# What is displaystyle1 sin theta    Next:Calculation of the optimal rotationUp:The quaternion for representation Previous:Derivation of the formula for

### Calculation of the unit quaternion

A vector supposed to be a vector around the angle be rotated. This rotation is made possible by the unit quaternion With and described. One way of finding the point to calculate is the method presented in the previous section, the quaternion of the vector With and to be multiplied (cf. (A.11)):

The quaternion multiplication (cf. (A.4)) is then applied and the result is:

Now, using some facts about vectors (that , and is) and includes the following trigonometric identities:

we get for (A.12):

The formula (A.13) is now proven by a graphical derivation. The left side of Figure A.1 shows a vector that around the angle is rotated. Be it a unit vector. The vectors and span a level. The vectors are on this plane and . The vector is perpendicular to this plane and has the same length as vector . The vector can be calculated by , there is a unit vector and is.  This allows the vector write as

As you can now see, the formula (A.14) corresponds exactly to the quaternion (A.13). This shows that the given description of the rotation by means of the unit quaternion (3.2), (3.3), (3.4), (3.5) actually represents a rotation around a given vector.    Next:Calculation of the optimal rotationUp:The quaternion for representation Previous:Derivation of the formula forAndreas Nüchter
2002-07-10