User talk:Aleric Inglewood

About Coordinate Systems and Rotations

The four coordinate systems

There are four coordinate systems that are related to LSL programming:

1. World coordinates
2. Region coordinates
3. Object coordinates (root prim coordinates)
4. Child prim coordinates (coordinates relative to the prim that the script is in, or refers to by link number)

The World coordinates refer to the map, and allow to include the sim in the coordinates, or refer to void water.

Region coordinates are relative to a given sim. The origin is in the South/West corner at height 0. The North/East corner then is 256, 256 and a Z coordinate for the height up to 4096 meter (on opensim you can go even higher).

Object coordinates are relative to the root prim. Hence, if the object is moved or rotated then the orientation of a child prim, when given in object coordinates, doesn't change. In LSL "local position" and "local rotation" refer to this coordinate system. "local" means relative to the root prim.

Prim Coordinates are relative to a given prim. If this prim is the root prim then the Prim Coordinates are the same as the Object Coordinates. For example, if a child prim is a cube with a size of 1,1,1 and one red surface where the center of that surface is at 1,0,0 then it will still be at 1,0,0 no matter how you move or rotate that child prim (relative to the other linked prims).

Positions

A different position of the origin of a coordinate system is easy to understand: You can think of positions as vectors that start in the origin of the coordinate system that they are given in and end in the point that they refer to. While the length of the vector is independent of the rotation of the coordinate system, the three coordinates are not; but a mental picture of an arrow doesn't have little numbers for three coordinates, so that picture works independent of the rotation too.

Since the rotation of the World Coordinate system and the Region Coordinate system is the same (X, Y and Z axis are parallel of both), and since World Coordinates aren't used in many LSL functions to begin with, we will ignore World Coordinates for now and only refer to Region Coordinates, or say "global" when we mean Region Coordinates.

Rotations

An LSL rotation internally stores a vector that is the axis around which to rotate and the angle of the rotation around that axis. Let V = <u, v, w> be the normalized vector (that is, with length 1) around which we rotate and let α (alpha) be the angle around which we have to rotate. Then the LSL rotation is a quaternion stored as r = <x, y, z, s> = < V * sin(α/2), cos(α/2) >. Thus, r.x = x * sin(α/2), and r.s = cos(α/2) etc. Note that the quaternion is also normalized. Also note that there is a duality here because inverting V (making it point the opposite way) and inverting the angle gives the same rotation; r and -r have different values but are the same rotation. Also note that if you don't rotate at all (α == 0) then it doesn't matter what axis V you pick, which is apparent because V drops out since sin(0) = 0. The quaternion <0, 0, 0, 1> is the ZERO_ROTATION quaternion.

The point of this technical story is to show that for an LSL rotation to make sense in terms of orientation, you need to be able to express a vector in three coordinates (u, v, w above): the axis around which we rotate is expressed relative to the X-, Y- and Z-axes of the coordinate system. Hence, it is the orientation of the X-, Y- and Z-axes that defines the meaning of a rotation in LSL.

In terms of a mental picture the origin with the (orientation of the) three axis, the red X-axis, the green Y-axis and the blue Z-axis is all the reference we need, combined with a vector for position or a quaternion for rotation.

When you edit an object, the viewer shows either 'World' or 'Local' axes, but really the 'World' axes show the wrong origin, shifted to an averaged center of the object, because if the origin was drawn at (0, 0, 0) you'd most likely not see it. The 'Local' ruler shows the correct coordinate system for the selected prim as its Prim Coordinate System. Selecting the root prim with 'Local' ruler on then shows the Object Coordinate System.

The dimension of rotations

As said before, given some coordinate system, any point in space can be represented with a vector. Obviously space is three dimensional, and thus vectors exists of three real values: one needs three distinct floating point numbers, the x coordinate, the y coordinate and the z coordinate to uniquely identify a position in a given coordinate system.

However, if one limits oneself to only normalized vectors, vectors with a length of one, then those represent all points on the surface of a sphere with radius 1. A surface is obviously two dimensional, so it should be possible to uniquely identify any point on the surface of such a sphere with only two floating point numbers.

One might think that selecting just two coordinates of the three of the vector will suffice because the third is fixed by the length requirement, but that only works for half spheres; for example, if x and y are known then z can still be either plus the square root of x squared plus y squared, or minus that value.

