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Quaternions

What is it?

The quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that the product of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors. Quaternions can also be represented as the sum of a scalar and a vector (wikipedia).

Any rotation in three dimensions can be represented as a combination of an axis vector and an angle of rotation. Quaternions give a simple way to encode this axis-angle representation in four numbers and apply the corresponding rotation to a position vector representing a point relative to the origin in R3(wikipedia).




A quaternion represents two things with four numbers:
  1. an axis of rotation
  2. an amount of rotation about the axis
With these four numbers, it is possible to build a matrix which will represent all the rotations. Of the four numbers, three have an imaginary component. i is defined as √-1. With quaternions, i = j = k = √-1. The quaternion itself is defined as q = w + xi + yj + zk. Where w, x, y, and z are all real numbers. w is the amount of rotation about the axis defined by [x, y, z]. Normalizing a quaternion isn't much harder than normalizing a vector. The magnitude of a quaternion is given by the formula 
magnitude = sqrt(w2 + x2 + y2 + z2)
The magnitude is one for the unit quaternion. If you have to normalize a quaternion: 
magnitude = sqrt(w2 + x2 + y2 + z2)
w = w / magnitude
x = x /  magnitude
y = y / magnitude
z = z / magnitude



Performing the above operations ensures that the quaternion is a unit quaternion.   There are many different possible unit quaternions - they actually describe a hyper-sphere, or a four dimensional sphere.   But because the end points for unit quaternions all lay on a hyper-sphere, multiplying one unit quaternion by another unit quaternion will result in a third unit quaternion.  



Quaternion Multiplication

          
One of the most important operations with a quaternion is multiplication.  When you have two quaternions, and you multiply them with each other, the rotations are merged into a single rotation. If you want to apply successive rotations to a model, you multiply all the respective quaternions representing these rotations with each other, and end up with a single quaternion.




Here is how the multiplication itself is performed:




Let Q1 and Q2 be two quaternions, which are defined, respectively, as (w1, x1, y1, z1) and (w2, x2, y2, z2).

(Q1 * Q2).w = (w1w2 - x1x2 - y1y2 - z1z2)
(Q1 * Q2).x = (w1x2 + x1w2 + y1z2 - z1y2)
(Q1 * Q2).y = (w1y2 - x1z2 + y1w2 + z1x2)
(Q1 * Q2).z = (w1z2 + x1y - y1x2 + z1w2


Quaternion multiplication is not commutative.  



What does the quaternion multiplication mean?  A quaternion stores an axis and the amount of rotation about that axis.  To change the rotation represented by a quaternion, a few steps are necessary.  First, you must generate a temporary quaternion, which will simply represent how you're changing the rotation.  If you're changing the current rotation by rotating backwards over the X-axis a little bit, this temporary quaternion will represent that.  By multiplying the two quaternions (the temporary and permanent quaternions) together, you will get a new permanent quaternion, which has been changed by the rotation described in the temporary quaternion.  



// generate a temp_rotation 
total = temp_rotation * total  //multiplication order matters on this line


To generate the temp_rotation, you'll need to have the axis and angle prepared, and this will convert them to a quaternion.  Here's the formula for generating the temp_rotation quaternion.


//axis is a unit vector
temp_rotation.w  = cosf( fAngle/2)
temp_rotation.x = axis.x * sinf( fAngle/2 )
temp_rotation.y = axis.y * sinf( fAngle/2 )
temp_rotation.z = axis.z * sinf( fAngle/2 )



Then, multiply temp_rotation by total.  Since you'll be multiplying two unit quaternions together, the result will be a unit quaternion.  You won't need to normalize it. 



Generate the matrix from the quaternion to rotate your points. 

 






w2+x2-y2-z2
       
      2xy-2wz
       
      2xz+2wy
       
      0
       
    



2xy+2wz
       
      w2-x2+y2-z2
       
      2yz+2wx
       
      0
       
    



2xz-2wy
       
      2yz-2wx
       
      w2-x2-y2+z2
       
      0
       
    



0
       
      0
       
      0
       
      1
       
    







And, since we're only dealing with unit quaternions, that matrix can be optimized a bit down to this:





1-2y2-2z2
       
      2xy-2wz
       
      2xz+2wy
       
      0
       
    



2xy+2wz
       
      1-2x2-2z2
       
      2yz+2wx
       
      0
       
    



2xz-2wy
       
      2yz-2wx
       
      1-2x2-2y2
       
      0
       
    



0
       
      0
       
      0
       
      1
       
    









For more information about quaternions:







 member WilliamAAdams created code to add quaternions to OpenSCAD. Using Adams's code will allow you to do the following:

myquat = quat(axis, angle);


To do a 30 degree rotation around the z-axis for example:

rotz = quat([0,0,1], 30);


You need to turn it into a matrix that OpenSCAD can easily use and apply as a transform:

rotzmatrix = quat_to_mat4(rotz);


Once you have this, you can use it with multmatrix()

multmatrix(rotzmatrix);



multmatrix(quat_to_mat4(quat([0,0,1],30)));


Combinations can be done like this:

q1 = quat([0,0,1],30);
q2 = quat([0,1,0], 15);
combo = quat_mult(q1, q2);

multmatrix(quat_to_mat4(combo));


This  combines the 30 degree rotation around the z-axis, followed by the 15 degree rotation about the y-axis, or visa versa.







Instructions


  1. Download maths.scad and test_maths.scad
    2. 

  2.  Play around with the test_maths.scad file to see how to use it 
    3. 



According to WilliamAAdams, "Quaternions are a big scary name. If you just change the name to rotation thingy life gets a lot easier."