Rotation/pl

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LSL'owy typ rotation jest używany do przedstawienia trójwymiarowej orientacji. Orientacja jest reprezentowana przez pojęcie matematyczne nazywane kwaternionem. Można myśleć o kwaternionie jako o czterech liczbach, z których pierwsze trzy reprezentują kierunek w którym obiekt jest skierowany (rozumiany jako wektor 3D), a czwarta z nich to nachylenie (kąt obrotu) w lewo lub w prawo w płaszczyźnie normalnej (prostopadłej) do kierunku skierowania. Podstawową zaletą kwaternionów jest to, ze nie powodują one błędów typu gimbal lock (związanych z równoległością prostych w przestrzeni trójwymiarowej). Jeżeli chcesz zapoznać się ze szczegółami matematycznymi, zobacz kawternion. Rotation Synopsis to lista użytecznych funkcji i zdarzeń (events) odnoszących się do obrotów. Jest też informacja o obracaniu tekstur, tutaj

Inne reprezentacje

Innym sposobem przedstawienia obrotów jest użycie trzech liczb, <X, Y, Z>, które reprezentują kąty (wyrażone w radianach) o które obiekt jest obracany dookoła każdej z osi. Ten rodzaj przedstawienia używany jest w okienku Edycji, i jest generalnie łatwy w wizualizacji. Uwaga, te trzy liczby to typ vector a nie rotation, choć reprezentują tę samą informację. Ten sposób przedstawienia rotacji nazywamy Eulerowskim.

Trzecim rodzajem reprezentacji jest użycie trzech wektorów, wskazujących gdzie jest skierowany przód, góra i lewa strona obiektu. W rzeczywistości potrzebne są tylko dwa z trzech wektorów, ponieważ z dowolnych dwóch da się trzeci wyliczyć.

Z pewnych powodów, takich jak np. możliwość składania obrotów, cztero-liczbowa wersja, rotation, jest lepsza, chociaż na początku może być trudniejsza to wyobrażenia sobie. Na szczęście bardzo rzadko występuje konieczność posługiwania się wewnętrzną reprezentacją obrotów, ponieważ są funkcje konwertujące różne sposoby reprezentacji.

Reguła prawej dłoni

W LSL'u wszystkie obroty są wykonywane zgodnie z regułą prawej dłoni. Ustawiamy dłoń z palcami wyprostowanymi wzdłuż osi OX. Następnie zginamy palce, z wyjątkiem kciuka, wskazując zwrot osi OY. Odwiedziny kciuk wskazuje zwrot osi OZ w układzie prawoskrętnym. Podczas edycji obiektu trzy kolorowe strzałki wskazują dodatni kierunek każdej osi (X: czerwona, Y: zielona, Z: czerwona).

Reguła Prawej Dłoni

Prawa ręka jeszcze się przyda, do ustalenia kierunku dodatniego obrotu. Palce pięści prawej dłoni, z kciukiem ustawionym w osi obrotu który nas interesuje, wskazują kierunek dodatnigo obrotu wokół tej osi. Obroty wzdłuż osi X, Y i Z są często określane jako "beczka", "nurkowanie" i "zboczenie" (Roll, Pitch, and Yaw), sczególnie dotyczy to pojazdów.

Kierunek obrotów wg Reguły Prawej Dłoni

Skłądanie obrotów

To combine rotations, you use the multiply and divide operators. Don't try to use addition or subtraction operators on rotations, as they will not do what you expect. The multiply operation applies the rotation in the positive direction, the divide operation does a negative rotation. You can also negate a rotation directly, just negate the s component, e.g. X.s = -X.s.

Unlike other types such as float, the order in which the operations are done, non-commutative, is important. The reason for this is simple: the order you do rotations in is important in RL. For example, if you had a dart with four feathers, started from rotation <0, 0, 0> with its tail on the origin, it would lie on the X axis with its point aimed in the positive X direction, its feathers along the Z and Y axes, and the axes of the dart and the axes of the world would be aligned. We're going to rotate it 45 degrees around X and 30 degrees around Y, but in different orders.

First, after rotating 45 deg around X the dart would still be on the X axis, unmoved, just turned along its long axis, so the feathers would be at 45 deg to the axes. Then rotating 30 deg around Y would move it in the XZ plane to point down 30 deg from the X axis (remember the right hand rule for rotations means a small positive rotation around Y moves the point down). The dart winds up pointing 30 deg down, in the same vertical plane it started in, but turned around its own long axis so the feathers are no longer up and down.

If you did it the other way, first rotating 30 deg in Y, the dart would rotate down in the XZ plane, but notice that it no longer is on the X axis; its X axis and the world's aren't aligned any more. Now a 45 degree rotation around the X axis would pivot the dart around its tail, the point following a 30 deg cone whose axis is along the positive world X axis, for 45 degrees up and to the right. If you were looking down the X axis, it would pivot from pointing 30 deg below the X axis, up and to the right, out of the XZ plane, to a point below the 1st quadrant in the XY plane, its feathers rotating as it went.

