Rotations in Three Dimensions

Date: 11/11/1999 at 21:11:14
From: Drew
Subject: Quaternions and 3D rotations

Hello:

I am trying to learn about quaternions and 3D rotations. I am trying 
to build my own 3D engine and ran into the classic problem concerning 
Euler angles. I understand the basics for quaternions, but don't 
understand how to use them to rotate something in 3D. I have read many 
books about it but they are all too complex. What if I want to rotate 
something (30 degrees about x, 20 about new y, -36 about new z)? None 
of the sources I have found are simple enough. Additionally, could you 
tell me how to rotate something about any general line in 3D? I know 
they are both connected, I just don't know how to do it.

Thanks a lot,
Drew

Date: 11/12/1999 at 08:27:56
From: Doctor Mitteldorf
Subject: Re: Quaternions and 3D rotations

Dear Drew,

I don't know enough about quaternions to tell you how to relate them 
to rotation matrices. I do know about rotation matrices, and I don't 
think they have to be very hard.

To rotate an angle a about the z axis, apply the matrix:

     [cos(a)  -sin(a)  0]
     [sin(a)   cos(a)  0]
     [  0       0      1]

From this, I am sure you can construct similar matrices for rotating 
about x and y.

To do two rotations in a row, multiply the second matrix on the left 
by the first matrix on the right. This should answer your question 
about rotating something about multiple axes. Just multiply the 3 
matrices together, with the last of the three operations on the left.

To rotate about a general line in 3D, here's your strategy: First, 
construct a rotation about z so that your line has no more y in it - 
it is purely x and z. Call this matrix A. Then rotate about the new 
y-axis until the new z axis is right along your line. Call this matrix 
B.  

Multiply BA = C, and calculate the inverse C^-1 = A^-1 * B^-1.

Now you have a matrix C that moves your z-axis to the line about which 
you wish to rotate. C^-1 * C clearly does nothing at all. But suppose 
that you sandwich another rotation in between: C^-1 * D * C, where D 
is a rotation about the z-axis. Now you have a matrix that first moves 
the z-axis to the line you've specified, then rotates about that line, 
then puts the z-axis back where it would have been.

This is the matrix you want: C^-1 * D * C is the rotation about the 
given line.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/   

Date: 12/2/2002
From: Jill
Subject: Re: Quaternions and 3D rotations

The preceding answer avoids this specific use of the quaternion. 
I believe some reasons for that may be: it is not used in the 
classic manner, and as a result is not encountered by math 
enthusiasts very often. This specific use is not as bad as it seems, 
it's just uncommon, unfamiliar, and applied a little differently.

With Euler angles the "Gimbal lock" problem is that certain
orientations are difficult or impossible to attain. And around these 
orientations any kind of precision is just as difficult. In some 
homebrew software you may have noticed the "top-over flip" effect, 
which is that as you pass over the vertical axis, the object or view 
suddenly flips by Pi degrees. The quaternion has 4 parts, basically 
three imaginary numbers and a scalar. Using the quaternion to 
overcome the "Gimbal lock" problem, the three imaginary numbers are 
used with the imaginary part simply removed [x,y,z]. As non-imaginary 
scalars they represent an arbitrary axis orientation in three 
dimensions, and the scalar [w] is used to represent a rotation around 
this axis. Note this also means there is more than one quaternion to
represent any particular orientation. This overcomes the "Gimbal lock" 
problem.  

There is more:

With these unique properties of the quaternion, it also makes the 
application and adding of rotational vectors much more intuitive for 
humans, and easier to code.

For example, when using Euler angles, if you move the joystick 
forward and backward you would expect a rotation around the x-axis, and 
if you move it left and right you would expect rotation around the 
z-axis, and similarly if you twist the handle of the joystick you would 
expect rotation around the y-axis. But if you move forward, then to the 
side, then twist, then undo each in the original order, you will see 
that [depending on which axis is calculated first] the last two angles 
calculated don't rotate around the expected axis, but around the 
projected axis after prior calculation(s). With the quaternion, the 
projection from the joystick can be more directly applied to the 
rotation of the object or view, and thus much more intuitive.

Another example is in physics. When applying your rotational velocity 
and/or rotational acceleration vectors, they can be added and applied 
much more directly and intuitively. It makes the calculations much 
more simple and thus much less complicated code, etc. When you grasp 
the concept, you may find, as I did, that you will want to use this 
form for all your 3D engine representations. However, I do have some 
transformation matrix algorithms available for Euler-quaternion 
conversion.