When studying space science, expressing the positions of stars, planets, spacecraft, and just about everything else in space requires coordinate systems. Without them, we would literally not know where anything is in space, let alone get anything man-made to rendezvous or land on these space objects. When dealing with coordinates, space scientists have to choose specific conventions in which to visualize a space object's position in space. For instance, the idealized Earth orbit is a plane with the Sun at its center. The Earth's position with respect to the Sun can be expressed in a specific coordinate system that is defined by the plane of Earth's orbit. This coordinate system is normally referred to as the Ecliptic Plane, or Ecliptical Coordinates. However, Earth's spin axis has an orientation that is not the same as the Ecliptical Coordinate system because Earth's axis is tilted at approximately 23.44 degrees to the Ecliptic Plane. Therefore, if an observer on the Earth measures the position of the Sun over the Earth's year with respect to the spin axis of the Earth (Equatorial Coordinate system), the Sun will not appear to be moving in a simple plane but will change both right ascension (RA) and declination (Dec.) over the year. However, if the observer could express his/her observations with respect to the Earth's orbit (Ecliptical Coordinates) instead of the Earth's spin axis (Equatorial Coordinates), then the Sun would appear to be moving in a simple plane.

2-Dimensional Coordinate Transformation

More on Ecliptical vs. Equatorial Coordinate systems later. Right now, let's begin with a simpler 2-dimensional example. For our example, let us imagine two people on a level and spinning merry-go-round looking at the rising Moon that is just above the observers' horizon. The two observers are at two different positions on the merry-go-round looking straight ahead in a direction directly opposite from the merry-go-round's center (Point C), as shown in Figure 1. Neither observer will move their head or eyes at any time. At some snapshot in time, one observer might see the Moon on the right side of center, while the other observer will see the Moon on the left side of center. Their two make-shift coordinate systems, based on their straight-ahead line of sight, differ enough from one another that they see the Moon at two different locations with respect to their line of sight. Now, let us say that the two observers know their positions with respect to one another on the merry-go-round. This knowledge is expressed as an angle φ measured from the center of the merry-go-round, as shown from directly above the merry-go-round as shown in Figure 1.

Figure 1: Two Observers Viewing the Rising Moon from Two Locations on a Merry-Go-Round

When asked, Observer #1 will say that the rising Moon is to his left and Observer #2 will say that the rising Moon is to her right. With respect to their respective coordinate systems, they are both correct. However, one observer will not be able to understand the other observer's observations without some method of transforming between the observers' coordinate systems. Now, let's take away the merry-go-round and look at the two observers' individual coordinate systems. Figure 2 shows that the observers' lines of sight correspond to their respective x-axes (x and x'). The y-axes (y and y') are 90 degrees counter-clockwise from the respective x-axes. Note how the angle between the y-axes also has to be the angle φ. The r-vector is in the direction of the Moon and has a magnitude (size) of the distance from the observers to the Moon.

Figure 2: The Two Coordinate Systems of the Observers. The x-y coordinate system corresponds to Observer #1, while the x'-y' coordinate system corresponds to Observer #2.

If Observer #1 wanted to go to the Moon by pacing along the x-axis and then along the y-axis, he would know how because he knows where the Moon is with respect to his own coordinate system. However, what if he was blindfolded and his position was changed to that of Observer #2? He would assume to walk the same amount of paces along x' and y', but completely miss the Moon because he is assuming the wrong coordinate system. In order for him to reach the Moon, he would have to know how to transform his own coordinate system into Observer #2's coordinate system.

Recall that the angle φ between the two observers was originally known. This means that there is a link between the two observers' coordinate systems. Using this one link, we can assist Observer #1 to reach the Moon using Observer #2's coordinate system. First, we assume that the x' and y' coordinates of the Moon will be some function of the x and y coordinates. First, let's deal with the x' coordinate.

Figure 3 shows the information required to determine the coordinate transformation between the x' and the x, y coordinate system. At first glance, it might seem difficult to follow, but studying Figure 3 will lead you to conclusion shown in Equation 1. Using algebraic manipulation, Equation 2 shows a cleaned-up version of Equation 1. Equations 3 and 4 continue to show the simplification of Equation 1. Equation 5 shows the final simplified version of Equation 1.

