



Quaterniones
Quaternions are similar to complex and hyper complex numbers, but their multiplication is not cummutative They have 3 non real parts, that are marked here with the letters j, k and l. The real part will not have an own letter here.

Description of rotation
Rotations with quaternions
A point P=(x,y,z) is represented with the quaternion p = ix + jy + kz . A rotation of the angle around the xaxis is represented with the quaternion .
The rotated point is .

Conversion of quaternion to rotation matrix

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Conversion of rotation matrix to quaternion
Trigonometric functions of quaternions
At the following procedure in a 1st step the quaternions will be converted to complex numbers. Then in a 2nd step the complex funktion is executed. In a 3rd step the complex result will be converted to a quaternion using the old quaternion. Because I didnot find a publication with a proper description about this topic, I don’t know whether this is correct and whether I can use this procedure for all function I used it. It would be nice to get a hint about this. (email: calc_3d@greuer.de)

Procedure
With the quaternion The function func(q) should be executed q can be seperated into the

real part


und den

imaginary part


With this a complex number is created. The function func(z) is executed with this complex number: Using the imaginary part of the former quaternion the result is created:


with


This procedure is used in Calc 3D for the following functions: sin; cos; tan; sinh; cosh; tanh; arcsin; arccos; arctan; arccot; arsinh; arcosh; artanh; arcoth; log; exp





