# 3D Rotations using complex numbers

1. Oct 25, 2013

### Septimra

I was thinking that if you could use quaternions to rotate an object using quaternion algebra that there might be a way to rotate an object using complex numbers in some fashion. I was looking at quaternion rotation of a vector and the amount of operations seemed to be a lot. Of course it levels out or even out performs matrices with linear interpolation and concatenation of rotations. But anyhow what i decided to do was rotate a vector (x ,y , z) = x + yi + zj

with the rotation complex numbers
(a + bi)
(a + bj)

a = cos

b = sin

Following this algorithm
(a + bi)(x + yi) = x' + y'i
(a + bj)(y' + zj) = y'' + z'j

the new rotated vector is x' + y''i + z'j;
This works, it rotates the object and preserves the magnitude of the vector.

What I don't understand is how it is rotating the object.

When I put in 90 degrees v(1,0,0) goes to v(0,0,1) to (0,-1,0) back to (1,0,0)

When I put in 90 degrees v(1,1,1) goes to v(-1,-1,1) to (1, -1, -1) to (1,1,1)

My question is: Is it rotating around an axis? Can it be changed? Is it Euler angles in disguise?

Why is it the first time it rotates by 90 degrees when I put in 90?

Then it rotates 180 when I still have 90 degrees in?

2. Oct 25, 2013

### UltrafastPED

Sure ... complex numbers are natural for 2D rotations, just as quaternions are natural for 3D rotations.

But you can also do rotations with matrices; see SO(3).

The question which you have found, what is the axis of rotation, is to be found in the construction of the above systems.