Figure 3: The x' Coordinate can be Expressed as a Function of the x, y Coordinate System

Eq. 1        x' = [ x / cos(φ) ] - [x tan(φ) - y] sin(φ)

Eq. 2        x' cos(φ) = x - [x sin(φ) - y cos(φ)] sin(φ)

Eq. 3        x' cos(φ) = x - x sin2(φ) + y cos(φ) sin(φ)

Eq. 4        x' cos(φ) = x cos2(φ) + y cos(φ) sin(φ)

Eq. 5        x' = x cos(φ) + y sin(φ)

Note how simple Equation 5 is when compared to Equation 1 originally derived. Similarly, the y' coordinate can be expressed in the x, y coordinate system, as Figure 4 clearly shows. Note how the y' coordinate is negative because it is in the opposite direction to the y' axis shown in Figure 2. Equation 6 results from the derivation. Equation 6 simplifies to Equation 7, then finally to its final form in Equation 8.

Figure 4: The y' Coordinate can be Expressed as a Function of the x, y Coordinate System

Eq. 6        y' = - [x tan(φ) - y] cos(φ)

Eq. 7        y' cos(φ) = - [x sin(φ) - y cos(φ)] cos(φ)

Eq. 8        y' = -x sin(φ) + y cos(φ)

In matrix form, the coordinate transformation between two 2-D coordinate systems, shown in Equation 9, is really simple, despite the more complicated derivation presented above. However, this derivation is simply a warm-up for the 3-D coordinate transformation equations that are next. Before we leave the 2-D case, suppose that only angles along the local horizon were used to convey the Moon's apparent position with respect to the two observers, as shown in Figure 5. These angles could be construed as the "Azimuth" coordinates since they are measured along the local horizon plane, but in this case the 0 degree "Azimuth" does not necessarily have to correspond to the "true north" direction. In fact, in this case, let us assume that the 0 degree case for each observer is along their respective x axes.

Eq. 9

Figure 5: The Perceived Angles of the Moon's Location as Viewed by both Observers

Although the relationship between the angles θ, θ' and φ appear very straight-forward, let's try to derive the relationship anyway using the coordinate transformation equations we had derived earlier. First, for simplicity, let us assume that the r-vector is also a unit vector so that we do not have to deal with the r-vector's magnitude. Second, let's show the basic relationship between the x, y coordinates and their equivalent polar coordinates. These are shown in Equation 10 (for the x coordinate) and Equation 11 (for the y coordinate). The same equations also hold for the x', y' coordinate system, but substituting the θ' angle for the θ angle.

Eq. 10        x = r cos(φ) = (1) cos(φ)

Eq. 11        y = r sin(φ) = (1) sin(φ)

Now, let's use the coordinate transformation equations, substituting the polar coordinates for the x, y coordinates, as shown in Equation 12 (for x'). Equation 12 simplifies to Equation 13, and so on until the final answer is reached at Equation 15. If you check Figure 5 against Equation 15, you might  notice an apparent contradiction. Figure 5 suggests that the θ and, θ' angles would add up to the φ angles. This would be true if the θ' angle was positive. However, notice that the θ' angle is negative because the y' coordinate is also negative. Therefore, Figure 5 and Equation 15 completely agree with one another. The angular relationships can also be verified using the y' coordinate too, but this is left to the reader to verify. Suffice it to say, it will involve the standard trigonometric identity for the sine of an angle difference.

Eq. 12        x' = (1) cos(θ') = x cos(φ) + y sin(φ) = cos(θ) cos(φ) + sin(θ) sin(φ)

Eq. 13        cos(θ') = cos(θ) cos(φ) + sin(θ) sin(φ)

Eq. 14        cos(θ') = cos(θ - φ)

Eq. 15        θ' = θ - φ

3-Dimensional Coordinate Transformation

In most space science applications, coordinate transformations of 3-D coordinate systems are the standard. This includes astrodynamics and orbital mechanics, despite orbits being confined to a 2-D plane. The most used form of coordinate transformation in 3-D is the Equatorial Coordinates to Alt-Az (or Az-El) Coordinates. This coordinate transformation allows the user to transform from RA-Dec. (inertial) to Az-El (fixed), and vice-versa, so that an object's apparent position can be expressed with respect to the spin axis of the Earth or with respect to the local horizon (or local zenith) of the observer's location on the Earth's surface. This allows an observer to predict where a satellite will be seen in their local sky. Two coordinate transformations are used here, beginning with the satellite's orbit plane, transforming to the Earth's spin axis (RA-Dec.), and finally transforming to the observer's local sky (Az-El).