Instead, a better choice would be the spherical coordinate system and express the unit vectors with the two the angular coordinates φ (phi) and θ (theta), where φ is the angle between the vector and the positive Z axis, and θ the angle that the projection of the vector on to the X,Y plane makes with the positive X axis. In other words, starting with a unit vector along the positive Z-axis, one can obtain the required point on the surface of the sphere by first rotating this vector around the Y axis (towards the positive X axis) by an angle of φ and then rotating the result around the Z-axis by an angle of θ. Note that in both cases the rotations are counter-clockwise when one looks at it from the positive side of the axis that one rotates around (towards the origin).

What we just did was expressing a unit vector in terms of two rotations: one around the Z axis and one around the Y axis; and indeed every rotation can be expressed as two rotations around those two axis (or really, any two arbitrary axes, as long as they are independent (not parallel)).

It is therefore possible to represent unit vectors (aka, a direction) with a rotation, as both are two dimensional. The mental picture here is that any unit vector can be expressed as some fixed unit vector (for example the one pointing along the positive Z axis, but any other would do as well) and the rotation needed to turn that 'starting vector' into the unit vector that one wants to represent. The other way around almost works as well, with the exception of the starting vector itself, as that can be expressed by any arbitrary rotation around itself, so that information was lost and it is not possible to know which of those rotations was used. In fact, any vector close to the starting vector would give inaccurate results in the light of floating point round off errors and is not a good way to represent a rotation. Of course, the two dimensional vector (θ, φ) would be excellent to represent both: the unit vector, as well as any rotation; but LSL stores vectors in x, y, z coordinates, not in spherical coordinates.

Converting between coordinate systems

Consider a child prim of an object that is not the root prim and express its position with the vector pos_oc, and its orientation with the rotation rot_oc, both relative to the Object Coordinates System.

A script inside that prim can easily obtain those values:

vector pos_oc = llGetLocalPos();
rotation rot_oc = llGetLocalRot();

The postfix '_oc' stands for Object Coordinates and is used to make clear relative to which coordinate system the x, y and z values of both variables are. Remember that also the rotation contains a vector (the axis around which we rotate), so from now on we'll talk about 'coordinates' relative to a coordinate system for both, the position as well as the rotation, where the coordinates refer to the x, y and z components of the position and the rotation. For example, the first sentence of this paragraph will read "...express its position with the vector pos_oc, and its orientation with the rotation rot_oc, both in Object Coordinates."

Note that one has to be careful here, because a script in the root prim of the object will not obtain Object Coordinates but instead Region Coordinates! The reason is that the 'Local' functions return coordinates relative to the 'parent' coordinate system, where the parent of a non-root prim is the root prim, but the parent of the root prim is the region (or avatar in case of an attachment). The position and rotation of the root prim in Object Coordinates is trivial (ZERO_VECTOR and ZERO_ROTATION respectively).

Back to the script in the child prim. It can also easily obtain its position and rotation in Region Coordinates:

vector pos_rc = llGetPos();
rotation rot_rc = llGetRot();

Both sets, (pos_oc, rot_oc) and (pos_rc, rot_rc) refer to the same thing, so it should be possible to convert them into each other.

One can do this by looking at the orientation (position and rotation) of one coordinate system relative to the other and expressed in the coordinates of that other. The position and rotation of the Object Coordinate System relative to the Region Coordinate System, and in Region Coordinates, could be obtained as follows:

vector oc_pos_rc = llGetRootPosition();
rotation oc_rot_rc = llGetRootRotation();

As indicated by the '_rc' postfix, these are again in Region Coordinates.

Having the relative orientation of the Object Coordinate System in Region Coordinates, it is possible to convert Object Coordinates into Region Coordinates. More abstractly put, if you have two coordinate systems A and B, and you have their relative orientation expressed in A then you can convert orientations expressed in B into A.

The formula is as follows:

vector   pos_A = pos_B * B_rot_A + B_pos_A;
rotation rot_A = rot_B * B_rot_A;

The order of the multiplications is important; swapping the B_rot_A to the front will not work!

Note that the B_*_A are the position and rotation of the coordinate system B expressed in terms of the coordinate system A. The reason that I use the prefix and postfix in the order is directly related to the order enforced for rotation multiplications. Written this way, the left-hand postfix of a multiplication must match the right-hand prefix and conviently short-circuits the notation:

something_A = something_B * B_rot_A

the two B's 'connect' and drop out (along with the '_rot_') to give something_ ... A. This only works when the postfix refers to the coordinate system that the variable is in, and the prefix is whatever is described by the variable, possibly another coordinate system.

