X-shear Regarding the Origin - 2-d and 3-d transformations
Suppose an object point P(x,y) be moved to P'(x',y') in the x-direction, via the given scale parameter 'a',that is, P'(x'y') be the outcome of x-shear of point P(x,y) through scale factor a regarding the origin, that is demonstrated in Figure 4.
Hence, the points P(x,y) and P'(x',y') have the subsequent relationship:
x' = x + ay
y' = y
= Sh_{x}(a) ------(11a)
Here 'a' is a constant (termed as shear parameter) which measures the degree of shearing. The shearing is in the opposite direction, if a is negative.
Remember that P(0,H) is taken into P'(aH,H). This obeys that the shearing angle A which is the angle by that the vertical edge was sheared; is specified by:
tan(A) = aH/H = a.
Consequently the parameter a is simply the tan of the shearing angle. Within matrix form of 2-D Euclidean system, we contain:
(12)
In terms of Homogeneous Coordinates, above equation (12) is:
-----------------------(13)
That is, P'_{h} = P_{h} Sh_{x}(a) --(14)
Here P_{h} and P'_{h} represent object points, before and after needed transformation, in Homogeneous Coordinates and Sh_{x}(a) is termed as homogeneous transformation matrix for x-shear along with scale parameter 'a' in the x-direction.