Equatorial to Ecliptical Coordinate Transformation

Thankfully, there is a relatively easy way to deal with 3-D coordinate transformations involving a single 2-D coordinate transformation about one of the axes. Let's begin with the simplest 3-D coordinate transformation: Equatorial Coordinates to Ecliptical Coordinates, as we had introduced at the beginning of this page. A convenient similarity between the two is that both of their respective x axes point to the First Point of Aries, or 0 degrees in RA, as shown in Figure 6. The angle θ in this case is the tilt angle of the Earth's spin axis or more commonly known as the "Obliquity of the Ecliptic". This angle is approximately equal to 23.44 degrees.

Figure 6: Equatorial Coordinates and Ecliptical Coordinates

In order to perform the coordinate transformation between Equatorial and Ecliptical coordinates, a simple 2-D rotation is performed in which the x-axis is the axis of rotation, as Figure 6 shows. In this case, the z and y axes will change but the x axes will be coincident during the entire rotation. This means that the x coordinate of the Equatorial coordinate system (xeq) will be the same as the x coordinate of the Ecliptical coordinate system (xecl). The y and z coordinates of the two coordinate systems will be linked by the 2-D matrix that was shown back in Equation 9. The entire transformation matrix is shown in Equation 16. The expanded form of the matrix is shown in Equations 17, 18 and 19.

Eq. 16

Eq. 17    xecl = xeq

Eq. 18    yecl = yeq cos(θ) + zeq sin(θ)

Eq. 19    zecl = - yeq sin(θ) + zeq cos(θ)

The RA and Dec. coordinates are in the Equatorial Coordinate system. The Geocentric (Earth at the center) Ecliptical Coordinate system is normally described by the Ecliptic Longitude (λ) and the Ecliptic Latitude (β). If we assume some unit r-vector (magnitude equal to 1) with coordinates xeq, yeq, zeq, we can determine its Ecliptical coordinates xecl, yecl, zecl using the matrix shown in Equation 16. However, Equatorial angles (RA/Dec. or more commonly α, δ) of the the r-vector can also be determined and therefore the Ecliptical angles (λ, β) can also be determined.

First, let us determine the RA and Dec. of the r-vector. The r-vector in the Equatorial Coordinate system and the corresponding RA/Dec. angles are shown in Figure 7. The RA (α) is measured along the xeq, yeq plane from the First Point of Aries (xeq=0). The Dec. (δ) is measured from the Equatorial Plane, beginning at the xeq, yeq line, to the zeq coordinate, as shown in Figure 7. The xeq, yeq, and zeq coordinates can be expressed in terms of the α, δ coordinates, as shown in Equations 20, 21, and 22. Similarly, the xecl, yecl, and zecl coordinates can be expressed in terms of the λ, β coordinates, as shown in Equations 23, 24, and 25.

Figure 7: The RA and Dec. Equatorial Angles shown in the xeq, yeq, zeq Cartesian Coordinate System

Eq. 20    xeq = cos(δ) cos(α)

Eq. 21    yeq = cos(δ) sin(α)

Eq. 22    zeq = sin(δ)

Eq. 23    xecl = cos(β) cos(λ)

Eq. 24    yecl = cos(β) sin(λ)

Eq. 25    zecl = sin(β)

The xeq, yeq, zeq, xecl, yecl, and zecl coordinates can be substituted in Equations 17 to 19 with their Equatorial and Ecliptical counterparts shown in Equations 20 to 25. The results are shown in Equations 26 to 28. These equations represent the cornerstone of all 3-Dimensional coordinate transformations, including RA/Dec. to Az/El transformations.

