GPGPU Programming for Games and Science

312 GPGPU Programming for Games and Science

}

}

else

{

if (r11 >=r22) // u1 ˆ2 l ar g es t −magnitude

{

U = (( r10 + r01 )/2 , r11 + 1 , ( r12 + r21 )/2 );

}

else // u2 ˆ2 la rg es t −magnitude

{

U = (( r20 + r02 )/2 , ( r21 + r12 )/2 , r22 + 1);

}

}

Normalize(U);

LISTING 6.35: Determining the largest-magnitude component of U from

the products of components.

Numerical bias is avoided by averaging the off-diagonal terms, ensuring the

results are the same for R or R

T

. It does not matter that the normalization

computes a positive entry in U corresponding to the specially computed diag-

onal entry. For a rotation by π radians, R(U,π)andR(−U,π)arethesame

rotation matrix.

6.3.6.5 Axis-Angle to Quaternion

This conversion is simple. The quaternion angle is half the rotation ...