3D shearing and matrix

Shearing is a transformation that distorts the shape of an object by sliding its layers over each other, as if the object were pushed sideways. One coordinate is kept fixed and the other two are changed in proportion to it. It is an affine transformation: straight lines stay straight, but angles change.

In 3D there are three kinds of shear, one for each axis whose coordinate is held fixed.

1. Shear along the x-axis (x fixed)

x' = x, y' = y + sh_y·x, z' = z + sh_z·x

| 1    0 0 0 |
| sh_y 1 0 0 |
| sh_z 0 1 0 |
| 0    0 0 1 |

2. Shear along the y-axis (y fixed)

x' = x + sh_x·y, y' = y, z' = z + sh_z·y

| 1 sh_x 0 0 |
| 0 1    0 0 |
| 0 sh_z 1 0 |
| 0 0    0 1 |

3. Shear along the z-axis (z fixed)

x' = x + sh_x·z, y' = y + sh_y·z, z' = z

| 1 0 sh_x 0 |
| 0 1 sh_y 0 |
| 0 0 1    0 |
| 0 0 0    1 |

The shear constants (sh_x, sh_y, sh_z) tell how much each coordinate is displaced per unit of the fixed coordinate. The general shear matrix is a combination of the three. Points on the fixed plane (for example x = 0 for x-shear) do not move.

Example: a unit cube sheared with the z-shear and sh_x = 1 becomes a slanted block whose top face (z = 1) moves one unit in x.