A complete example script follows: <lsl> default {

   touch(integer num_detected)
   {
       llSay(0, "---------");

       // Get the pos and rot in Object Coordinates.
       vector pos_oc = llGetLocalPos();
       rotation rot_oc = llGetLocalRot();
       
       // Fix these values when this script is in the root prim!
       if (llGetLinkNumber() == 0)
       {
           pos_oc = ZERO_VECTOR;
           rot_oc = ZERO_ROTATION;
       }
       
       // Get the pos and rot in Region Coordinates.
       vector pos_rc = llGetPos();
       rotation rot_rc = llGetRot();
       
       // Get the pos and rot of the Object Coordinate System in Region Coordinates.
       vector oc_pos_rc = llGetRootPosition();
       rotation oc_rot_rc = llGetRootRotation();
       
       // Print the values of the positions (the rotations are too hard to understand).
       llSay(0, "oc_pos_rc = " + (string)oc_pos_rc);
       llSay(0, "pos_os = " + (string)pos_os);
       llSay(0, "pos_rc = " + (string)pos_rc);
       
       // Calculate pos_rc and rot_rc from pos_oc and rot_oc : converting between _oc and _rc.
       vector   pos2_rc = pos_oc * oc_rot_rc + oc_pos_rc;
       rotation rot2_rc = rot_oc * oc_rot_rc;
       
       // Print the result for the position, and show the difference with what llGetPos() and llGetRot() returned.
       llSay(0, "pos2_rc = " + (string)pos2_rc);
       // These values should be ZERO_VECTOR and ZERO_ROTATION (or -ZERO_ROTATION).
       llSay(0, "pos_rc - pos2_rc = " + (string)(pos_rc - pos2_rc));
       llSay(0, "rot_rc / rot2_rc = " + (string)(rot_rc / rot2_rc));
   }

} </lsl>

Obviously there is no preference for one coordinate system over the other for this conversion; one can equally as well convert orientations in A into B:

vector   pos_B = pos_A * A_rot_B + A_pos_B;
rotation rot_B = rot_A * A_rot_B;

The question arises then how to obtain the A_*_B values. For example, there is no LSL function that returns directly the position or rotation of the region relative to a prim (aka, rc_*_oc).

Intuitively one would say that it has to be possible to express A_*_B in B_*_A on grounds of symmetry, and that is indeed the case.

To show how this works, lets just take the formula for pos_A and rot_A (see above) and rework them to get pos_B and rot_B on the left-hand side. This results first in

pos_B * B_rot_A = pos_A - B_pos_A
rot_B * B_rot_A = rot_A

Then divide both sides by B_rot_A to get:

pos_B = pos_A / B_rot_A - B_pos_A / B_rot_A
rot_B = rot_A / B_rot_A

from which we can conclude that

A_pos_B = - B_pos_A / B_rot_A
A_rot_B = ZERO_ROTATION / B_rot_A

Note how A_rot_B is the inverse of B_rot_A: A_rot_B * B_rot_A = ZERO_ROTATION. As per our notation convention you'd expect A_rot_B * B_rot_A to be A_rot_A, and that is actually the case as A_rot_A means the relative orientation of coordinate system A expressed in A's coordinates, which is the trivial ZERO_VECTOR and ZERO_ROTATION as we saw before for the root prim orientation expressed in Object Coordinates.

And for fun, note that if we do this inversion again we get:

B_pos_A = - A_pos_B / A_rot_B = - (- B_pos_A / B_rot_A) / (ZERO_ROTATION / B_rot_A) = B_pos_A / B_rot_A * B_rot_A = B_pos_A
B_rot_A = ZERO_ROTATION / A_rot_B = ZERO_ROTATION / (ZERO_ROTATION / B_rot_A) = B_rot_A

as expected.

We can now also convert Region Coordinates to Object Coordinates.

