Y-shear about the origin - 2-d and 3-d transformations, Computer Graphics

y-shear about 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, through the specified scale parameter 'b'. that is, P'(x'y') be the result of y-shear of point P(x,y) through scale factor 'b' regarding the origin, that is demonstrated in Figure 5(a).

1739_y-shear about the origin - 2-d and 3-d transformations.png

Figure: 5(a)

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

x' = x

 

y' = y+bx

= Shy(b)

 

 

Here 'b' is a constant (termed as shear parameter) which measures the degree of shearing. In matrix form, we contain:

2465_y-shear about the origin - 2-d and 3-d transformations 1.png

In terms of Homogeneous Coordinates, above equation (16) is:

479_y-shear about the origin - 2-d and 3-d transformations 2.png

There is, P'h = Ph.Shy(b)

Here Ph  and P'h  represents object points, before and after needed transformation, in Homogeneous Coordinates and Shy(b) is termed as homogeneous transformation matrix for y-shear along with scale factor 'b' in the y-direction.

Posted Date: 4/3/2013 5:33:17 AM | Location : United States







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