GPGPU Programming for Games and Science

Linear and Affine Algebra 307

Let the initial body axes be the orthonormal right-handed basis

{U

0

, U

1

, U

2

}. The vectors are unit-length, mutually perpendicular, and

U

0

× U

1

= U

2

. Let the rotation angles be θ

i

for 0 ≤ i ≤ 2 and define

s

i

=sinθ

i

and c

i

=cosθ

i

. The rotation matrix by an angle θ corresponding

to an axis with unit-length direction V is R(V,θ).

The illustration assumes the vector-on-the-right multiplication convention.

The first rotation is by angle θ

0

about the body axis U

0

; let the rotation

matrix be denoted R

0

= R(U

0

,θ

0

). The second rotation is by angle θ

1

about

the rotated body axis R

0

U

0

; the rotation matrix is R

1

= R(R

0

U

0

,θ

1

). The

third rotation is by angle θ

2

about the twice-rotated body axis R

1

R

0

U

2

;

the rotation matrix is R

2

= R(R

1

R

0

U

2

,θ

2

). The composition ...