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Quaternions and spatial rotation 136
Performance comparisons with other rotation methods
This section discusses the performance implications of using quaternions versus other methods (axis/angle or
rotation matrices) to perform rotations in 3D.
Results
Storage requirements
Method Storage
Rotation matrix 9
Quaternion 4
Angle/axis 3*
* Note: angle-axis can be stored as 3 elements by multiplying the unit rotation axis by the rotation angle, forming the
logarithm of the quaternion, at the cost of additional calculations.
Used methods
Performance comparison of rotation chaining operations
Method # multiplies # add/subtracts total operations
Rotation matrices 27 18 45
Quaternions 16 12 28
Performance comparison of vector rotating operations
Method # multiplies # add/subtracts # sin/cos total operations
Rotation matrix 9 6 0 15
Quaternions 18 12 0 30
Angle/axis 23 16 2 41
There are three basic approaches to rotating a vector :
1. Compute the matrix product of a 3x3 rotation matrix R and the original 3x1 column matrix representing . This
requires 3*(3 multiplications + 2 additions) = 9 multiplications and 6 additions, the most efficient method for
rotating a vector.
2. Using the quaternion-vector rotation formula derived above of , the rotated vector can be
evaluated directly via two quaternion products from the definition. However, the number of multiply/add
operations can be minimised by expanding both quaternion products of into vector operations by twice
applying . Further applying a number of
vector identities yields which requires only 18 multiplies and 12 additions
to evaluate. As a second approach, the quaternion could first be converted to its equivalent angle/axis
representation then the angle/axis representation used to rotate the vector. However, this is both less efficient and
less numerically stable when the quaternion nears the no-rotation point.
3. Use the angle-axis formula to convert an angle/axis to a rotation matrix R then multiplying with a vector.
Converting the angle/axis to R using common subexpression elimination costs 14 multiplies, 2 function calls (sin,
cos), and 10 add/subtracts; from item 1, rotating using R adds an additional 9 multiplications and 6 additions for a
total of 23 multiplies, 16 add/subtracts, and 2 function calls (sin, cos).