Eq. 26    cos(β) cos(λ) = cos(δ) cos(α)

Eq. 27    cos(β) sin(λ) = cos(δ) sin(α) cos(θ) + sin(δ) sin(θ)

Eq. 28    sin(β) = - cos(δ) sin(α) sin(θ) + sin(δ) cos(θ)

Equatorial to Azimuth-Elevation (Az-El) Coordinate Transformation

The Az-El (Az, E) Coordinate system is very different from the Ecliptical Coordinate system in two important ways. First, the orientation of the Az-El coordinate system depends on where the observer is on the surface of the Earth (latitude and longitude). Second, the orientation of the Az-El coordinate system changes with respect to the RA/Dec. coordinate system as the Earth spins on its axis. This means that the xeq axis and the xAE axis do not coincide. Third, the Azimuth (Az) angle increases in the opposite direction (left-handed system) to the RA angle (right-handed system).

The Az-El coordinate system is a wonderful example of a "Fixed" coordinate system, in which the coordinate system stays fixed with respect to the stationary observer. The latitude-longitude coordinate system of the Earth is another fixed coordinate system in which a stationary observer's latitude and longitude do not change as the Earth spins.

The RA/Dec. (Equatorial) coordinate system is a wonderful example of an "Inertial" coordinate system, in which the coordinates continue to move with respect to a stationary observer as the Earth spins. Although the stars will have specific Equatorial coordinates as the Earth spins, the Equatorial Coordinate system will appear to move to a "stationary" observer (at one location on the Earth) as the Earth spins.

The Az-El Coordinate system changes with respect to the Equatorial Coordinate system due to the observer's latitude and time. The node point shown in Figure 6 will not be the intersection of two literal x-axes as was the case for the Equatorial / Ecliptical case, but will correspond to the due east point in the observer's sky intersecting with some RA coordinate. As the Earth spins, the RA of the due east point will constantly change but will cycle through all of the RAs in the Earth's sidereal spin period (sidereal day) of 23h 56m 4s. It will be important to know what RA will correspond to the due east point in order to set up the Az-El's virtual x-axis at that point.

The true Az-El Coordinate system compared to the Equatorial Coordinate system is shown in Figure 8. The RA (α) coordinate that coincides with the observer's due south (Az=180 degrees) meridian line is called the "Sidereal Time" (αside). The due east point is exactly 6 hours (90 degrees) in RA greater than the Sidereal Time. The Sidereal Time constantly changes with respect to the stationary observer and completes one cycle every sidereal day. Equations 26 to 28 can be altered to determine the Azimuth and Elevation of an object from the object's RA and Dec., the observer's sidereal time, the time of day, and the observer's latitude on the Earth's surface.

Figure 8: Equatorial Coordinates, Az-El Coordinates, and Sidereal Time

First, let's deal with the observer's latitude (φ). The Earth's latitude is measured from the Equatorial plane. On the equator, the latitude is 0 degrees. At the North Pole, the latitude is +90 degrees. At the South Pole, the latitude is -90 degrees. At the North Pole (φ = +90 degrees), the zenith axis (zAE) is coincident with the Earth's spin axis (zeq). At the equator (φ = 0 degrees), the zAE is 90 degrees from zeq. At the South Pole (φ = -90 degrees), zAE is 180 degrees from zeq. Therefore, we can conclude from this trend that the 2-D rotation angle between the z-axes and the y-axes (the angle θ) is simply the complementary angle to the observer's latitude, as shown in Equation 29. Equations 30 and 31 show what happens to sines and cosines when using complementary angles.

Eq. 29    θ = 90° - φ

Eq. 30    sin(90° - φ) = cos(φ)

Eq. 31    cos(90° - φ) = sin(φ)

Next, we tackle the RA (α) coordinate. We will need to find the RA angle measured from the due east intersection point (αeast) in order to use Equations 26 to 28. We can know the RA angle measured from the Sidereal Time (αside) coordinate once the Sidereal Time is known. The angle from the due east point is 90 degrees subtracted from this difference, as shown in Equation 32. This can be rearranged to be a complementary angle, as shown in Equation 33. The quantity (αside - α) is also known as the "hour angle" (h), as shown in Equation 34. The final form is shown in Equation 35.

