Transformations - Transformations and Projections in Computer Graphics - page 49

1.4.6 Concatenating Rotations
Sometimes, we have to perform two consecutive rotations on an object. This turns out
to be easy and numerically stable with a quaternion representation.
If q 1 and q 2 are unit quaternions representing the two rotations, then associativity
of quaternion multiplication implies that the combined rotation of q 1 followed by q 2 is
represented by the quaternion q 2 ·

q 1 . The proof is

( q 1 · P · q − 1 · q − 1

2

( q − 1
1

· q − 1
2

( q 2 · q 1 ) − 1 . q 2 · ) =( q 2 · q 1 ) · P · )=( q 2 · q 1 ) · P ·

1

Quaternion multiplication involves fewer operations than matrix multiplication, so
combining rotations by means of quaternions is faster. Performing fewer multiplications
also implies better numerical accuracy.
In general, we use 4

4 transformation matrices to express three-dimensional trans-
formations, so we would like to be able to express the rotation P ∗ = q [0 , P ] q − 1 as P ∗ =
PM ,where M is a 4

×

4 matrix. Given the two quaternions q 1 = w 1 + x 1 i + y 1 j + z 1 k =
( w 1 ,x 1 ,y 1 ,z 1 )and q 2 = w 2 + x 2 i + y 2 j + z 2 k =( w 2 ,x 2 ,y 2 ,z 2 ), their product is

×

q 1 · q 2 =( w 1 w 2 − x 1 x 2 − y 1 y 2 − z 1 z 2 )+( w 1 x 2 + x 1 w 2 + y 1 z 2 − z 1 y 2 ) i

+( w 1 y 2 − x 1 z 2 + y 1 w 2 + z 1 x 2 ) j +( w 1 z 2 + x 1 y 2 − y 1 x 2 + z 1 w 2 ) k .

The first step is to realize that each term in this product depends linearly on the
coe cients of q 1 . This product can therefore be expressed as

⎛ ⎞

w 1 z 1 − y 1 − x 1

− z 1 w 1 x 1 − y 1

⎝ ⎠

q 1 · q 2 = q 2 · L ( q 1 )=( x 2 ,y 2 ,z 2 ,w 2 ) .

y 1 − x 1 w 1 − z 1

x 1 y 1 z 1 w 1

When L ( q 1 ) multiplies the row vector q 2 , the result is a row vector representation for
q 1 · q 2 . Each term also depends linearly on the coe cients of q 2 , so the same product

can also be expressed as

⎛ ⎞

w 2 − z 2 y 2 − x 2

z 2 w 2 − x 2 − y 2

⎝ ⎠

q 1 · q 2 = q 1 · R ( q 2 )=( x 1 ,y 1 ,z 1 ,w 1 ) .

− y 2 x 2 w 2 − z 2

x 2 y 2 z 2 w 2

When R ( q 2 ) multiplies the row vector q 1 , the result is also a row vector representation
for q 1 · q 2 .

We can now write the triple product q q − 1

· [0 , P ] · in terms of the matrices L ( q )

and R ( q ):

q [0 , P ] q − 1 = q ([0 , P ] q − 1 )= q ([0 , P ] R ( q − 1 ))

=([0 , P ] R ( q − 1 )) L ( q )=[0 , P ]( R ( q − 1 ) L ( q ))
=[0 , P ] M ,

·