One could first calculate,

vector rc_pos_oc = - llGetRootPosition() / llGetRootRotation();
rotation rc_rot_oc = ZERO_ROTATION / llGetRootRotation();

and then convert as usual:

vector   pos_oc = pos_rc * rc_rot_oc + rc_pos_oc;
rotation rot_oc = rot_rc * rc_rot_oc;

or if you don't want to calculate the intermediate rc_*_oc values, you could do immediately:

vector   pos_oc = (pos_rc - llGetRootPosition()) / llGetRootRotation();
rotation rot_oc = rot_rc / llGetRootRotation();

To verify this, you could extend the above script with <lsl>

       // Calculate pos_oc and rot_oc from pos_rc and rot_rc.
       vector   pos2_oc = (pos_rc - llGetRootPosition()) / llGetRootRotation();
       rotation rot2_oc = rot_rc / llGetRootRotation();
       
       llSay(0, "pos2_oc = " + (string)pos2_oc);
       // These values should again be ZERO_VECTOR and +/- ZERO_ROTATION.
       llSay(0, "pos_oc - pos2_oc = " + (string)(pos_oc - pos2_oc));
       llSay(0, "rot_oc / rot2_oc = " + (string)(rot_oc / rot2_oc));

</lsl>

Cascading coordinate systems

Imagine you have a castle object where the floor is the root prim. Everything is linked because you want it to be easy to move the castle around. One of the child prims is a sculpty representing the hinges around which you want to rotate a large gate. The gate has a rotating wheel on it.

You want that the object (castle) to keep working when it is moved and/or rotated, but also when it is edited and the hinges are moved and/or rotated. This means that the state of the gate prims must be stored relative to the hinges, which is by far the easiest thing to do to begin with, after all, the gate rotates around the hinges and has no relation with the rest of the castle.

Hence, we have the Region Coordinate System, the Object Coordinate System (the castle) and the Hinges Coordinate System. The latter is our first example of a Prim Coordinates System, but I'll use the postfix '_hc' for "Hinges Coordinates" instead of '_pc'. Finally we have the Spill Coordinate System (or Gate Coordinates) where the spill is a prim around which we want to rotate the wheel, and I'll used '_sc' as postfix for that.

The variables used then would be an orientation for the wheel in *_sc coordinates, the orientation of the spill, and other gate prims, in *_hc coordinates and the orientation of the hinges in *_oc coordinates. Finally we could have variables for the orientation of the castle (root prim) in *_rc coordinates of course.

vector   castle_pos_rc; // The position of the castle in Region Coordinates.
rotation castle_rot_rc; // The rotation of the castle in Region Coordinates.

vector   hinges_pos_oc; // The position of the hinges in Object Coordinates.
rotation hinges_rot_oc; // The rotation of the hinges in Object Coordinates.

vector   spill_pos_hc;  // The position of the spill in Hinges Coordinates.
rotation spill_rot_hc;  // The rotation of the spill in Hinges Coordinates.

vector   wheel_pos_sc;  // The position of the wheel in Spill Coordinates.
rotation wheel_rot_sc;  // The rotation of the wheel in Spill Coordinates.

Now because we used existing prims orientations as coordinate system for the next prim, the position and rotation of those prims are the position and rotation of that coordinate system. Just like before we saw that llGetPos/llGetRot returns the global orientation of the root prim when used in a script in the root prim, while llGetRootPos/llGetRootRot gave us the relative orientation of the Object Coordinate System in Region Coordinates, but is in fact the same as what llGetPos/llGetRot in the root prim returns: the position/rotation of the root prim is the orientation of the Object Coordinate System relative to the Region.

Likewise, in this case, castle_*_rc is the relative orientation of the Object Coordinate System in Region Coordinates as well as the orientation of the root prim in Region Coordinates; they are the same thing.

The same then applies to the *_oc variables, which we defined as the orientation of the hinges in Object Coordinates, but at the same time are the relative orientation of the Hinges Coordinate System in Object Coordinates. The *_hc variables are the orientation of spill in Hinges Coordinates, but also the relative orientation of the Spill Coordinates in Hinges Coordinates.

Lets write it out using the same notation convention as used above, then we have:

vector   oc_pos_rc = castle_pos_rc; // The Object Coordinate System orientation in Region Coordinates.
rotation oc_rot_rc = castle_rot_rc;

vector   hc_pos_oc = hinges_pos_oc; // The Hinges Coordinate System orientation in Object Coordinates.
rotation hc_rot_oc = hinges_rot_oc;

vector   sc_pos_hc = spill_pos_hc;  // The Spill Coordinate System orientation in Hinges Coordinates.
rotation sc_rot_hc = spill_rot_hc;

