Rotation in Three Dimensions - 3D Math Primer for Graphics and Game Development - page 274

• Rotating points between coordinate spaces (object and upright)

◦ Matrix : Possible; can often by highly optimized by SIMD instructions.

◦ Euler Angles : Impossible (must convert to rotation matrix).

◦ Exponential Map : Impossible (must convert to rotation matrix).

◦ Quaternion : On a chalkboard, yes. Practically, in a computer, not really. You might as well

convert to rotation matrix.

• Concatenation of multiple rotations

◦ Matrix : Possible; can often be highly optimized by SIMD instructions. Watch out for matrix
creep.
◦ Euler Angles : Impossible.
◦ Exponential Map : Impossible.
◦ Quaternion : Possible. Fewer scalar operations than matrix multiplication, but maybe not as

easy to take advantage of SIMD instructions. Watch out for error creep.

• Inversion of rotations

◦ Matrix : Easy and fast, using matrix transpose.

◦ Euler Angles : Not easy.

◦ Exponential Map : Easy and fast, using vector negation.

◦ Quaternion : Easy and fast, using quaternion conjugate.

• Interpolation

◦ Matrix : Extremely problematic.

◦ Euler Angles : Possible, but Gimbal lock causes quirkiness.

◦ Exponential Map : Possible, with some singularities, but not as troublesome as Euler angles.

◦ Quaternion : Slerp provides smooth interpolation.

• Direct human interpretation

◦ Matrix : Difficult.

◦ Euler Angles : Easiest.

◦ Exponential Map : Very difficult.

◦ Quaternion : Very difficult.

• Storage efficiency in memory or in a file

◦ Matrix : Nine numbers.

◦ Euler Angles : Three numbers that can be easily quantized.

◦ Exponential Map : Three numbers that can be easily quantized.

◦ Quaternion : Four numbers that do not quantize well; can be reduced to three by assuming

fourth component is always nonnegative and quaternion has unit length.

•

Unique representation for a given rotation

◦ Matrix : Yes.

◦ Euler Angles : No, due to aliasing.

◦ Exponential Map : No, due to aliasing, but not as complicated as Euler angles.

◦ Quaternion : Exactly two distinct representations for any angular displacement, and they are

negatives of each other.

• Possible to become invalid

◦ Matrix : Six degrees of redundancy inherent in orthogonal matrix. Matrix creep can occur.

◦ Euler Angles : Any three numbers can be interpreted unambiguously.

◦ Exponential Map : Any three numbers can be interpreted unambiguously.

◦ Quaternion : Error creep can occur.

Table 8.1. Comparison of matrices, Euler angles, exponential maps, and quaternions