
y = -x2 sin q + y2 cos q
z = z2

In matrix form, the rotation can be expressed as

X = R3 X2

where

 

The Combined Three-Dimensional Rotation

Final combined expression for the rotation is X = R3 R2 R1 X' = RX'           (a)


where R is  .

where

Note that R is an orthogonal matrix which has the property R-1 = RT. Using this property, Equation (a) can be rewritten as

X' = R-1X = R TX         (b)

Equation (b) is the final form of the rotation matrix.

 

View animation

Directions for Viewing 3d Animation of the Rotational System

  • You will need a VRML 2.0 viewer to see the animation to the left. Download and install the free Cortona vrml plug-in. 
  • Select the graphic in the left screen or the link below it.
  • Select θx to view the rotation about the x axis.
  • Select θy to view the rotation about the y axis.
  • Select θz to view the rotation about the z axis.
  • The remaining buttons are for a complete 3D transformation discussed elsewhere.
  • While viewing the animation, you can select the "study" and "turn" buttons on the left panel to roll the entire image so that the rotations can be viewed from any perspective. The R button resets the image to its original orientation.

Fundamentals of Geodetic Astronomy

Since the distance from the earth to the nearest star (excluding the sun) is 109 larger that the earth's radius, the dimensions of the earth are negligible. Also, due to this distance, the stars are almost immobile. Thus to an observer on the earth, it stars and galaxies appear to be sitting on a transparent celestial sphere where the earth is considered a dimensionless point at the center of the sphere. As shown in the figure 1, directions on the earth can be extended to the celestial sphere. The earth's instantaneous spin axis is projected out onto this sphere, creating the north and south celestial poles. The north celestial pole is now commonly referred to as the celestial ephemeris pole. Similarly, the instantaneous position of the plane of the equator is projected outwards to create the celestial equator. A plane that is parallel to the equator and is extended outward to intersect the celestial sphere is called the celestial parallel. Any great circle that intersects the celestial poles is called a astronomical meridian (aka - celestial meridian). The point where the gravity vector of an observer extended upward intersects the celestial sphere is known as the observer's zenith. The point opposing zenith on the celestial sphere is known as the observer's nadir. The great circle that is perpendicular to the observer's gravity vector at the observing station is called the celestial horizon, and any small circle that is parallel to the celestial horizon is called an almucantar. The vertical plane perpendicular to the meridian is the prime vertical.

FIGURE 1

The position of any star on the celestial sphere can be defined by the polar coordinates (r, f, l). If r is taken as a unit vector, then the position can be effectively defined by two angles, f and l. Coordinate systems used in celestial work are the right ascension system, and the local astronomical system. All of these systems are use a two angle coordinate system to define the position of a star.

 

Commonly Used Coordinate Systems

Right Ascension System

The right ascension coordinate system Shown in Figure 2 is a heliocentric (sun) centered system, with the sun at its origin, the z axis pointing to the celestial ephemeris pole CEP which also called the north celestial pole (NCP). The x axis points to the vernal point ^, and the y axis creating a right-handed system. Recall that the vernal point is the location where the sun apparently crosses the celestial equator when going from the southern hemisphere to the northern hemisphere on the first day of Spring. The declination d to the star is the angle from the celestial equatorial plane to the line from the sun to the star in the astronomical meridian of the star. The right ascension a to the star is the angle measured counterclockwise (as seen from the NCP) in the celestial equatorial plane from the astronomical meridian of ^ (called the equinoctial colure) to the celestial meridian of the star. Since the distance to the star is irrelevant in this system, we describe positions of celestial objects by the angles of declination δ and right ascension α. Using a distance of 1 to the star, the unit vector describing the direction to the star in this system is

The relationship between the angles and the Cartesian coordinates is

d = sin-1 zRA
(1)

where the

  1. First part of the equation requires a determination of the proper quadrant of the angle.
  2. Second part of the equation is positive from 0° to 180°, and negative from 180° to 360°.

Of course since the NCP moves with respect to the stars as a function of time, the values of a and d also change with time. Thus, observations made on celestial objects must be referenced in time. Star catalogues usually use a right ascension system that precesses but does not nutate. This is known as the mean right ascension system [MRA(t0)].

Conventional Terrestrial Coordinate System

The Conventional Terrestrial System (CTS) is the closest practical approximation of a natural geocentric coordinate system, and is probably the most important system in geodesy. This system is is a three-dimensional Cartesian coordinate system with x, y, and z axes.

In this coordinate system,

 

View animation

Directions for Viewing 3d Image of CTS Coordinate System

  • You will need a VRML 2.0 viewer to see the animation to the left. Download and install the free Cortona vrml plug-in. 
  • While viewing the animation, you can select the "study" and "turn" buttons on the left panel to roll the entire image so that the rotations can be viewed from any perspective. the "restore" button in the lower panel will reset the image to its original orientation.

