I'm in the process of reading up on 3d matrices and trying to following this powerpoint and had a few questions as to how to exactly determine the type of transformation from a matrix.

If I have a matrix such as...

a 0 0
0 b 0
0 0 1


1 0 a
0 1 b
0 0 1

...would these be considered a translation by (a, b) vector?

Or a matrix such as...

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

... would this be considered rotation about Z-axis through the angle a?

Is there a better way to visualize these?

  • \$\begingroup\$ Do you want to represent linear transformations in 3D or affine transformations in 2D? If linear in 3D, there is no translation! if affine in 2D, there is no Z-axis! \$\endgroup\$ – Ivan Kuckir Mar 29 '13 at 13:57

The best way to gain intuition about how a matrix behaves is by determining its effect on the standard basis vectors:

     1        0        0
e1 = 0   e2 = 1   e3 = 0
     0        0        1

Since any 3D vector can be written as a combination of a*e1 + b*e2 + c*e3, if we know how a matrix changes these three vectors, we know how a matrix changes any vector.

So, to start off, let's examine the first matrix you have:

    a 0 0
M = 0 b 0
    0 0 1

We can see that

         a              0              0
e1 * M = 0     e2 * M = b     e3 * M = 0
         0              0              1

Hence, we see that e1 gets scaled by a, e2 gets scaled by b, and e3 stays the same. Since we don't know whether or not a equals b, or even if they are equal to one, this is a skew matrix.

If we perform a similar analysis to the matrix

    1 0 a
M = 0 1 b
    0 0 1

we see that e1 and e2 do not change, but e3 changes drastically. This means that every vector in the XY plane will remain the same, but every vector that has a nonzero z coordinate will become distorted! So, to answer your question, neither of these are translation matrices.

As your intuition might tell you by this point, you cannot translate a 3D vector using only 3x3 matrices. For that you need to use homogeneous coordinates.

I would suggest you go through these steps with the last matrix you have written. Spoiler alert: it is indeed a rotation about the z-axis.

  • 2
    \$\begingroup\$ The second one could be a 2D translation in homogeneous cords, an affine transformation. \$\endgroup\$ – luser droog Mar 29 '13 at 5:44

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