Clearly this is a different result from the first rotation, but the order of rotation is the only thing changed.

To do a constant rotation you need to define a rotation value which can be done by creating a vector with the X, Y, Z angles in radians as components (called an Euler angle), then converting that to a rotation by using the llEuler2Rot function. You can alternately create the native rotation directly: the real part is the cosine of half the angle of rotation, and the vector part is the normalized axis of rotation multiplied by the sine of half the angle of rotation. To go from a rotation to an Euler angle vector use llRot2Euler.

NOTE: angles in LSL are in radians, not degrees, but you can easily convert by using the built-in constants RAD_TO_DEG and DEG_TO_RAD. For a 30 degree rotation around the X axis you might use:

rotation rot30X	= llEuler2Rot(<30, 0, 0> * DEG_TO_RAD);	// convert the degrees to radians, then convert that vector into a rotation, rot30x
vector vec30X	= llRot2Euler(rot30X );	// convert the rotation back to a vector (the values will be in radians)

Order of rotation for Euler Vectors

From the above discussion, it's clear that when dealing with rotations around more than one axis, the order they are done in is critical. In the Euler discussion above this was kind of glossed over a bit, the individual rotations around the three axis define an overall rotation, but that begs the question: What axis order are the rotations done in? The answer is Z, Y, X in global coordinates. If you are trying to rotate an object around more than one axis at a time using the Euler representation, determine the correct Euler vector using the Z, Y, X rotation order, then use the llEuler2Rot function to get the rotation for use in combining rotations or applying the rotation to the object.

Local vs Global (World) rotations

It is important to distinguish between the rotation relative to the world, and the rotation relative to the local object itself. In the editor, you can switch back and forth from one to the other. In a script, you must convert from one to the other to get the desired behavior.

Local rotations are ones done around the axes embedded in the object itself forward/back, left/right, up/down, irrespective of how the object is rotated in the world. Global rotations are ones done around the world axes, North/South, East/West, Higher/Lower. You can see the difference by rotating a prim, then edit it and change the axes settings between local and global, notice how the colored axes arrows change.

In LSL, the difference between doing a local or global rotation is the order the rotations are evaluated in the statement.

This does a local 30 degree rotation by putting the constant 30 degree rotation to the left of the object's starting rotation (myRot). It is like the first operation in the first example above, just twisting the dart 30 degrees around its own long axis.

rotation localRot =

rot30X * myRot;

// do a local rotation by multiplying a constant rotation by a world rotation

To do a global rotation, use the same rotation values, but in the opposite order. This is like the second operation in the second example, the dart rotating up and to the right around the world X axis. In this case, the existing rotation (myRot) is rotated 30 degrees around the global X axis.

rotation globalRot

= myRot * rot30X;

// do a global rotation by multiplying a world rotation by a constant rotation

Another way to think about combining rotations

You may want to think about this local vs global difference by considering that rotations are done in evaluation order, that is left to right except for parenthesized expressions.

In the localRot case, what happened was that starting from <0, 0, 0>, the rot30X was done first, rotating the prim around the world X axis, but since when it's unrotated, the local and global axes are identical it has the effect of doing the rotation around the object's local X axis. Then the second rotation myRot was done which rotated the prim to its original rotation, but now with the additional X axis rotation baked in. What this looks like is that the prim rotated in place, around its own X axis, with the Y and Z rotations unchanged, a local rotation.

In the globalRot case, again starting from <0, 0, 0>, first the object is rotated to its original rotation (myRot), but now the object's axes and the world's axes are no longer aligned! So, the second rotation rot30x does exactly what it did in the local case, rotates the object 30 degrees around the world X axis, but the effect is to rotate the object through a cone around the world X axis since the object's X axis and the world's X axis aren't the same this time. What this looks like is that the prim pivoted 30 degrees around the world X axis, hence a global rotation.

Division of rotations has the effect of doing the rotation in the opposite direction, multiplying by a 330 degree rotation is the same as dividing by a 30 degree rotation.

Using Rotations

You can access the individual components of a rotation R by R.x, R.y, R.z, & R.s (not R.w). The scalar part R.s is the cosine of half the angle of rotation. The vector part (R.x,R.y,R.z) is the product of the normalized axis of rotation and the sine of half the angle of rotation. You can generate an inverse rotation by negating the x,y,z members (or by making the s value negative). As an aside, you can also use a rotation just as a repository of float values, each rotation stores four of them and a list consisting of rotation is more efficient than a list consisting of floats, but there is overhead in unpacking them.

rotation rot30X	= llEuler2Rot(<30, 0, 0> * DEG_TO_RAD );	// Create a rotation constant
rotation rotCopy	= rot30X;	// Just copy it into rotCopy, it copies all 4 float components
float X	= rotCopy.x;	// Get out the individual components of the rotation
float Y	= rotCopy.y;
float Z	= rotCopy.z;
float S	= rotCopy.s;
rotation anotherCopy	= <X, Y, Z, S>;	// Make another rotation out of the components

There is a built in constant for a zero rotation ZERO_ROTATION which you can use directly or, if you need to invert a rotation R, divide ZERO_ROTATION by R. As a reminder from above, this works by first rotating to the zero position, then because it is a divide, rotating in the opposite sense to the original rotation, thereby doing the inverse rotation.

rotation rot330X	= <-rot30X.x, -rot30X.y, -rot30X.z, rot30X.s>;	// invert a rotation - NOTE the s component isn't negated
rotation another330X	= ZERO_ROTATION / rot30X;	// invert a rotation by division, same result as rot330X
rotation yetanother330X	= <rot30X.x, rot30X.y, rot30X.z, -rot30X.s>;	// not literally the same but works the same.

