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# trvec2tform

Convert translation vector to homogeneous transformation

## Syntax

``tform = trvec2tform(trvec)``

## Description

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````tform = trvec2tform(trvec)` converts the Cartesian representation of the translation vector `trvec` to the corresponding homogeneous transformation `tform`. When using the transformation matrix, premultiply it by the coordinates to be transformed (as opposed to postmultiplying).```

## Examples

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```trvec = [0.5 6 100]; tform = trvec2tform(trvec)```
```tform = 4×4 1.0000 0 0 0.5000 0 1.0000 0 6.0000 0 0 1.0000 100.0000 0 0 0 1.0000 ```

## Input Arguments

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Cartesian representation of a translation vector, specified as an n-by-2 matrix if `tform` is a 3-by-3-by-n array and an n-by-3 matrix if `tform` is a 4-by-4-by-n array. n is the number of translation vectors. Each vector is of the form [x y] or [x y z].

Example: `[0.5 6 100]`

## Output Arguments

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Homogeneous transformation, returned as a 3-by-3-by-n array or 4-by-4-by-n array. n is the number of homogeneous transformations. When using the rotation matrix, premultiply it with the coordinates to be rotated (as opposed to postmultiplying).

Example: ```[0 0 1 0; 0 1 0 0; -1 0 0 0; 0 0 0 1]```

2-D homogeneous transformation matrices are of the form:

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

3-D homogeneous transformation matrices are of the form:

`$T=\left[\begin{array}{cccc}{r}_{11}& {r}_{12}& {r}_{13}& {t}_{1}\\ {r}_{21}& {r}_{22}& {r}_{23}& {t}_{2}\\ {r}_{31}& {r}_{32}& {r}_{33}& {t}_{3}\\ 0& 0& 0& 1\end{array}\right]$`

## More About

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### Homogeneous Transformation Matrices

Homogeneous transformation matrices consist of both an orthogonal rotation and a translation.

2-D Transformations

2-D transformations have a rotation θ about the z-axis:

`${R}_{z}\left(\theta \right)=\left[\begin{array}{cc}\mathrm{cos}\theta & -\mathrm{sin}\theta \\ \mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right]$`

, and a translation along the x and y axis:

`$t=\left[\begin{array}{c}x\\ y\end{array}\right]$`

, resulting in the 2-D transformation matrix of the form:

`$T=\left[\begin{array}{cc}R& t\\ {0}_{1×2}& 1\end{array}\right]=\left[\begin{array}{cc}{I}_{2}& t\\ {0}_{1×2}& 1\end{array}\right]·\left[\begin{array}{cc}R& 0\\ {0}_{1×2}& 1\end{array}\right]$`

3-D Transformations

3-D transformations contain information about three rotations about the x-, y-, and z-axes:

`${R}_{x}\left(\varphi \right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& \mathrm{cos}\varphi & \mathrm{sin}\varphi \\ 0& -\mathrm{sin}\varphi & \mathrm{cos}\varphi \end{array}\right],{R}_{y}\left(\psi \right)=\left[\begin{array}{ccc}\mathrm{cos}\psi & 0& \mathrm{sin}\psi \\ 0& 1& 0\\ -\mathrm{sin}\psi & 0& \mathrm{cos}\psi \end{array}\right],{R}_{z}\left(\theta \right)=\left[\begin{array}{ccc}\mathrm{cos}\theta & -\mathrm{sin}\theta & 0\\ \mathrm{sin}\theta & \mathrm{cos}\theta & 0\\ 0& 0& 1\end{array}\right]$`

and after multiplying become the rotation about the xyz-axes:

`$\begin{array}{l}{R}_{xyz}={R}_{x}\left(\varphi \right){R}_{y}\left(\psi \right){R}_{z}\left(\theta \right)=\\ \left[\begin{array}{ccc}\mathrm{cos}\varphi \mathrm{cos}\psi \mathrm{cos}\theta -\mathrm{sin}\varphi \mathrm{sin}\theta & -\mathrm{cos}\varphi \mathrm{cos}\psi \mathrm{sin}\theta -\mathrm{sin}\varphi \mathrm{cos}\theta & \mathrm{cos}\varphi \mathrm{sin}\psi \\ \mathrm{sin}\varphi \mathrm{cos}\psi \mathrm{cos}\theta +\mathrm{cos}\varphi \mathrm{sin}\theta & -\mathrm{sin}\varphi \mathrm{cos}\psi \mathrm{sin}\theta +\mathrm{cos}\varphi \mathrm{cos}\theta & \mathrm{sin}\varphi \mathrm{sin}\psi \\ -\mathrm{sin}\psi \mathrm{cos}\theta & \mathrm{sin}\psi \mathrm{sin}\theta & \mathrm{cos}\psi \end{array}\right]\end{array}$`

and a translation along the x-, y-, and z-axis:

`$t=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$`

, resulting in the 3-D transformation matrix of the form:

`$T=\left[\begin{array}{cc}R& t\\ {0}_{1x3}& 1\end{array}\right]=\left[\begin{array}{cc}{I}_{3}& t\\ {0}_{1x3}& 1\end{array}\right]·\left[\begin{array}{cc}R& 0\\ {0}_{1x3}& 1\end{array}\right]$`

## Version History

Introduced in R2015a

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