The Conventional Terrestrial Pole is commonly referred to as the Earth's North Pole. However it should be remembered that the Earth's polar axis precesses and nutates (see lesson on Motions of the Earth). Thus the position of the "instantaneous" pole is given in seconds of arc from the CTP. As previously studied, the International Earth Rotation Service (IERS) tracks the position of the pole in relation to the CTP as a function of time.

An example of a CT system is the International Terrestrial Reference Frame (ITRF) where stations are located with reference to the GRS 80 ellipsoid using VLBI and SLR techniques. This world-wide datum takes into account the temporal effects such as plate tectonics and tidal effects. Thus it is regularly updated and the date of the update is appended to its name. For example, ITRF 00 is the datum as defined in J2000.0. Previous versions were ITRF 97, ITRF 96, and ITRF 94. The datum known as WGS 84 (not to be confused with the WGS 84 ellipsoid) is another example of a TRF system of coordinates. Both of these systems of points with coordinates are known as worldwide datums. Since NAD 83 uses points only on the North American continent, it is known as a local datum. NAD 83 is also called a regional datum.

Local Astronomical Coordinates

Local Astronomical Coordinate. Description in text.

Surveyors work in the three-dimensional Cartesian system called the Local Astronomical (LA) coordinates to describe positions in reference to their own location.

In this coordinate system:

Note that unless the observer is at the North Pole, the direction of the U axis (local astronomical z axis) will not align with the Z axis in the CT system. 

The local astronomical (LA) system shown in Figure 3 is the one in which observations to stars (S) are made. Note that the origin of this system is at the surface of the earth, at the observer's station, and is thus called a topocentric system. The direction vector to the star S is

where the angles are related to the Cartesian coordinates

v = ½ p - z = sin-1 zLA

The observations are made in a topocentric system which is spinning as well as revolving with the earth. Since the positions of the stars are usually given in the mean right ascension system MRA(t0). This is accomplished with a series of coordinate systems.

 

Local Geodetic Coordinates

Another commonly used local coordinate system has its x axis (N) defined by the position of the CTP. This system z axis (U) is along the normal to the ellipsoid going away from the Earth at the observer's station, and the y axis creates a left-handed coordinate system. This system relates surveying observations of geodetic azimuth α, vertical angles v, and slope distance S to changes in northing, easting, and up (Δn, Δe, Δu). Note that the surveyor can go from the local astronomical coordinate system to the local geodetic coordinate system by coordinate correcting azimuth for the instantaneous position of the pole and observed azimuth and vertical (zenith) angle observations for deflection of the vertical. Once these observations are corrected, the changes in the local geodetic system are determined as

where z is the corrected zenith angle at the observer's station, v is the corrected vertical angle, and a the geodetic azimuth. Note that these equations are similar to the previously defined equations for the LA system with the inclusion of the slope, also known as the slant distance in geodesy. The inverse relationships are

Note that the results of the inverse tangent function in the azimuth formula, α, must be placed in its proper quadrant. The first part of the formula is handled the same as was taught in SUR 111. The second part of the formula is the atan2 function which yields a negative results for azimuths from 180° to 360° and positive for azimuths from 0° to 180°. Thus if the results are negative for α from the atan2 function, simply adding 360° to obtain the azimuth of the line.

Aligning The Local Geodetic and Conventional Terrestrial Coordinate Systems

To align local geodetic coordinates (LG) with a conventional terrestrial coordinate system (CTS), you will need to perform mathematical conversions which "rotate" the LG coordinates around two axes.

The mathematical results of these rotations are

Note that the Δ symbol has been dropped in this equation.

View animation

Directions for Viewing 3d Animation

  • You will need a VRML 2.0 viewer to see the animation to the left. Download and install the free Cortona vrml plug-in. 
  • Select the graphic in the left screen or the link below it.
  • Please press the cube labeled 1 to view the first rotation of 90 − φ about the y (E) axis.
  • The cube labeled 2 will animate the second rotation of 180 −  λ about the once rotated z (U) axis. Note that after the second rotation, the NEU is aligned with the XYZ axis. However the E axis is pointing opposite the direction of the y axis. This discrepancy is corrected with the introduction of a negative sign to E.
  • The cube labeled "R" will reset it to the original rotation so that you can view the rotations another time.
  • While viewing the animation, you can select the "study" and "turn" buttons on the left panel to roll the entire image so that the rotations can be viewed from any perspective. the "restore" button in the lower panel will reset the image to its original orientation.

Conventional Celestial Reference System

The Conventional Celestial Reference System (CR) is defined similar to the conventional terrestrial system. In the CR, the

Points referenced in the system are part of the Celestial Reference Frame (CRF). The most noteworthy of these systems is that developed by IERS which is defined by reference stations positioned using approximately 500 extragalactic objects (quasars and galactic nuclei). This system is known as ICRF where the I represents the source of IERS.