Single or Root Prims vs Linked Prims vs Attachments

The reason for talking about single or linked prim rotations is that for things like doors on vehicles, the desired motion is to move the door relative to the vehicle, no matter what the rotation of the overall vehicle is. While possible to do this with global rotations, it would quickly grow tedious. There are generally three coordinate systems a prim can be in: all alone, part of a linkset, or part of an attachment. When a prim is alone, i.e., not part of a linkset, it acts like a root prim; when it is part of an attachment, it acts differently and is a bit broken.

Getting and setting rotations of prims

Function	Ground (rez'ed) Prims		Attached Prims
	Root	Children	Root	Children
llGetRot llGPP:PRIM_ROTATION llGetObjectDetails	global rotation of prim	global rotation of prim	global rotation of avatar	global rotation of avatar * global rotation of prim (Not Useful)
llGetLocalRot llGPP:PRIM_ROT_LOCAL	global rotation of prim	rotation of prim relative to root prim	rotation of attachment relative to attach point	rotation of prim relative to root prim
llGetRootRotation	global rotation of prim	global rotation of root prim	global rotation of avatar	global rotation of avatar
llSetRot* llSPP:PRIM_ROTATION*	set global rotation	complicated, see llSetRot	set rotation relative to attach point	set rotation to root attachment rotation * new_rot.
llSetLocalRot* llSPP:PRIM_ROT_LOCAL*	set global rotation	set rotation of prim relative to root prim	set rotation relative to attach point	set rotation of prim relative to root prim
llTargetOmega† ll[GS]PP:PRIM_OMEGA	spin linkset around prim's location	spin prim around its location	spin linkset around attach point	spin prim around its location

*	Physical objects which are not children in a linkset will not respond to setting rotations.
†	For non-Physical objects llTargetOmega is executed on the client side, providing a simple low lag method to do smooth continuous rotation.

Rotating Vectors

In LSL, rotating a vector is very useful if you want to move an object in an arc or circle when the center of rotation isn't the center of the object.

This sounds very complex, but there is much less here than meets the eye. Remember from the above discussion of rotating the dart, and replace the physical dart with a vector whose origin is the tail of the dart, and whose components in X, Y, and Z describe the position of the tip of the dart. Rotating the dart around its tail moves the tip of the dart through and arc whose center of rotation is the tail of the dart. In exactly the same way, rotating a vector which represents an offset from the center of a prim rotates the prim through the same arc. What this looks like is the object rotates around a position offset by the vector from the center of the prim.

rotation rot6X	= llEuler2Rot(<30, 0, 0> * DEG_TO_RAD );	// create a rotation constant, 30 degrees around the local X axis
vector offset	= <0, 1, 0>;	// create an offset one meter in the global positive Y direction
vector rotatedOffset	= offset * rot6X;	// rotate the offset to get the motion caused by the rotations
vector newPos	= llGetPos() + (offset - rotatedOffset) * llGetRot();	// move the prim position by the rotated offset amount
rotation newRot	= rot6X * llGetRot();	// change rot to continue facing offset point

<lsl> //-- same as: llSetPrimitiveParams( [PRIM_POSITION, llGetPos() + (gPosOffset - gPosOffset * gRotArc) * llGetRot(),

                      PRIM_ROTATION, gRotArc * llGetRot() );

</lsl> Nota Bene: Doing this is a move, so don't forget about issues of moving a prim off world, below ground, more than 10 meters etc.

To get a point relative to the objects current facing (such as used in rezzors)

rotation rotFacing	= llGetRot();	// get the objects current rotation
vector offset	= <0, 0, 1>;	// create an offset one meter in the global positive Z direction
vector rotatedOffset	= offset * rotFacing;	// rotate the offset to be relative to object rotation
vector correctOffset	= llGetPos() + rotatedOffset;	// get the global position of the local offset

<lsl> //-- same as: vector correctOffset = llGetPos() + offset * llGetRot(); </lsl>

Constants

ZERO_ROTATION

ZERO_ROTATION = <0.0, 0.0, 0.0, 1.0>;
A rotation constant representing a Euler angle of <0.0, 0.0, 0.0>.

DEG_TO_RAD

DEG_TO_RAD = 0.01745329238f;
A float constant that when multiplied by an angle in degrees gives the angle in radians.

RAD_TO_DEG

RAD_TO_DEG = 57.29578f;
A float constant when multiplied by an angle in radians gives the angle in degrees.