2-dimensional xy-shearing transformation, as explained in equation (19), can also be simply extended to 3-dimensional case. All coordinates are translated as a function of displacements of another two coordinates, which is,
Sh_{xyz}= x'=x+a.y+b.z
y'=y+c.x+d.z
z'=z+e.x+f.y--------(46)
Here a,b,c,d,e and f are the shearing factors in the respective directions. Equation (46) in terms of homogeneous HC system, is:
That is, P'_{h} = P_{h}.Sh_{xyz} --------------(47)
Here remember that the off-diagonal terms in the upper left 3x3 sub-matrix of the generalized 4x4 transformation matrix in equation (31) produce shear in 3-dimensions.