Quaternions
(continued from
rotation)
To set the elements of the rotation
type directly, compute the
quaternion as follows. To represent a rotation of a given angle around a given
axis, first, convert the axis to a
unit vector (a vector of
magnitude 1). Take the
x,
y, and
z components of the unit vector and multiply them by the
sine of half the angle (these are the
x,
y, and
z elements of the
quaternion). The
s element of the
quaternion is equal to the
cosine of half the angle.
The following
function performs this in
LSL:
rotation make_quaternion( vector axis, float angle )
{
vector unit_axis = llVecNorm( axis );
float sine_half_angle = llSin( angle/2 );
float cosine_half_angle = llCos( angle/2 );
rotation quat;
quat.x = sine_half_angle * unit_axis.x;
quat.y = sine_half_angle * unit_axis.y;
quat.z = sine_half_angle * unit_axis.z;
quat.s = cosine_half_angle;
return quat;
}
This example duplicates the function
llAxisAngle2Rot. Note that the angle must be specified in
radians rather than
degrees.
For a rotation around just the x-axis, the
quaternion becomes simpler to compute, because in this case the unit axis is <1,0,0>:
rotation x_rot = <llSin( angle/2 ), 0, 0, llCos( angle/2 )>; // rotate around x-axis
Similarly, a rotation around just the y-axis or just the z-axis is also simpler to compute:
rotation y_rot = <0, llSin( angle/2 ), 0, llCos( angle/2 )>; // rotate around y-axis
rotation z_rot = <0, 0, llSin( angle/2 ), llCos( angle/2 )>; // rotate around z-axis
A rotation of angle 0 around any axis is represented the same way. Consider that the sine of 0 is 0, while the cosine of 0 is 1. As a result, no matter what axis is chosen, the
null rotation is equal to
<0,0,0,1>.
To find the inverse of a rotation, negate the last element of the
quaternion:
inverse_quat = quat;
inverse_quat.s = -quat.s;
To find the inverse, negate the first 3 elements:
inverse_quat = < -quat.x, -quat.y, -quat.z, quat.s >;
If you negate all 4 elements of a
quaternion, you end up with the same effective rotation.
"Quaternions are the things that scare all manner of mice and men. They are the things that go bump in the night. They are the reason your math teacher gave you an F. They are all that you have come to fear, and more. Quaternions are your worst nightmare." -- Confuted
This is so inappropriate. I understand why folks think this is funny, but come on. Why scare foks off? Quanternions aren't out of reach for most of us. Just takes a little patience. Note "Confuted" goes on to say "Okay, not really. They aren't actually that hard. I just wanted to scare you. "
Additional Resources:
Mathworld
Delphi3D
GameDev.net
Programming Rotations with Quaternions
Michel Isner's guide to Quaternions - arguably the best and clearest tutorial on rotation representations.
Rotation |
Types