# Transformation matrix

## Background

Transformation matrix class is an affine homogeneous class that can be used to rotate objects like images and VOIs. It can constructed as a 2D(3x3) or 3D(4x4) homogeneous matrix for transform (rotation, translation, skew and zoom) images/VOIs. Skew is not commonly used.

The MIPAV 3D model for 4 X 4 transformations is:

```        [ m00 m01 m02 m03 ]   [ x ]   [ x' ]
[ m10 m11 m12 m13 ] . [ y ] = [ y' ]
[ m20 m21 m22 m23 ]   [ z ]   [ z' ]
[ m30 m31 m32 m33 ]   [ w ]   [ w' ]

x' = m00*x + m01*y + m02*z + m03*w
y' = m10*x + m11*y + m12*z + m13*w
z' = m20*x + m21*y + m22*z + m23*w
w' = m30*x + m31*y + m32*z + m33*w

```

However, because the transform type is limited, we can always set the third row to [ 0 0 0 1] and write instead:

```        [ m00 m01 m02 ]   [ x ]   [ t0 ]   [ x' ]
[ m10 m11 m12 ] . [ y ] + [ t1 ] = [ y' ]
[ m20 m21 m22 ]   [ z ]   [ t2 ]   [ z' ]

x' = m00*x + m01*y + m02*z + t0
y' = m10*x + m11*y + m12*z + t1
z' = m20*x + m21*y + m22*z + t2

```

We still represent 4x4 m with a WildMagic Matrix4f, because of the existing implementations that assume a 4x4 matrix, and the use of the upper 3x3 matrix as a 2D transform.

We considered representing m with a WildMagic Matrix3f, and t with a Vector3f, as encapsulated in the WildMagic Transformation class, but it does not match typical Mipav usage.

ORDER OF TRANSFORMATIONS = TRANSLATE, ROTATE, ZOOM or TRANSLATE, ROTATE, SKEW, ZOOM Row, Col format - right hand rule 2D Example

```         zoom_x    theta    tx
theta     zoom_y   ty
0         0         1
represented in 3D by leaving Z the identity transform:
zoom_x    theta    0    tx
theta     zoom_y   0    ty
0         0        1    0
0         0        0    1
```

Note that for 2D, the tx and ty components are stored in M02 and M12, and not in M03 and M13, as might be guessed by the use of a Matrix4f to store both 2D and 3D transforms.

Note for 3D - ref. Foley, Van Dam p. 214

```         Axis of rotation            Direction of positive rotation is
x                               y to z
y                               z to x
z                               x to y

```
```         Order of rotation is important (i.e. not commutative)
Rx Ry Ry != Ry Rx Rz
```