Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis.

A unit quaternion is a quaternion of norm one. Dividing a nonzero quaternion q by its norm produces a unit quaternion Uq called the versor of q: = ‖ ‖.

Quaternions are very efficient for analyzing situations where rotations in R3 are involved. A quaternion is a 4-tuple, which is a more concise representation than a rotation matrix. Its geo-metric meaning is also more obvious as the rotation axis and angle can be trivially recovered.

This paper provides a basic introduction to the use of quaternions in 3D rotation applications. We give a simple definition of quaternions, and show how to convert back and forth between quaternions, axis-angle representations, Euler angles, and rotation matrices.

We use quaternions with zero “real’’ part to represent vectors. So the vector r is represented by ˚r =(0,r). Consider the transformation of r to r performed by ˚r =q˚˚r˚q∗ where ˚r is a “purely imaginary’’ quaternion (i.e. ˚r = (0,r)) and ˚q is a unit quaternion (i.e. ˚q ·q˚ =1).

where ˚r is a “purely imaginary” quaternion (i.e. ˚r =(0, r)) and q˚ is a unit quaternion (i.e. q˚ ˚ ·q =1). Applying the above rule for multiplication of quaternions twice we find first that the “real” part of the result is zero, so that we can write ˚r =(0, r), and second r …

But there are many more unit quaternions than these! i, j, and k are just three special unit imaginary quaternions. Take any unit imaginary quaternion, u = u1i + u2j + u3k. That is, any unit vector.

2021年10月10日 · The unit quaternions, denoted \(U(\mathbb{H})\), is the set of quaternions with modulus 1. 1 Analogous to complex numbers, quaternions can be expressed in polar form .

Unit quaternions make it easy to compose rotations (unlike, e.g., axis-and-angle notation). Unit quaternions do not suffer from singularities (as do, e.g., Euler angles when two axes line up – see gimbal lock). Unit quaternions, while redundant (four parameters for three degrees of freedom), have only one constraint on their components

To define the quaternions, we first introduce the symbols i, j, k. These sym-bols satisfy the following properties: ki = j. kx = xk. You can work out other rules from these properties. For example, suppose you want to compute the mystery symbol T = ji. Note that. T i = jii = j(−1) = (−1)j = −j = −ki. Cancelling the i gives T = −k.