Pivot Point Rotation

Pivot point rotation, also known as arbitrary axis rotation, is a type of rotation transformation in computer graphics that allows us to rotate an object around an arbitrary point in 3D space, rather than just around the origin.

To perform a pivot point rotation, we first need to translate the object so that the pivot point is at the origin. We then perform a standard rotation around one of the coordinate axes, such as the x-axis, y-axis, or z-axis, depending on the desired orientation of the rotation axis. Finally, we translate the object back to its original position.

Here are the steps to perform a pivot point rotation about a point `P = (px, py, pz)` by an angle `theta` around an axis defined by a vector `V = (vx, vy, vz)`:

1. Translate the object so that the pivot point is at the origin:

```

T1 = [ 1  0  0  -px ]

     [ 0  1  0  -py ]

     [ 0  0  1  -pz ]

     [ 0  0  0   1  ]

```

2. Compute a rotation matrix `R` that rotates around one of the coordinate axes, such as the x-axis, y-axis, or z-axis, depending on the desired orientation of the rotation axis. For example, if we want to rotate around the z-axis, we can use the following matrix:

```

Rz = [ cos(theta) -sin(theta)  0  0 ]

     [ sin(theta)  cos(theta)  0  0 ]

     [     0           0       1  0 ]

     [     0           0       0  1 ]

```

3. Compute a rotation matrix `R'` that rotates `V` onto the z-axis. This can be done using a combination of rotation and reflection transformations. The exact method depends on the orientation of `V`. One possible method is as follows:

```

d = sqrt(vx^2 + vy^2)

Rx = [ 1   0    0     0 ]

     [ 0  vx/d -vy/d  0 ]

     [ 0  vy/d  vx/d  0 ]

     [ 0   0    0     1 ]

Ry = [  dz/d  0  vx/d  0 ]

     [   0    1   0    0 ]

     [ -vx/d  0  dz/d  0 ]

     [   0    0   0    1 ]

Rz = [ cos(phi) -sin(phi)  0  0 ]

     [ sin(phi)  cos(phi)  0  0 ]

     [    0         0      1  0 ]

     [    0         0      0  1 ]

Ry_inv = [  dz/d  0  -vx/d  0 ]

         [   0    1    0    0 ]

         [  vx/d  0  dz/d   0 ]

         [   0    0    0    1 ]

Rx_inv = [ 1   0    0    0 ]

         [ 0  vx/d  vy/d 0 ]

         [ 0 -vy/d  vx/d 0 ]

         [ 0   0    0    1 ]

R' = Ry_inv * Rx_inv * Rz * Rx * Ry

```

4. Combine the translation, rotation, and inverse