The rotation angle in the equatorial plane about the Z axis between CRS and CTS is called the Greenwich Apparent Sidereal Time (GAST) and is often designated with a capital omega, W.

It should be noted that the astronomical meridian plane of the observer in the LA system contains both the gravity vector of the observer and the CR coordinate system. Thus it is parallel to the conventional spin axis but does not necessarily contain the mass center of the earth.

To transfer between CRF and TRF coordinate systems, we must account for polar motion (xp, yp), sidereal time W, precession, and nutation. 

Converting CR Coordinates to LA Coordinates

The LA coordinate system can be rotated into the CR coordinate system using the astronomical coordinates of the observer's station. This transformation is similar to the transformation from the LG system to the CT system except that the astronomical position of the observer's station must be known. The astronomical coordinates are (1) astronomical latitude F and (2) astronomical longitude L. To take a vector from the local astronomical system into the celestial reference system, the XZLA plane is rotated about the YLA axis by an amount of ½p - F, and then the once rotated XYLA plane is rotated about the ZLA axis by an amount of p - L, or

 

Accounting for Polar Motion

A coordinate system based on the instantaneous position of the pole is known as the International Terrestrial Reference System (IT). In this system the Z axis corresponds to the instantaneous position of the pole, the X axis goes from the origin (mass center of the Earth) through the vernal point ^, and the Y axis forms a right-handed coordinate system. The only difference between the CR and IT systems is that the instantaneous pole does not coincide with the CTP (CTP). In Figure 4, and as stated earlier, the xP and yP coordinates are published in arc seconds by the IERS for specific epochs in time. See previous lesson on Motions of the Earth.

To rotate the CR coordinate system into the instantaneous terrestrial (IT) system the XZCT plane must be rotated about the YCT axis by an amount of xP, and the YZCT plane must be rotated about the XCT axis by an amount of yP. This transformation can be expressed as

Since xP and yP are small, and since the since of a small angle is approximately the angle in radian units, and the cosine of a small value is approximately equal to 1, a sufficiently close approximation of the previous transformation is

This rotation will designated as RM in the remainder of this course.

Figure 9

Accounting for Sidereal Time

The final transformation is the taking the IT axis at epoch t into the apparent place (AP) system at the same epoch. The AP(t) is a geocentric system in which the

In this transformation, all that is needed is to take the IT system into the AP system is to rotate the xyIT plane about the ZIT by an angular amount known as the Greenwich Apparent Sidereal Time (W). This is the same rotation needed to take the conventional terrestrial coordinate system into the instantaneous coordinate syste. This rotation angle is shown in Figure 9 and is mathematically expressed as

where

This rotation will be designated as RS in the remainder of this course.

Accounting for Precession

Since the Earth's pole wobbles, any coordinate system referenced to the instantaneous pole must be corrected to the conventional terrestrial pole. The wobble is broken into two components, precession and nutation where precession is the major component of the poles motion and nutation is the smaller minor wobbles due to things such as differential snow load, lunar and solar tides, and so on. These motions are shown in Figure 10. Both of these values are referenced to some reference epoch τ0.

Figure 10

The amount of precession P(t0, t) occurring in this time interval is given by three precessional constants, (z0, q, z) as shown in the following figure.

where

Figure 11

The angles (½p - z0) and (½ p + z) are the right ascensions of the ascending node of the mean Equator t, measured respectively in the two mean systems (at t0 and t). The q is the inclination between the mean equators at the two epochs. The transformation is accomplished by rotating the XY plane in the MRA(t0) system about the ZMRA(t0) axis by an amount of -z0. Then taking this once-rotated system and rotating the XZ plane in the MRA(t0) system about the YMRA(t0) axis by an amount of q. Finally the twice-rotated XY plane in the MRA(t0) system about the ZMRA(t0) axis by an amount of -z. MRA is called the Mean Right Ascension system. The resulting transformation is

 

 

View animation

Directions for Viewing 3d Animation

  • You will need a VRML 2.0 viewer to see the animation to the left. Download and install the free Cortona vrml plug-in. 
  • Select the graphic in the left screen or the link below it.
  • Please press the cube labeled 1 to view the first rotation of −ζ0 about the ZMRA(t0) axis.
  • The cube labeled 2 will animate the second rotation of θ about the once-rotated YMRA(t0) axis.
  • Note that after the second rotation, the twice-rotated mean equatorial plane coincident with the instantaneous equatorial plane. Also note that the Z axes are now aligned. However the X axis is of the twice-rotated mean system does not coincide with the instantaneous position for the right ascension. This discrepancy is corrected by rotating the twice-rotated mean system about the ZMRA(t0) by an amount of −z.
  • The cube labeled "R" will reset it to the original rotation so that you can view the rotations another time.
  • While viewing the animation, you can select the "study" and "turn" buttons on the left panel to roll the entire image so that the rotations can be viewed from any perspective. the "restore" button in the lower panel will reset the image to its original orientation.

