# Transformation Matrix

### From Math Images

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- | A '''transformation matrix''' is a special matrix that can describe transformations. | + | A '''transformation matrix''' is a special matrix used in linear algebra and computer graphics that can describe 2d and 3d transformations. If ''T'' is a linear transformation mapping '''R'''<sup>''n''</sup> to '''R'''<sup>''m''</sup> and ''x'' is a [[column vector]] with ''n'' entries, then |

- | + | :<math>T( \vec x ) = \mathbf{A} \vec x</math> | |

+ | |||

+ | for some ''m''×''n'' matrix '''A''', called the '''transformation matrix of ''T'''''. | ||

===Examples in 2D Graphics=== | ===Examples in 2D Graphics=== |

## Revision as of 14:50, 25 August 2009

A **transformation matrix** is a special matrix used in linear algebra and computer graphics that can describe 2d and 3d transformations. If *T* is a linear transformation mapping **R**^{n} to **R**^{m} and *x* is a column vector with *n* entries, then

for some *m*×*n* matrix **A**, called the **transformation matrix of T**.

## Contents |

### Examples in 2D Graphics

In 2D graphics Linear transformations can be represented by 2x2 matrices. Most common transformations such as rotation, scaling, shearing, and reflection are linear transformations and can be represented in the 2x2 matrix. Other affine transformations can be represented in a 3x3 matrix.

#### Rotation

For rotation by an angle θ clockwise about the origin, the functional form is x' = xcosθ + ysinθ and y' = − xsinθ + ycosθ. Written in matrix form, this becomes:

Similarly, for a rotation counterclockwise about the origin, the functional form is and and the matrix form is:

#### Scaling

For scaling we have and . The matrix form is:

#### Shearing

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

A shear parallel to the *y* axis has and , which has matrix form:

### Composing transformations

### 2D Affine Transformations

Affine transformations are represented by (x,y,1) and a 3x3 matrix instead of a 2x2 matrix.