It will therefore probably not surprise you that

vector   hc_pos_rc = hc_pos_oc * oc_rot_rc + oc_pos_rc; // The Hinges Coordinate System orientation in Region Coordinates.
rotation hc_rot_rc = hc_rot_oc * oc_rot_rc;

vector   sc_pos_rc = sc_pos_hc * hc_rot_rc + hc_pos_rc; // The Spill Coordinate System orientation in Region Coordinates.
rotation sc_rot_rc = sc_rot_hc * hc_rot_rc;

And finally, using the latter, we can express the wheel orientation in region coordinates:

vector   wheel_pos_rc = wheel_pos_sc * sc_rot_rc + sc_pos_rc;
rotation wheel_rot_rc = wheel_rot_sc * sc_rot_rc;

Although this was nothing else but applying the coordinate system conversion formula three times, it might give some insight to get rid of the intermediate variables and express the wheel orientation in region coordinates without them, giving:

wheel_pos_rc = ((wheel_pos_sc * sc_rot_hc + sc_pos_hc) * hc_rot_oc + hc_pos_oc) * oc_rot_rc + oc_pos_rc
wheel_rot_rc = wheel_rot_sc * sc_rot_hc * hc_rot_oc * oc_rot_rc

Applying rotations and translations

Applying rotations and translations (moving) is very much the same as coordinate system transformations described above. To show this consider the following: imagine you have a prim 'pa' with its own coordinate system A. Then it trivially has position ZERO_VECTOR and rotation ZERO_ROTATION relative to A, per definition.

vector   pa_pos_A = ZERO_VECTOR;
rotation pa_rot_A = ZERO_ROTATION;

Now consider a prim 'pb' that has some relative position pb_pos_A and rotation pb_rot_A given in A coordinates. For example,

vector   pb_pos_A = <0.5, 0.6, 0.7>;
rotation pb_rot_A = <0.38, 0.26, 0.34, 0.82>;

Note that 0.38² + 0.26² + 0.34² + 0.82² = 1, as it must be since rotations are normalized.

As seen before, the orientation of pb relative to pa is the same as the orientation of pb relative to A, and the same as the orientation of B relative to A. pa and A are interchangeable as long as pa has ZERO_VECTOR and ZERO_ROTATION relative to A. Thus, with B being the coordinate system where pb currently is, we also have:

vector   B_pos_A = pb_pos_A; // The orientation of coordinate system B, in coordinates of A.
rotation B_rot_A = pb_rot_A;

Now suppose we want move the prim pa (without moving the coordinate system A), by applying the translation 'B_pos_A', after all a translation is stored in a vector and B_pos_A is a vector. Likewise, we want to rotate pa (again without influencing A) by applying the rotation B_rot_A. Then the most logical order to this in is by first applying the rotation and then applying the translation, because then the meaning of the translation is preserved and will match the meaning of B_pos_A in that pa will end up where pb is now. If you'd first apply the translation so that pa ends up where pb is, and then apply the rotation (around A!) then pa would swoop around the origin of A and move away from its desired position.

Doing this correctly should leave us with pa_pos_A == pb_pos_A and pa_rot_A == pb_rot_A, so that pa ended up precisely where pb is now. We can't just assign those values however, because we're looking for a formula that works in general; a way to apply B_rot_A and then B_pos_A to any prim with -say- orientation pc_pos_A, pc_rot_A.

Thus, first we apply the rotation:

pc_rot_A *= B_rot_A;
pc_pos_A *= B_rot_A;

And then apply the translation:

pc_pos_A += B_pos_A;

Now lets check if indeed this would cause pa to end up where pb is. The effect on pa_*_A would be:

pa_rot_A = pa_rot_A * B_rot_A = ZERO_ROTATION * B_rot_A = B_rot_A = pb_rot_A;
pa_pos_A = pa_pos_A * B_rot_A = ZERO_VECTOR * B_rot_A = ZERO_VECTOR;

and then the translation

pa_pos_A = pa_pos_A + B_pos_A = ZERO_VECTOR + B_pos_A = B_pos_A = pb_pos_A;

So yes, pa_*_A ends up as pb_*_A!

The real test however would be if we first applied some transformation to pa to bring it to some arbitrary orientation (aka, pc), then apply the transformation B_rot_A,B_pos_A and then reverse the first transformation but now relative to B, and then find that we indeed end up at pb.

Let us first define a 'transformation', which exists of a vector for the translation and a rotation for the rotation. We will have two transformation during this test: t1: pa -> pc, and t2: pa -> pb.

