Shear in Computer Graphic

A Transformation that slants the shape of an object is called the shear transformation. Two common shearing transformations are used. One shifts coordinate values and other shift y co- ordinate values. However in both the cases only one co-ordinate (xory)changes its coordinates and other preserves its values.

X- Shear

The x shear preserves the y coordinates, but changes the x values which cause vertical lines to tilt right or left as shown in figure

The Transformations matrix for x-shear is

which transforms the coordinates as

x’ =x+ shx .y ;y’ =y

Y - Shear

The y shear preserves the x coordinates, but changes the y values which cause horizontal lines which slope up or down The Transformations matrix for y-shear is

which transforms the coordinates as x’ = x

y’ = y + y sh_x .x

XY - Shear

The transformation matrix for xy-shear

=  

which transforms the coordinates as

x’ = x +xsh_x.y y’ = y +ysh_x

Shearing Relative to other reference line

We can apply x shear and y shear transformations relative to other reference lines. In x shear transformations we can use y reference line and in y shear we can use x reference line.

X - Shear with y reference line

We can generate x-direction shears relative to other reference lines with the transformation matrix

which transforms the coordinates as

x’ = x+xsh_x (y_ref y)

y’ = y

Example   

Sh_x = ½ and Y_ref = -1

Y - Shear with x reference line

We can generate y-direction shears relative to other reference lines with the transformation matrix which transforms the coordinates as

Example

x’ = x

Y’ = sh_y (x – x _ref) + y Sh_y = ½ and x _ref = -1