Eq. 32    αeast = α - αside - 90°

Eq. 33    αeast = - (αside - α) - 90°

Eq. 34    h = αside - α

Eq. 35    αeast = - [90° - (-h)]

Finally, we will need to deal with the offset between the due east point and the Azimuth (Az) angle. The Azimuth is 0 degrees at due North, 90 degrees at due East, 180 degrees at due South and 270 degrees at due West. If the Azimuth is found using Equations 26 to 28, the problem will be that the Azimuth will be measured in a right-handed system just like the RA angle is. However, the true accepted Azimuth coordinate is measured in a left-handed system. However, we can find the common trend by comparing the major compass directions, as we did for the observer's latitude. Note that if the Azimuth from the due east point (Azeast) is 0 degrees, the true Azimuth (Az) is 90 degrees. If Azeast is 90 degrees, the Az is 0 degrees. If Azeast is 180 degrees, the Az is 270 degrees (-90 degrees). Finally, if Azeast is 270 degrees, the Az is 180 degrees (-180 degrees). From this trend, a direct relationship between Azeast and Az can be determined, as shown in Equation 36.

Eq. 36    Azeast = 90° - Az

Now that the angles θ, α, δ, and λ have been properly defined, they can substitute the appropriate variables in Equations 26 to 28 to produce the Equatorial to Az-El coordinate transformation. The intermediate steps are shown in Equations 37 to 42. The final equations are shown in Equations 43 to 45.

Eq. 37    cos(E) cos(Azeast) = cos(δ) cos(αeast)

Eq. 38    cos(E) sin(Azeast) = cos(δ) sin(αeast) cos(θ) + sin(δ) sin(θ)

Eq. 39    sin(E) = - cos(δ) sin(αeast) sin(θ) + sin(δ) cos(θ)

Eq. 40    cos(E) cos (90° - Az) = cos(δ) cos{ -[90° - (-h)] }

Eq. 41    cos(E) sin (90° - Az) = cos(δ) sin{ -[90° - (-h)] } cos(90° - φ) + sin(δ) sin(90° - φ)

Eq. 42    sin(E) = - cos(δ) sin{ -[90° - (-h)] } sin(90° - φ) + sin(δ) cos(90° - φ)

Eq. 43    cos(E) sin(Az) = - cos(δ) sin(h)

Eq. 44    cos(E) cos(Az) = - cos(δ) cos(h) sin(φ) + sin(δ) cos(φ)

Eq. 45    sin(E) = cos(δ) cos(h) cos(φ) + sin(δ) sin(φ)

Of course, the opposite coordinate transformation, from Az-El to Equatorial can be determined by using the same steps shown above, but in reverse. It is left to the reader to show that Equations 46, 47, and 48 are true. Note that Equation 46 is the same as Equation 43 because the coordinate rotation is about the x-axes in both cases.

Eq. 46    cos(δ) sin(h) = - cos(E) sin(Az)

Eq. 47    cos(δ) cos(h) = - cos(E) cos(Az) sin(φ) + sin(E) cos(φ)

Eq. 48    sin(δ) = cos(E) cos(Az) cos(φ) + sin(E) sin(φ)

In some literature Equation 44 (or 47) are not mentioned. This is because it is not technically required if the user is wary of the quadrant the determined Azimuth coordinate resides in. However, the user is encouraged to use both the sine and cosine of the Azimuth coordinate in order to check the quadrant so that they will be assured of the proper coordinates every time. The determined Elevation coordinate returns the correct sign every time. Technically, only equations 43, 45, and 48 are required to perform the transformations in both directions. Finally, Equations 43, 45, and 48 are shown as Equations 49, 50, and 51 for convenience. To transform from RA-Dec to Az-El, use Equation 50 to find E, then use E, h, and δ in Equation 49 to find the Az (and possibly confirm the Azimuth's quadrant using Equation 44). To transform from Az-El to RA-Dec, use Equation 51 to find δ, then use δ, E and Az to find h (and therefore α) (and possibly checking the Hour Angle's quadrant by using Equation 47).

Eq. 49    cos(E) sin(Az) = - cos(δ) sin(h)

Eq. 50    sin(E) = cos(δ) cos(h) cos(φ) + sin(δ) sin(φ)

Eq. 51    sin(δ) = cos(E) cos(Az) cos(φ) + sin(E) sin(φ)

It is not too much of a stretch to generalize these three equations such they can be used for anything, including the apparent solar Az and El angles on a reflective surface of a spinning satellite, as long as the satellite's spin axis orientation is known and the orientation of the reflective surface on the spinning satellite, relative to the satellite's spin axis, is also known. There is much more to talk about with respect to coordinate transformations, but these basics are the cornerstone of all space science, from orbital mechanics to attitude control systems on board the most expensive and important satellites in orbit.