As we just saw (or we hope this is going to be the case), t2 can be defined as:

vector   t2_translation = B_pos_A;
rotation t2_rotation = B_rot_A;

where as t1 is arbitrary, just some values t1_translation and t1_rotation.

Applying t1 to pa, we get (this is thus the same as above, but now with the 't1' notation):

pa_rot_A *= t1_rotation;
pa_pos_A *= t1_rotation;
pa_pos_A += t1_translation;

Next we apply translation t2 to that:

pa_rot_A *= t2_rotation;
pa_pos_A *= t2_rotation;
pa_pos_A += t2_translation;

Finally we want to reverse t1, but in the coordinate system of B. Therefore, we first convert our coordinates to that of B:

pa_pos_B = pa_pos_A * A_rot_B + A_pos_B;
pa_rot_B = pa_rot_A * A_rot_B;

where

A_pos_B = - B_pos_A / B_rot_A;
A_rot_B = ZERO_ROTATION / B_rot_A;

and then reverse t1. Note now we have to do the translation first!

pa_pos_B -= t1_translation;
pa_pos_B /= t1_rotation;
pa_rot_B /= t1_rotation;

And if everything worked out then pa now should be at ZERO_VECTOR, ZERO_ROTATION relative to B!

As pa_pos_A,pa_rot_A started as ZERO_VECTOR,ZERO_ROTATION, the first step of applying t1 resulted in pa_rot_A = t1_rotation and pa_pos_A = t1_translation.

After applying t2 we get pa_rot_A = t1_rotation * t2_rotation, and pa_pos_A = t1_translation * t2_rotation + t2_translation.

Converting that to B gives

pa_pos_B = (t1_translation * t2_rotation + t2_translation) * A_rot_B + A_pos_B
pa_rot_B = (t1_rotation * t2_rotation) * A_rot_B

where, remembering that t2_translation = B_pos_A and t2_rotation = B_rot_A,

A_pos_B = - B_pos_A / B_rot_A = - t2_translation / t2_rotation
A_rot_B = ZERO_ROTATION / B_rot_A = ZERO_ROTATION / t2_rotation

so that we get

pa_pos_B = (t1_translation * t2_rotation + t2_translation) / t2_rotation - t2_translation / t2_rotation = t1_translation
pa_rot_B = (t1_rotation * t2_rotation) / t2_rotation = t1_rotation

and finally reversing t1 gives:

pa_pos_B = (t1_translation - t1_translation) / t1_rotation = ZERO_VECTOR
pa_rot_B = t1_rotation / t1_rotation = ZERO_ROTATION

So that we can conclude that if you have two prims pa and pb, each representing their own coordinate system A and B respectively, and you express the orientation of B relative to A as the pair (B_pos_A, B_rot_A) — then the following holds.

The transformation pair (t_translation, t_rotation) equals (B_pos_A, B_rot_A) as long as you first apply the rotation and then the translation while working in the coordinate system of A.

The latter is also very practical, because you normally won't know anything about the target coordinate system B until you actually did the transformation.

Now remember that we found that in order to convert variables from being relative to a coordinate system B to a coordinate system A, we did:

pos_A = pos_B * B_rot_A + B_pos_A
rot_A = rot_B * B_rot_A

which is thus exactly the same as first applying the rotation t1_rotation and then the translation t1_translation on the pair (pos_B, rot_B) where that pair is relative to A! That isn't too weird because if you consider a prim in the origin of A with no rotation, aka (ZERO_VECTOR, ZERO_ROTATION) relative to A and then apply the transformation then you expect to end up in the origin of B with no rotation relative to B. Aka pos_B = ZERO_VECTOR and rot_B = ZERO_ROTATION. So now work backwards: assume you already have the prim in B like that, then what are its coordinates in A? That would be the above conversion thus, with pos_B = ZERO_VECTOR and rot_B = ZERO_ROTATION.

Hence, doing the conversion with ZERO as input gives the same results as doing the t transformation with ZERO as input! And the final formula for a transformation just looks exactly the same:

target_pos_A = source_pos_A * t_rotation + t_translation
target_rot_A = source_rot_A * t_rotation

LSL functions

What all of the above taught you is mainly a coding style: use prefixes and postfixes for your rotations (and translations)! It's all in the names of your variables. A 'rotation' type can be the rotation part of a transformation (not tight to a particular coordinate system, but rather rotating around whatever coordinate system you apply it to), it can be the representation of a unit vector, it can be the rotation part of the orientation of a prim relative to a given coordinate system, or it can represent the rotation part of a coordinate system conversion. If you don't use a consistent way to reflect all that in your variable names then you will get confused.

