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

• r₁₁ = cos(q₂) cos(q₃)
• r₁₂ = sin(q₁) sin(q₂) cos(q₃) + cos(q₁) sin(q₃)
• r₁₃ = -cos(q₁) sin(q₂) cos(q₃) + sin(q₁) sin(q₃)
• r₂₁ = -cos(q₂) sin(q₃)
• r₂₂ = -sin(q₁) sin(q₂) sin(q₃) + cos(q₁) cos(q₃)
• r₂₃ = cos(q₁) sin(q₂) sin(q₃) + sin(q₁) cos(q₃)
• r₃₁ sin(q₂)
• r₃₂ = -sin(q₁) cos(q₂)
• r₃₃ = cos(q₁) cos(q₂)

Note that R is an orthogonal matrix which has the property R^-1 = R^T. 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 z^RA
(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,

• The origin (0,0,0) corresponds with the mass center of the Earth.
• The X axis is parallel to the Equator and points through the Greenwich Meridian (0° longitude). The Greenwich Meridian is also known as the prime meridian.
• The Z axis is coincident to the Conventional Terrestrial Pole (CTP) which was the mean position of the Earth's rotational axis between 1900 and 1905.
• The Y axis lies in the Earth's equatorial plane and is perpendicular to the X and Z axes and creates a right-handed Cartesian 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

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:

• The origin (0,0,0) corresponds with location of the instrument used to make surveying measurements on the surface of the Earth: from now on called the observer's station.
• The x axis (N) points from the origin towards astronomical north and is a tangent with the curvature of the Earth. Remember from a previous lesson that astronomical north and geodetic north differ by the deflection of the vertical components.
• The z axis (U) points away from the surface of the Earth opposite the direction of gravity towards the observer's zenith. Its negative axis points in the direction of gravity and the observer's nadir. Thus, the U axis aligns with measurements used to determine orthometric heights.
• The y axis (E) creates a left-handed Cartesian coordinate system by being perpendicular to both the x and z axes and pointing east from the observer's station. This axis is tangent to the curvature of the Earth at the observer's station. The xy plane forms the observer's horizon and is perpendicular to the gravitational equipotential surface at the observer's station.

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 z^LA

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 first rotation is around the E axis (LG, y axis) and "pushes" the N axis (x axis) down until the NE plane is parallel with the Earth's equator. The U axis (z axis) is now parallel with the CTP (North pole) and the N axis in the LG system is now pointing into the earth. The amount of rotation is ½p - f about the y axis (east) of the LG coordinate system.
• The second rotation is around the U axis in the LG system and aligns the once-rotated LG N axis with the CT x axis in equatorial plane. The amount of rotation for this transformation is p - l about the once-rotated z axis of the LG system.
• Because the LG coordinate system is a left-handed coordinate system and the CT coordinate system is a right-handed coordinate system, the East axis will be pointing 180° degrees away from the CT y axis (that is, west from the observer).
• To make the systems identical, a negative sign is introduced in the LG y axis.

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

• Z axis in this system corresponds to the position of the Earth's spin axis at the beginning of 2000. This is known as the standard reference epoch J2000.0 where the J represents Julian day and 2000.0 means January 1st at midnight (0:00:00) universal time (UT) in that year. Thus the Z axis represents the instantaneous position of the pole at J2000.0.
• X axis goes from the origin (mass center of the Earth) through the vernal point ^
• Y axis forms a right-handed coordinate system.

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 (x_p, y_p), 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 XZ^LA plane is rotated about the Y^LA axis by an amount of ½p - F, and then the once rotated XY^LA plane is rotated about the Z^LA 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 x_P and y_P 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 XZ^CT plane must be rotated about the Y^CT axis by an amount of x_P, and the YZ^CT plane must be rotated about the X^CT axis by an amount of y_P. This transformation can be expressed as

Since x_P and y_P 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 R^M 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

• z axis coincides with the z^IT axis, That is, the instantaneous position of the pole
• x axis points to the vernal point ^
• y axis completes the right-handed system

In this transformation, all that is needed is to take the IT system into the AP system is to rotate the xy^IT plane about the Z^IT 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

• W = 1.0027379093 UT1 + φ₀ + Dψ cos ε

• φ₀ = 24110.54841^s+ 8640184.812866^s T₀ + 0.093104^s T₀² − (6.2 x 10⁻⁶)^sT₀³

• T₀ is the difference in time span expressed in Julian centuries between the standard epoch of J2000.0 and the day of the observations at 0 hours UT. For example, November 4, 2006 at 0:00:00 UT time is 2006.84325804244 Julian years since Nov 4 is the 308th day of the year. Thus T = (2006.84325804244 − 2000.0)/100 =0.0684325804244 centuries.

This rotation will be designated as R^S 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

• z₀" = 2306.2181 T + 0.30188 T² + 0.017998 T³
• z" = 2306.2181 T + 1.09468 T² + 0.018203 T³
• q" = 2004.3109 T - 0.42665 T² - 0.041833 T³
• T represents the time span expressed in Julian centuries of 36525 mean solar days from the standard epoch of J2000.0.

