Transformation for 3-D Shearing

2-dimensional xy-shearing transformation, as explained in equation, 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

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

Here remember that the off-diagonal terms in the upper left 3x3 sub-matrix of the generalized 4x4 transformation matrix in equation produce shear in 3-dimensions.