In matrix terms, how is a translation represented?

In matrix terms, a translation is represented using homogeneous coordinates, which allows for translations to be performed using matrix operations. Specifically, translation can be represented by augmenting the transformation matrix with an additional row that accounts for the translation components.

In a two-dimensional space, for example, a standard transformation matrix for rotation, scaling, or shearing is a 2x2 matrix. To include translation, an additional row is added to convert this into a 3x3 matrix. This additional row allows for the inclusion of translation by modifying the third homogeneous coordinate.

Thus, in homogeneous coordinates, a point (x, y) is represented as (x, y, 1). A translation by (tx, ty) can then be expressed in matrix form as follows:

[

\begin{pmatrix}

1 & 0 & tx \

0 & 1 & ty \

0 & 0 & 1

\end{pmatrix}

]

When this matrix is multiplied by the homogeneous coordinate vector of a point, it translates the point appropriately.

Therefore, the representation of a translation in matrix form involves the use of an additional row to accommodate the translation values, confirming that the correct representation is achieved with the addition of a row