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 Z^MRA(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 Y^MRA(t₀) axis by an amount of q. Finally the twice-rotated XY plane in the MRA(t₀) system about the Z^MRA(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 Z^TRA axis coincides with the instantaneous spin axis of the earth and the true vernal equinox defines the positive X^TRA 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 X^MRA axis by and amount of e, then rotating this once rotated system's XY plane about the Z^MRA axis by an amount of -Dy, and finally rotating the twice rotated system's YZ plane about the X^MRA axis by an amount -(e + De). This can be expressed as

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

• ε = 23°26'21.448" − 46.8150" T − 0.00059" T² + 0.001813 T³
• Δε = 9.2" cos Ω_m + ... (64 terms)
• Δψ = −17.2" sin Ω_m + ... (106 terms)

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 + L^IT

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 ±0.7^s 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

• IAT = GPS + 19.000
• ITS = UTC + 1.000 n where is was 32 in June of 2000.
• UTC = GPS + 13.000
  
  

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 + L^IT - 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)

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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.

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Alabaster	White, grays, beige, orange, yellows, reds, white translucent, variegated	Very Soft

Note: The Sculpture Studio does not sell stone or tools.

Man has been creating art from stone ever since he found he could shape it by striking a softer stone with a harder one. Over the millennia, a few types of stone have become popular with sculptors. Here are the most common ones used for carving from the three different types of rock: igneous, sedimentary, and metamorphic. Igneous: The characteristics of igneous rocks result from the way in which they were formed. Deep in the earth, under the intense heat of volcanic action, magma was forced up through the older solid rock. It then cooled, forming granite, basalt, and diorite. Granite is made of quartz crystals and feldspar which contain silica. Prolonged breathing of silica can cause lung damage. Wear a good respirator when around granite dust. Granite is very hard stone and difficult to carve. In fact 'carving' might not be the right word, since you are not pushing the chisel through the material like you would with the limestone or marble, but are pulverizing the stone, trying to break off chips of crystals (which can be as sharp as glass). Your carving tools must be carbide-tipped to stand up to the wear and tear of working granite. Fine detailed carving is better left to other types of stone. Diamond saws and cup grinders cut through granite quite easily and can speed up the carving and finishing process. Granite comes in a wide variety of colors, and the crystal size can vary from large and coarse to very fine and dense. Granite takes a high polish and holds up very well outdoors. Sedimentary: As rocks began to erode from wind, rain, and sun, the particles were washed into low lying areas where the sediment accumulated. In the sea, small plant and animal forms died and drifted to the bottom, adding to the sediment. Over thousands of years, the pressure of these layers cemented the sediment onto limestone or sandstone. Limestone formed on the sea floor from sediment and the bodies of primitive sea creatures. As you are working, you can find small fossilized creatures like crynoids and braciopods in the stone. Limestone is composed primarily of the mineral calcite, or calcium carbonate. Limestone is easy to carve and will hold small detail work, but is also strong enough to support undercutting. While it does not have the innate beauty of the more colorful stones, a wide variety of textures can be created that can really bring the stone to life. There is a definite 'grain' or bed (formed as the layers of sediment stacked upon each other on the sea floor) in limestone. The stone breaks fairly predictably along the bed lines, but less cooperatively going across the bed. (Think of opening a phone book with the pages, compared to trying to tear it across the pages.) Limestone can be polished, but the polish will not last long outdoors. Limestone seems to survive acid rain better than marble, so is a good choice for outdoor sculpture. Sandstone is formed from sedimentary sand held together by silica or calcium carbonate. Sandstone also has a distinct bed direction, and large chips can be broken off when going with the bed. Sandstone tends to wear out your tools quickly. Sandstone contains silica, so a respirator must be worn. Metamorphic: Metamorphic rocks are formed when a sedimentary layer is exposed to heat and pressure and undergoes a chemical change which forms a new crystalline material. After metamorphism, limestone becomes marble. Marble has been the most preferred stone for carving since the time of the ancient Greeks. Marble is moderately hard to work. It will hold very fine detail. Marbles from the United States comes in over 250 colors. When brought to a high polish, its crystalline structure sparkles. It does not hold up well outdoors. Acid rain begins to deteriorate the surface within a few years. Alabaster is a very soft stone for carving and tends to flake and split along hidden cracks in the stone. It will take a high polish, which brings out its incredible colors and patterns. In fact the stone is so beautiful that the viewer may overlook your sculptural forms and only admire the stone. The dust from alabaster may cause an allergic reaction in some people, so wear a respirator. Soapstone, or steatite, is soft enough to carve with a knife. It is composed of talc and has a slippery, soapy feel. It will take a polish and hold fine texture detail. It is a good choice for your first stone carving. Prolonged exposure to talc dust can cause respiratory problems, so wear a respirator.