The LSL functions mostly deal with two types: orientations (position + rotation) of prims relative to some coordinate system, and simply transformations (rotation and/or translation).

The table below shows the coordinate systems involved.

		Coordinates of the prim containing the script (sc_)
Coordinate System	Abbreviation	script in child prim	script in root prim
World/Map Coordinates	wc	sc_pos_wc = sc_pos_rc + llGetRegionCorner() sc_rot_wc = sc_rot_rc
Region Coordinates	rc	sc_pos_rc = llGetPos() sc_rot_rc = llGetRot()
Object/Root Coordinates	oc	sc_pos_oc = llGetLocalPos() sc_rot_oc = llGetLocalRot()	sc_pos_oc = ZERO_VECTOR sc_rot_oc = ZERO_ROTATION
Self Coordinates	sc	sc_pos_sc = ZERO_VECTOR sc_rot_sc = ZERO_ROTATION

Where 'in the root prim' can be detect by testing that llGetLinkNumber() returns a value <= 1.

Alternatively, you can set link = llGetLinkNumber() and use the following table which shows how to obtain the coordinates of a prim with link number 'link' from a script anywhere in the same object. Note that this doesn't work when the object exist of a single prim: then llGetLinkNumber() returns 0, while 'link' must be larger than 0 here. If you need to get Object Coordinates (_oc) then do not use the negative value LINK_THIS. You can use LINK_THIS for the other coordinate systems, provided the object exists of at least two prims.

		Coordinates of a linked prim (link_)
Coordinate System	Abbreviation	link > 1	link = 1
World/Map Coordinates	wc	link_pos_wc = link_pos_rc + llGetRegionCorner() link_rot_wc = link_rot_rc
Region Coordinates	rc	link_pos_rc = llList2Vector(llGetLinkPrimitiveParams(link, [PRIM_POSITION]), 0) link_rot_rc = llList2Rot(llGetLinkPrimitiveParams(link, [PRIM_ROTATION]), 0)
Object/Root Coordinates	oc	link_pos_oc = llList2Vector(llGetLinkPrimitiveParams(link, [PRIM_POS_LOCAL]), 0) link_rot_oc = llList2Rot(llGetLinkPrimitiveParams(link, [PRIM_ROT_LOCAL]), 0)	link_pos_oc = ZERO_VECTOR link_rot_oc = ZERO_ROTATION
Self Coordinates	sc	link_pos_sc = (link_pos_oc - sc_pos_oc) / sc_rot_oc link_rot_sc = link_rot_oc / sc_rot_oc

While the above table works for link = LINK_ROOT, it is probably more convenient to use the following table:

		Coordinates of the root prim (oc_)
Coordinate System	Abbreviation	script in any prim
World/Map Coordinates	wc	oc_pos_wc = oc_pos_rc + llGetRegionCorner() oc_rot_wc = oc_rot_rc
Region Coordinates	rc	oc_pos_rc = llGetRootPosition() oc_rot_rc = llGetRootRotation()
Object/Root Coordinates	oc	oc_pos_oc = ZERO_VECTOR oc_rot_oc = ZERO_ROTATION
Self Coordinates	sc	oc_pos_sc = - sc_pos_oc / sc_rot_oc oc_rot_sc = ZERO_ROTATION / sc_rot_oc

As shows from the above table, LSL does not directly support _sc coordinates, and I used conversions to fill in the tables. All in all, we listed four coordinate systems (wc, rc, oc and sc; not including the 'link_' stuff, which isn't really a separate coordinate system) and thus could write out four times four is sixteen ways to convert one into the other. Four of those would be trivial, and of the remaining twelve we can express six as the inverse of the other six.

However, lets ignore the World Coordinates for now, which are just the same as the Region Coordinates with an offset for the position. Then we have three coordinate systems (rc, oc and sc), leading to nine conversion, three of which are trivial, and of the remaining six three can be expressed as the inverse of the other three. Those three are: oc_*_rc, sc_*_rc and sc_*_oc, which are all listed above. The trivial one, rc_*_rc, oc_*_oc and sc_*_sc are all ZERO_* (though the latter two have been listed in the above tables anyway). Finally, the inverse values, rc_*_oc, rc_*_sc and oc_*_sc can be expressed with the inversion formula given in #Converting between coordinate systems.