Explain Shearing and reflection as a technique of 2d transformation?

Answer 1

ShearingFor shear mapping (visually similar to slanting), there are two possibilities. For a shear parallel to the x axis has x' = x + ky and y' = y; the shear matrix, applied to column vectors, is:

A shear parallel to the y axis has x' = xand y' = y + kx, which has matrix form:

ReflectionTo reflect a vector about a line that goes through the origin, let be a vector in the direction of the line:

To reflect a point through a plane ax + by + cz = 0 (which goes through the origin), one can use , where is the 3x3 identity matrix and is the three-dimensional unit vector for the surface normal of the plane. If the L2 norm of a,b, and c is unity, the transformation matrix can be expressed as:

Note that these are particular cases of a Householder reflection in two and three dimensions. A reflection about a line or plane that does not go through the origin is not a linear transformation; it is an affine transformation.