Difference between revisions of "User talk:Aleric Inglewood"

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== About Coordinate Systems and Rotations ==
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=== 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 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 any position in a given coordinate system,
any point in space.
 
However, if one limits oneself to only normalized vectors, vectors
with a length 1, then those represent all the 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 position on
the surface of such a sphere with only two numbers. One might think
that selecting just two coordinates of the three of the vector will
suffice because the third is fixed by the requirement that the sum
of the squares of the coordinates is 1, but that would only work
on a half sphere. For example, if x and y are known then z can still
be either plus the square root of x squared plus y squares, ''or'' minus
that value. Instead, one could better choose the spherical coordinate system
and express the unit vectors in the angular coordinates φ and θ,
where φ is the angle of the vector with the positive Z axis, and
θ the angle of the projection of the vector on to the X,Y plane
with the positive X axis.

Latest revision as of 18:02, 27 March 2014

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