# transformPointsInverse

Apply inverse geometric transformation

## Syntax

``[u,v] = transformPointsInverse(tform,x,y)``
``[u,v,w] = transformPointsInverse(tform,x,y,z)``
``U = transformPointsInverse(tform,X)``

## Description

example

````[u,v] = transformPointsInverse(tform,x,y)` applies the inverse transformation of 2-D geometric transformation `tform` to the points specified by coordinates `x` and `y`.```
````[u,v,w] = transformPointsInverse(tform,x,y,z)` applies the inverse transformation of 3-D geometric transformation `tform` to the points specified by coordinates `x`, `y`, and `z`.```
````U = transformPointsInverse(tform,X)` applies the inverse transformation of `tform` to the input coordinate matrix `X` and returns the coordinate matrix `U`. `transformPointsInverse` maps the kth point `X`(k,:) to the point `U`(k,:).```

## Examples

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Create an `affine2d` object that defines the transformation.

```theta = 10; tform = affine2d([cosd(theta) -sind(theta) 0; sind(theta) cosd(theta) 0; 0 0 1])```
```tform = affine2d with properties: T: [3x3 double] Dimensionality: 2```

Apply forward transformation of 2-D geometric transformation to an input point.

`[X,Y] = transformPointsForward(tform,5,10)`
```X = 6.6605 Y = 8.9798```

Apply inverse transformation of 2-D geometric transformation to output point from the previous step to recover the original coordinates.

`[U,V] = transformPointsInverse(tform,X,Y)`
```U = 5.0000 V = 10 ```

Specify the packed (x,y) coordinates of five input points. The packed coordinates are stored in a 5-by-2 matrix, where the x-coordinate of each point is in the first column, and the y-coordinate of each point is in the second column.

`XY = [10 15;11 32;15 34;2 7;2 10];`

Define the inverse mapping function. The function accepts and returns points in packed (x,y) format.

`inversefn = @(c) [c(:,1)+c(:,2),c(:,1)-c(:,2)]`
```inversefn = function_handle with value: @(c)[c(:,1)+c(:,2),c(:,1)-c(:,2)] ```

Create a 2-D geometric transform object, `tform`, that stores the inverse mapping function.

`tform = geometricTransform2d(inversefn)`
```tform = geometricTransform2d with properties: InverseFcn: @(c)[c(:,1)+c(:,2),c(:,1)-c(:,2)] ForwardFcn: [] Dimensionality: 2 ```

Apply the inverse geometric transform to the input points.

`UV = transformPointsInverse(tform,XY)`
```UV = 5×2 25 -5 43 -21 49 -19 9 -5 12 -8 ```

Create an `affine3d` object that defines the transformation.

`tform = affine3d([3 1 2 0;4 5 8 0;6 2 1 0;0 0 0 1])`
```tform = affine3d with properties: T: [4×4 double] Dimensionality: 3```

Apply forward transformation of 3-D geometric transformation to an input point.

`[X,Y,Z] = transformPointsForward(tform,2,3,5)`
```X = 48 Y = 27 Z = 33```

Apply inverse transformation of 3-D geometric transformation to output point from the previous step to recover the original coordinates.

`[U,V,W] = transformPointsInverse(tform,X,Y,Z)`
```U = 2.0000 V = 3 W = 5.0000```

Specify the packed (x,y,z) coordinates of five input points. The packed coordinates are stored as a 5-by-3 matrix, where the first, second, and third columns contain the x-, y-, and z- coordinates,respectively.

`XYZ = [5 25 20;10 5 25;15 10 5;20 15 10;25 20 15];`

Define an inverse mapping function that accepts and returns points in packed (x,y,z) format.

`inverseFcn = @(c) [c(:,1)+c(:,2),c(:,1)-c(:,2),c(:,3).^2];`

Create a 3-D geometric transformation object, `tform`, that stores this inverse mapping function.

`tform = geometricTransform3d(inverseFcn)`
```tform = geometricTransform3d with properties: InverseFcn: @(c)[c(:,1)+c(:,2),c(:,1)-c(:,2),c(:,3).^2] ForwardFcn: [] Dimensionality: 3 ```

Apply the inverse transformation of this 3-D geometric transformation to the input points.

`UVW = transformPointsInverse(tform,XYZ)`
```UVW = 5×3 30 -20 400 15 5 625 25 5 25 35 5 100 45 5 225 ```

## Input Arguments

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Geometric transformation, specified as a geometric transformation object.

For 2-D geometric transformations, `tform` is an `affine2d`, `projective2d`, `geometricTransform2d`, `LocalWeightedMeanTransformation2D`, `PiecewiseLinearTransformation2D`, or `PolynomialTransformation2D` geometric transformation object.

For 3-D geometric transformations, `tform` can be an `affine3d` object, or a `geometricTransform3d` or `rigid3d` geometric transformation object.

x-coordinates of points to be transformed, specified as an m-by-n or m-by-n-by-p numeric array. The number of dimensions of `x` matches the dimensionality of `tform`.

Data Types: `single` | `double`

y-coordinates of points to be transformed, specified as an m-by-n or m-by-n-by-p numeric array. The size of `y` must match the size of `x`.

Data Types: `single` | `double`

z-coordinates of points to be transformed, specified as an m-by-n-by-p numeric array. `z` is used only when `tform` is a 3-D geometric transformation. The size of `z` must match the size of `x`.

Data Types: `single` | `double`

Coordinates of points to be transformed, specified as an l-by-2 or l-by-3 numeric array. The number of columns of `X` matches the dimensionality of `tform`.

The first column lists the x-coordinate of each point to transform, and the second column lists the y-coordinate. If `tform` represents a 3-D geometric transformation, `X` has size l-by-3 and the third column lists the z-coordinate of the points to transform.

Data Types: `single` | `double`

## Output Arguments

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x-coordinates of points after transformation, returned as an m-by-n or m-by-n-by-p numeric array. The number of dimensions of `u` matches the dimensionality of `tform`.

Data Types: `single` | `double`

y-coordinates of points after transformation, returned as an m-by-n or m-by-n-by-p numeric array. The size of `v` matches the size of `u`.

Data Types: `single` | `double`

z-coordinates of points after transformation, returned as an m-by-n-by-p numeric array. The size of `w` matches the size of `u`.

Data Types: `single` | `double`

Coordinates of points after transformation, returned as a numeric array. The size of `U` matches the size of `X`.

The first column lists the x-coordinate of each point after transformation, and the second column lists the y-coordinate. If `tform` represents a 3-D geometric transformation, the third column lists the z-coordinate of the points after transformation.

Data Types: `single` | `double`