Introduction to Quaternion

There are two main types of rotation in graphics ： Euler rotation and angular rotation .

Euler once proved , Any rotation can be expressed as the rotation of an object about its three axes . So Euler's rotation is the object in a particular order , Rotation around its three axes in turn . But this rotation is about the second axis 90° You lose a degree of freedom , Cause universal joint lock .

Shaft angle type , As the name suggests, it rotates a certain angle around a certain axis , By quaternion sum Rodriguez The formula can express this rotation process .

The plural number can be expressed as a+bi, among i²=-1, The multiplication of complex numbers and the multiplication of matrices can be regarded as equivalent . Quaternions are similar to complex numbers but not complex numbers . Definition of quaternion a+bi+cj+dk, among i²=j²=k²=ijk=-1. Let vector V=（b,c,d）, Then quaternion can also be expressed as [a,V]. Quaternions are like matrices , Commutative law that does not satisfy multiplication .

For a vector V, Express it as v=[0,V], Make q=[cos(θ/2),sin(θ/2)U], among U For the axis of rotation ,θ Is the angle of rotation about the axis . be v'=qvq*,q* by q Conjugation of ,v' Namely V Around the U Shaft rotation θ After the results of the .

Another advantage of quaternion is that it can be easily interpolated , This produces a smoother continuous rotation . Interpolation methods are ：Lerp,NLerp,Slerp.