Maths - Rotations using quaternions - message from minorlogic

By: minorlogic ( Michaele Norel )
RE: quternion from 2 vec  
2003-08-27 12:23

Found in GDA mailing list archives, very simple trip.
Here we get result, how create quaternion of the half rotation. We get that we can just add the 1 to w component.

The same we can get if we will lineary lerp from identity quaternion ( w== 1) to given by half.

0.5*q + 0.5*q_ident -> scale it by 2

we get the same result without the complicated math. That is so simple.

By: martinbaker ( Martin Baker )
RE: quternion from 2 vec  
2003-08-27 16:16

Hi minorlogic,

Can I just check what you are saying? As an example assume I am rotating 90' about the z-axis.
The would be represented by quat = k

Half that angle would be given by quat = cos(45') + k sin(45')
= 0.707 + k 0.707

Are you saying that I can derive this from the first equation by adding 1 to give
1 + k
and then normalising?

That's very interesting, do you know why this is so? Do you have a derivation? Can you think of a way to say, divide the angle into 3 parts?

Martin

By: minorlogic ( Michaele Norel )
RE: quternion from 2 vec  
2003-08-28 08:20

From dot and cross product we can get the quaternion of DOUBLE angle rotation

q( x*sin a, y*sin a, z*sin a, cos a);
but we need the quat of angle rotation
q( x*sin a/2, y*sin a/2, z*sin a/2, cos a/2);

So we need to get quaternion from double angle -> to angle.

This quaternion "lay between" identity quaternion
q(0,0,0,1 ) and given double angle quaternion. We can just lerp this two quaternions.

So as we find bisector of two unit vectors just adding tham, we add to double angle quaternion ( must be same magnitude ) identity quaternion and normalize it.

lets play with your example:
quaternion double_a_q( 0*sin(90), 0*sin(90), 1*sin(90), cos(90) )

double_a_q( 0, 0, 1, 0 )
identity_q( 0,0,0,1);

Add them :

result_q( 0,0,1,1);
normalize it :
mag = sqrt(0*0 + 0*0+ 1*1 +1*1) = sqrt(2)

result_q( 0,0,1/sqrt(2),1/sqrt(2));
result_q( 0,0,0.707,0.707);

And to derive this we don't need trigonometry.

By: martinbaker ( Martin Baker )
RE: quternion from 2 vec  
2003-08-28 17:49

Yes, I made a silly mistake in not dividing the angle by 2 when calculating the quaternion, thank you for correcting me.

I think I understand now, I have written this down at the following webpage:
https://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/transforms/halfAngle.htm

Is this alright?

Thanks,

Martin