### 2D Shear

Shear is the translation along an axis (say, X axis) by an amount that increases linearly with another axis (Y). It produces shape distortions as if objects were composed of layers that are caused to slide over each other [].

Shear transformations are very useful in creating italic letters and slanted letters from regular letters.

### 2D Shear along X-direction

Shear in X direction is represented by the following set of equations.

where h is the negative or positive fraction of Y coordinate of P to be added to the X coordinate. can be any real number.

The matrix of form of shear in x-direction is given by

### 2D Shear along Y Direction

Similarly, shear along y-direction is given by

Combining the shear in X and Y directions,

where g is a non-zero fraction of to be added to the Y coordinate.

#### General matrix form of shear

The general matrix form of shear is

Shear will reduce to a pure shear in the y-direction, when h=0.

The inverse of Shear is given by

### Example

1. If , then shear along X direction of the point is obtained by substituting these values in (0.3).

Shear in Y direction is

2. Consider a square of side = 2. Show the effect of shear when (1)

### 3D Shear

Shear along any pair of axes is proportional to the third axis. For instance, to shear along z in 3D, x and y values are altered by an amount proportional to the value of z, leaving z unchanged. Let , is the shear due to z along x and y directions respectively and are real values. Then the matrix representation is

Shear for x , y axis is similar to that of z. The general form of shear is given by

Priya Asokarathinam /ADVISOR Shoaff