 

Accounting for Nutation

Figure 12

Up to this point only the precession component of the earth's wobble has been considered. We must also take into account the motion caused by nutation of the spin axis. This defines the true right ascension system at epoch t known as TRA(t). The ZTRA axis coincides with the instantaneous spin axis of the earth and the true vernal equinox defines the positive XTRA axis. 

Figure 13

Nutation is usually defined by two terms - nutation in longitude Dy and nutation in obliquity De as shown in the figure to the right. The transformation of a and d from the MRA(t) to the TRA(t) systems is done by rotating the YZ plane about the XMRA axis by and amount of e, then rotating this once rotated system's XY plane about the ZMRA axis by an amount of -Dy, and finally rotating the twice rotated system's YZ plane about the XMRA axis by an amount -(e + De). This can be expressed as

The values for ε, De, and Dy can be found using the formulas:

Converting between a Celestial Reference Frame and a Terrestrial Reference Frame

To convert celestial reference frame (CRF) coordinates to terrestrial reference frame (TRF) coordinates the following equation is used.

 

TIME SYSTEMS

The last concept essential in astronomical positioning is the concept of time. The hour angle h of the star is the angle between the astronomical meridian of the star and that of the observer. The local apparent sidereal time (LAST) is the hour angle of the true vernal equinox. GAST (W) is the hour angle of the true vernal equinox ^ as seen at Greenwich. LAST and GAST can be linked together by the equation

LAST = GAST + LIT

In practice, GAST is measured through universal time (UT) which differs from every day standard time by an integral number of hours dependent on the hour angle. Below are the different version of UT that are used.

  1. UT reflects the actual non-uniform rotation of the earth. It is affected by to polar motion since local astronomical meridians are slightly displaced.
  2. UT1, also depicts the non-uniform rotation of the earth, but does not account for polar motion. UT1 corresponds to GAST and is needed for transforming the true right ascension (TRA) system to the instantaneous (IT) system.
  3. UTC (universal coordinated time) is the broadcast time that represents a smooth rotation of the earth. (It does not account for propagation delays.) UTC is kept to within ±0.7s of UT1 by the introduction of leap seconds.
  4. UT2 is the smoothest time, and has all corrections applied to it.
  5. International Atomic Time (IAT) is based on an atomic second. To keep IAT and UT1 close, leap seconds are introduced.
  6. GPS time is also based on an atomic second. It coincided with UTC time on January 6, 1980 at 0.0 hours. With the introduction of leap seconds to IAT, there is now a constant offset of 19 seconds between GPS time and IAT.

Relationships in Time Standards

Astronomical Azimuth

Astronomical positioning has been the primary method of point positioning for centuries. However, since the introduction of GPS, its importance has greatly declined. Today, only astronomical azimuth observations are made, and even these are being supplanted with GPS.

where t is the hour angle of the star as given by the equation t = GAST + LIT - a. The hour-angle to the star can often be determined by a polar sketch of the situation at the time of the observation depicting the GAST, LAST, the meridian of the observer, and the right ascension angle to the star. To minimize errors in timing, it is best to observe the star at elongation. However, to minimize errors in astronomical latitude, it is best to observe the star close to the meridian. Since both conditions can not be simultaneously met with a single star, the most precise results are obtained by observing star-pairs, where one star meets the first condition and the second the latter.

Astronomical Observation Handbook (pdf file)

Conventional Terrestrial Pole (CTP)

The conventional terrestrial pole is the mean position of the Earth's spin axis between 1900.0 and 1905.0. All instantaneous positions of the pole are referenced to this position. The International Earth Rotation Service (IERS) maintains a record of the instantaneous pole in relation to the CTP in units of seconds.

Right- and left-handed coordinate systems:

A three-dimensional coordinate system is often referred to as right-handed (most common) or left-handed. The right-handed coordinate system is formed as follows:

Turn your right-hand so that you are looking at the palm of your hand then perform the following steps as shown in the figure below.

  1. Make a fist with your hand with your thumb pointing to the right. This is the positive X axis.

  2. Straighten your index finger so that it is pointing straight up and 90° from your thumb. This is the positive Y axis.

  3. Straighten your middle finger so that it is pointing at you. This is the positive Z axis.

A left-handed system is formed in a similar manner with your left-hand. Note that the difference between the left-handed system and the right-handed system is the direction of the positive X axis is in the opposite direction.

Last updated: October 28, 2008
Created by Charles Ghilani, Ph.D.
Penn State Surveying Program, Copyright © 2000 - 2008