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Alan,
Ok thanks, this does help, I just assumed that in GA terms it would be a simple multiplication like:
L = [I]w
which maps bivectors to bivectors and I could not work out how any multiplication apart from just a scalar could map bivectors to bivectors?
I'm now starting to see that the inertia tensor in GA terms is a function:
I(Omega) = sum d^3 xpx ^ (x.Omega)
which maps...
08:23AM UTC on Jun 24 2009 in mjbWorld
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> So 'Omega' is what you were describing on the page, the
> vector representing the angular velocity put in a quaternion,
> and 'omega' is the same but a reversed velocity.
I think that, for practical purposes, that would be my interpretation. If you are interested in the theory then in Geometric Algebra equations like Omega = I*omega represent a powerful duality between different...
08:13AM UTC on Jun 23 2009 in mjbWorld
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I think, if I can try and summerise, there are two possible approaches:
1) If we know the angular velocity as a function of time we can get the orientation as a function of time.
2) If the steps are very small then dq/dt could be added and renormalised for each frame.
Since the angular velocity is constant in this case then it is much more accurate to use option 1.
So the quaternion...
06:15PM UTC on Jun 22 2009 in mjbWorld
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Hi Diggory,
I think that, in a program, it is usually best to represent angular velocities by ordinary 3D vectors, if you plan to do physics a possible alternative is to use angular momentum, but either way you only need 3D vectors.
This is because finite rotations are combined using quaternion multiplication but angular velocities (or angular momentum) are combined using vector addition.
09:34AM UTC on Jun 22 2009 in mjbWorld
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Hi Dan,
The code for Quaternion to AxisAngle is on the page above the page you mention:
http://www.euclideanspace.com/maths/geometry/rotations/conversions/quaternionToAngle/index.htm
I think is better to get the code from here since the applet has all sorts of other stuff mixed in with it.
Martin.
07:35AM UTC on Jun 21 2009 in mjbWorld
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Ethan,
I have done what you suggested and updated the page.
I have worked through some examples and convinced myself that what you suggest is better.
When I get some time, I think I will return to this and see if I can fill in some of the gaps in the theory, especially on what assumptions we can make about an orthogonal matrix. Its a long time since the page was originally written and...
07:26AM UTC on Jun 21 2009 in mjbWorld
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Yes, I do like the approach of using the largest diagonal like this:
float tr1 = 1.0 + m00 - m11 - m22
float tr2 = 1.0 - m00 + m11 - m22
float tr3 = 1.0 - m00 - m11 + m22
if (tr1 > tr2)&(tr1 > tr3)) {
float S = sqrt(tr1) * 2; // S=4*qx
qw = (m21 - m12) / S;
qx = 0.25 * S;
qy = (m01 + m10) / S;
qz = (m02 + m20) / S;
} else if (tr2 > tr1)&(tr2 >...
07:34AM UTC on Jun 20 2009 in mjbWorld
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Hi Ethan,
Thanks, yes I agree with what you say, I guess there are two issues:
1) danger of division by zero or square root of a negative number.
if m11 and m22 are very close to m00 it may pass the (m00 > m11)&(m00 > m22) test but the square root may still give zero or negative due to floating point errors. I don't know how likely this is but its better to keep clear.
2...
08:51AM UTC on Jun 18 2009 in mjbWorld
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Hi Ben,
I am hoping that Noel Hughes sees this and replies as I don't have any practical experience of accelerometers and I'm sure will know much more about that than me.
The equations:
real = cos (θ/2)
imaginary i = x / sin(θ/2)
imaginary j = y / sin(θ/2)
imaginary k = z / sin(θ/2)
convert an axis-angle representation of rotation to a quaternion representation of rotation...
06:13PM UTC on Jun 16 2009 in mjbWorld
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I think I have got them all now, but those minus signs can be hard to see, so please let me know if you notice any more.
I am keen to improve the web pages so it doesn't bug me, I appreciate your help.
Cheers,
Martin.
07:10AM UTC on May 12 2009 in mjbWorld
