Rotation matrix derived from quaternion is opposite of expected direction

Question

When I calculate a rotation matrix from a quaternion, it seems to be in the opposite direction. For instance:

For a rotation of +45 degrees about the Y-axis, I would expect to get the following matrix:

# Pesudo Code
m1 = Matrix()
m1.RotateEulerY(45)
m1 = 
| 0.71  0.00 0.71 0.00  |
| 0.00  1.00 0.00 0.00  |
| -0.71 0.00 0.71 0.00  |
| 0.00  0.00 0.00 1.00  |   

However the rotation matrix calculated from a quaternion with the same rotation is

# Pesudo Code
axis_angle = AxisAngle((0,1,0), 45)
quat = Quaternion(axis_angle)
quat = { 0.0,0.375,0.0,0.927 }

rotate_matrix = quat.ToRotationMatrix() 
rotate_matrix = 
| 0.72 0.00 -0.69 0.00 |
| 0.00 1.00 0.00  0.00 |
| 0.69 0.00 0.72  0.00 |
| 0.00 0.00 0.00  1.00 |

Which happens to be a rotation of -45 about the Y-axis. I have tried to get the rotation matrix from the angle axis and it the same as the expected result.

Is this the expected behaviour or have I made a mistake/assumption somewhere? (Yes, the answers are not exactly the same; for now I am assuming it is a rounding error since the answers are so close).

Background info: I am using column major matrix, i.e. OpenGL style. The algorithms used to calculate the rotation matrix from the quaternion comes from http://www.j3d.org/matrix_faq/matrfaq_latest.html, Q54.

UPDATE #1

As teodron suggested, , I have tried to do a Quaternion(q) to Rotation Matrix(M) and back to Quaternion combination (i.e. q->M->q'). I also did a M->q->M' conversion for sanity check.

While q == q' and M == M', the conversion between q and M is wrong!

The rotation used in the example is the same as before, a rotation of +45 around the Y-axis. The matrix is right hand orientated, using column major vector, a.k.a openGL conformance. Matrix and Quaternion format is shown below:

#       | 0 4 8  12 |
# M =   | 1 5 9  13 |
#       | 2 6 10 14 |
#       | 3 7 11 15 |
#
# q =   {i,j,k,w}

First convert M to q then back to q.

#M->q->M'

# I am confident M is the correct representation for the rotation. 
M =
| 0.71 0.00 0.71 0.00 |
| 0.00 1.00 0.00 0.00 |
| -0.71 0.00 0.71 0.00 |
| 0.00 0.00 0.00 1.00 |

# Here I would expect the quaternion to be all positive, but the
# j component is negative!

q =  
{ 0.00, -0.38,0.00, 0.92}

# M' == M
M' = 
| 0.71 0.00 0.71 0.00 |
| -0.00 1.00 0.00 0.00 |
| -0.71 -0.00 0.71 0.00 |
| 0.00 0.00 0.00 1.00 |

second part: from q to M to q'

#q->M->q'

# I have created the quaternion from both an axis angle and euler angle
# and both approaches give me the same answer
q = 
{ 0.00, 0.37,0.00, 0.93}

# Here M is different from the Matrix in the first part
M = 
| 0.72 0.00 -0.69 0.00 |
| 0.00 1.00 0.00 0.00 |
| 0.69 0.00 0.72 0.00 |
| 0.00 0.00 0.00 1.00 |

# but q' == q!
q' = 
{ 0.00, 0.37,0.00, 0.93}

Since the q<->M is consistently wrong, I thought I would focus on q->M for now.

I had check the formula to convert a quaternion to matrix across different sources and they are all consistent with one another. Basically the normal maths style, row-major matrix is as follows: (taken from http://www.euclideanspace.com/maths/geometry/rotations/conversions/quaternionToMatrix/index.htm)

| 1 - 2(kk + jj)    2(ij - kw)       2(jw + ik)    |
| 2(ij + kw)        1 - 2(ii + kk)   2(jk - iw)    |
| 2(ik - jw)        2(jk + iw)       1- 2(jj + ii) |

Hence the openGL column major matrix equivalent should be

| 1 - 2(kk + jj)    2(ij + kw)      2(ik - jw)      |
| 2(ij - kw)        1 - 2(ii + kk)  2(jk + iw)      |
| 2(jw + ik)        2(jk - iw)      1- 2(jj + ii)   |

Comparing this with the column major rotation matrix as generated via a rotation of angle A around Y axis.

| cos(A)    0   sinA    |
| 0         1   0       |
| -sinA     0   cosA    |

Now the problem is more obvious to me: Looking at M[2] for both matrices, the quaternion matrix ,(2(jw + ik)), will produce a positive while the rotation matrix, -sinA, will produce a negative value!

I am at a loss here; I cannot think of a reason why these two matrices differ and I don't know what should I investigate next.Any ideas?

UPDATE #2

This is an update to explain the solution to the problem. Basically it boils down to how I ordered the elements in the matrices. For my case I did not need to transpose the original quaternion to matrix formula; some websites do perform the transpose because they have ordered it differently.

Answer 1

You're getting the transpose of the matrix you wanted, so you probably just have a row-vector vs. column-vector issue; that is, you're using row vectors and the source where you found the quat-to-matrix conversion formula was using column vectors, or vice versa.

Answer 2

Quaternions, when converting to rotation matrices, can be treated as either left-handed or right handed. One rotates in one direction, and obviously the other in the opposite direction. If you mix formulas from different sources you will often find the results don't always line up -- though one would hope if using only equations from one source it'd be consistent.

So in short, this is a typical issue that arises. It is a matter of interpretation and neither result is more correct, just perhaps not consistent in your system. Either take the time to make all your formulas line up (which is often non-trivial), or simply negate the angle if you want something quick.