Difference between revisions of "User talk:Aleric Inglewood"
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An LSL rotation internally stores a vector that is the axis around which | An LSL rotation internally stores a vector that is the axis around which | ||
to rotate and the angle of the rotation around that axis. | 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 | Let '''V''' = <u, v, w> be the normalized vector (that is, with length 1) around | ||
which we rotate and let a be the angle around which we have to rotate. | which we rotate and let a 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(a/2), cos(a/2) >. | Then the LSL rotation is a quaternion stored as r = <x, y, z, s> = < '''V''' * sin(a/2), cos(a/2) >. | ||
Thus, r.x = x * sin(a/2), and r.s = cos(a/2) etc. Note that the quaternion | Thus, r.x = x * sin(a/2), and r.s = cos(a/2) etc. Note that the quaternion | ||
is also normalized. | is also normalized. | ||
Also note that there is a duality here because inverting | 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 | opposite way) and inverting the angle gives the same rotation; r and -r have | ||
different values but are the same rotation. | different values but are the same rotation. | ||
Also note that if you don't rotate at all (a == 0) then it doesn't matter | Also note that if you don't rotate at all (a == 0) then it doesn't matter | ||
what axis V you pick, which is apparent because V drops out since sin(0) = 0. | 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 quaternion <0, 0, 0, 1> is the ZERO_ROTATION quaternion. | ||
Revision as of 18:02, 15 March 2014
About Coordinate Systems and Rotations
The four coordinate systems
There are four coordinate systems that are related to LSL programming:
- World coordinates
- Region coordinates
- Object coordinates (root prim coordinates)
- Prim coordinates
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 a 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(a/2), cos(a/2) >. Thus, r.x = x * sin(a/2), and r.s = cos(a/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 (a == 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 and 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.