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 :
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x' = x +ay
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y' = y+bx
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= Shxy(a,b)
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(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 = Ph.Shxy(a,b) ----------(22)
Here Ph and P'h represent object points, before and after needed transformation, in Homogeneous Coordinates and Shxy(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.