xy-Shear about the Origin - 2-d and 3-d transformations

Suppose an object point P(x,y) be moved to P'(x',y') as a outcome of shear transformation in both x- and y-directions along with shearing factors a and b, respectively, as demonstrated in

The points P(x,y) and P'(x',y') have the subsequent relationship :

(19)

Here ′ay′ and ′bx′ are shear factors in x and y directions, respectively. The xy-shear is also termed as shearing for short or simultaneous shearing.

In matrix form, we contain:

-------------(20)

In terms of Homogeneous Coordinates, we contain:

---------(21)

It is, P'_h = P_h.Sh_xy(a,b)  ----------(22)

Here P_h   and P'_h represent object points, before and after needed transformation, in Homogeneous Coordinates and Sh_xy(a,b) is termed as homogeneous transformation matrix for xy-shear in both x- and y-directions along with shearing factors a and b, respectively, particular case: while we put b=0 in above equation (21), we contain shearing in x-direction, and while a=0, we have Shearing in the y-direction, correspondingly.