# rigidtform2d

2-D rigid geometric transformation

## Description

A `rigidtform2d` object stores information about a 2-D rigid geometric transformation and enables forward and inverse transformations.

## Creation

### Syntax

``tform = rigidtform2d``
``tform = rigidtform2d(Translation)``
``tform = rigidtform2d(RotationAngle,Translation)``
``tform = rigidtform2d(R,Translation)``
``tform = rigidtform2d(A)``
``tform = rigidtform2d(tformIn)``

### Description

````tform = rigidtform2d` creates a `rigidtform2d` object that performs an identity transformation.```
````tform = rigidtform2d(Translation)` creates a `rigidtform2d` object that performs a rigid transformation consisting only of translation. The specified property `Translation` indicates the amount of translation in the x- and y-directions.If you want a transformation that performs only 2-D translation, consider using the `transltform2d` object.```

example

````tform = rigidtform2d(RotationAngle,Translation)` creates a `rigidtform2d` object that performs a rigid transformation based on the specified values of the RotationAngle and `Translation` properties. These properties indicate the rotation angle and the amount of translation in the x- and y- directions.```
````tform = rigidtform2d(R,Translation)` creates a `rigidtform2d` object that performs a rigid transformation based on the specified values of the `R` and `Translation` properties. These properties indicate the rotation matrix and the amount of translation in the x- and y- directions.```
````tform = rigidtform2d(A)` creates a `rigidtform2d` object and sets the property `A` as the specified 2-D rigid transformation matrix.```
````tform = rigidtform2d(tformIn)` creates a `rigidtform2d` object from another geometric transformation object, `tformIn`, that represents a valid 2-D rigid geometric transformation.```

### Input Arguments

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Rigid 2-D geometric transformation, specified as an `affinetform2d` object, `rigidtform2d` object, `simtform2d` object, `transltform2d` object, or `projtform2d` object.

## Properties

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Forward 2-D rigid transformation, specified as a nonsingular 3-by-3 numeric matrix. When you create the object, you can also specify `A` as a 2-by-3 numeric matrix. In this case, the object concatenates the row vector `[0 0 1]` to the end of the matrix, forming a 3-by-3 matrix. The default value of `A` is the identity matrix.

The matrix `A` transforms the point (u, v) in the input coordinate space to the point (x, y) in the output coordinate space using the convention:

`$\left[\begin{array}{c}x\\ y\\ 1\end{array}\right]=Α×\left[\begin{array}{c}u\\ v\\ 1\end{array}\right]$`

For a rigid transformation, `A` has the form:

`$Α=\left[\begin{array}{ccc}cosd\left(r\right)& -\mathrm{sind}\left(r\right)& {t}_{x}\\ \mathrm{sind}\left(r\right)& cosd\left(r\right)& {t}_{y}\\ 0& 0& 1\end{array}\right]$`

where r is the rotation angle and corresponds to the property `RotationAngle`. tx and ty are the amount of translation in the x- and y- directions, respectively, and correspond to the property `Translation`.

Data Types: `double` | `single`

Rotation matrix, specified as a 2-by-2 numeric matrix. The matrix must have the form

` R = [cosd(r) -sind(r); sind(r) cosd(r)]`
where r is the value of the `RotationAngle` property.

Rotation angle about the origin, in degrees, specified as a numeric scalar. The rotation angle corresponds to the value r in the transformation matrix defined by `A`, and in the rotation matrix defined by `R`.

Data Types: `double` | `single`

Amount of translation, specified as a 2-element numeric vector of the form [tx ty]. These amounts of translation correspond to the values tx and ty in the rigid transformation matrix defined by `A`.

Data Types: `double` | `single`

Dimensionality of the geometric transformation for both input and output points, specified as `2`.

Data Types: `double`

## Object Functions

 `invert` Invert geometric transformation `outputLimits` Find output spatial limits given input spatial limits `transformPointsForward` Apply forward geometric transformation `transformPointsInverse` Apply inverse geometric transformation

## Examples

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Specify a rotation angle and an amount of translation.

```theta = 30; translation = [10 20.5];```

Create a `rigidtform2d` object that performs the specified rotation and translation.

`tform = rigidtform2d(theta,translation)`
```tform = rigidtform2d with properties: Dimensionality: 2 RotationAngle: 30 Translation: [10 20.5000] R: [2x2 double] A: [3x3 double] ```

Examine the value of the `A` property.

`tform.A`
```ans = 3×3 0.8660 -0.5000 10.0000 0.5000 0.8660 20.5000 0 0 1.0000 ```

## Version History

Introduced in R2022b

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