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

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

323_xy-Shear about the Origin - 2-d and 3-d transformations.png

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

x' = x +ay

 

y' = y+bx

= Shxy(a,b)

 

(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:

541_xy-Shear about the Origin - 2-d and 3-d transformations 1.png

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

In terms of Homogeneous Coordinates, we contain:

397_xy-Shear about the Origin - 2-d and 3-d transformations 2.png

---------(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.

Posted Date: 4/3/2013 5:39:49 AM | Location : United States







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