SHEAR TRANSFORMATIUONS:
To complete the set of transformations discussed in the notes I should really add the 3D shear operations.
Shearing in 2D:
A shear along the x direction in 2D changes a rectangle (with lower right corner at the origin) into a parallelogram as follows:
Y ^ Y ^ | | P |___________________ | P' _____________________ | | | / / | | | / / | | |A / / | | | / / |___________________|____ X |/____________________/______ X
The transformation has the properties: a) The y coordinate of any point (x,y) is unchanged b) the x coordinate is stretched in a linear way, based on the height of the point above the x axis - i.e. on y. Thus the coordinate change has the form:
x' = x + ay y' = y
where a is a constant that measures the degree of shearing. If a is negatrive then the shearing is in the opposite direction.
Note that P(0,H) is taken into P'(aH,H). It follows that the shearing angle A (the angle through which the vertical edge was sheared) is given by:
tan(A) = aH/H = a.
So the parameter a is just the tan of the shearing angle.
A suitable multiplicative matrix description of the transform is given by:
SHx(a) = | 1 a 0 | | 0 1 0 | | 0 0 1 |
We can similarly introduce a shearing along the y axis, which would have the form:
x' = x y' = y + bx
and in this case a suitable matrix descrription is given by:
SHy(b) = | 1 0 0 | | b 1 0 | | 0 0 1 |
Shearing in 3D:
In 3D we can shear along the x-axis, y-axis or z-axis. AS an example, the shear along the z axis leaves the z coordinate of points the same but changes the x and y coordinates by amounts proportional to their z coordinate:
x' = x + az y' = y + bz z' = z
and can be represented by the matrix:
SHxy(a,b) = | 1 0 a 0 | | 0 1 b 0 | | 0 0 1 0 | | 0 0 0 1 |
The shears along the x and y directions have similar forms.