# se2

## Description

The `se2`

object represents an SE(2) transformation as a
2-D homogeneous transformation matrix consisting of a translation and rotation.

For more information, see the 2-D Homogeneous Transformation Matrix section.

This object acts like a numerical matrix, enabling you to compose poses using multiplication and division.

## Creation

### Syntax

### Description

#### Rotation Matrices, Translation Vectors, and Transformation Matrices

`transformation = se2`

creates an SE(2) transformation
representing an identity rotation with no translation.

$$transformation=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$$

`transformation = se2(`

creates an
SE(2) transformation representing a pure rotation defined by the orthonormal rotation
`rotation`

)`rotation`

with no translation. The rotation matrix is represented
by the elements in the top left of the `transformation`

matrix.

$$rotation=\left[\begin{array}{cc}{r}_{11}& {r}_{12}\\ {r}_{21}& {r}_{22}\end{array}\right]$$

$$transformation=\left[\begin{array}{ccc}{r}_{11}& {r}_{12}& 0\\ {r}_{21}& {r}_{22}& 0\\ 0& 0& 1\end{array}\right]$$

`transformation = se2(`

creates an SE(2) transformation representing a rotation defined by the orthonormal
rotation `rotation`

,`translation`

)`rotation`

and the translation
`translation`

. The function applies the rotation matrix first, then
the translation vector, to create the transformation.

$$rotation=\left[\begin{array}{cc}{r}_{11}& {r}_{12}\\ {r}_{21}& {r}_{22}\end{array}\right]$$, $$translation=\left[\begin{array}{c}{t}_{1}\\ {t}_{2}\end{array}\right]$$

$$transformation=\left[\begin{array}{ccc}{r}_{11}& {r}_{12}& {t}_{1}\\ {r}_{21}& {r}_{22}& {t}_{2}\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& {t}_{1}\\ 0& 1& {t}_{2}\\ 0& 0& 1\end{array}\right]\xb7\left[\begin{array}{ccc}{r}_{11}& {r}_{12}& 0\\ {r}_{21}& {r}_{22}& 0\\ 0& 0& 1\end{array}\right]$$

`transformation = se2(`

creates an SE(2) transformation representing a translation and rotation as defined by
the homogeneous transformation `transformation`

)`transformation`

.

#### Other 2-D Rotations and Transformation Representations

`transformation = se2(`

creates SE(2) transformations `angle`

,"theta")`transformation`

from rotations around
the *z*-axis, in radians. The transformation contains zero
translation.

`transformation = se2(`

creates SE(2) transformations from rotations around the `angle`

,"theta",`translation`

)*z*-axis, in
radians, with translations `translation`

.

`transformation = se2(`

creates an SE(2) transformation from the translation vector
`translation`

,"trvec")`translation`

.

`transformation = se2(`

creates an SE(2) transformation from the 2-D compact pose
`pose`

,"xytheta")`pose`

.

### Input Arguments

## Object Functions

## Examples

## Algorithms

## Extended Capabilities

## Version History

**Introduced in R2022b**