# Documentation

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# `plot`::`Transform2d`

Affine linear transformation of 2D objects

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## Syntax

```plot::Transform2d(<`b2d`>, `A2d`, `obj1`, <`obj2,...`>, <`a = amin .. amax`>, `options`)
```

## Description

`plot::Transform2d(b, A, objects)` with a vector b and a matrix A applies the affine linear transformation $x\to Ax+b$ to 2D objects.

The transformation matrix `A` can be specified by a list of lists, with the sublists representing the rows.

```[[A1, 1, A1, 2, …], [A2, 1, A2, 2, …], …]```

A plain list ```[A1, 1, A1, 2, A2, 1, A2, 2]``` represents the matrix row by row.

Transform objects can transform several graphical objects simultaneously. Plotting the transform object renders all graphical objects inside.

Transformed objects have a tendency to overestimate their `ViewingBox`. In such cases, specify a suitable `ViewingBox` explicitly.

Transformation objects can be used inside transformation objects. If they are animated, the animations run simultaneously.

Animated transform objects are rather “cheap” concerning computing and storing costs. For more complex graphical objects, it is more efficient to use an animated transform object than to redefine the object for each frame.

The function `op` allows to extract the graphical objects inside a transformation object.

## Attributes

AttributePurposeDefault Value
`AffectViewingBox`influence of objects on the `ViewingBox` of a scene`TRUE`
`Frames`the number of frames in an animation`50`
`Matrix2d`transformation matrices[`1`, `0`, `0`, `1`]
`Name`the name of a plot object (for browser and legend)
`ParameterEnd`end value of the animation parameter
`ParameterName`name of the animation parameter
`ParameterBegin`initial value of the animation parameter
`ParameterRange`range of the animation parameter
`Shift`shift vector[`0`, `0`]
`ShiftX`shift vector`0`
`ShiftY`shift vector`0`
`TimeEnd`end time of the animation`10.0`
`TimeBegin`start time of the animation`0.0`
`TimeRange`the real time span of an animation`0.0` .. `10.0`

## Examples

### Example 1

Visualize a linear transformation $x\to Ax$ without shift:

```x1 := plot::Arrow2d([0, 0], [1/3, 1], Color = RGB::Red): x2 := plot::Arrow2d([0, 0], [1, 1], Color = RGB::Green): x3 := plot::Arrow2d([0, 0], [1, 1/3], Color = RGB::Blue): A := matrix([[1, -2], [-2, 1]]): plot(plot::Scene2d(x1, x2, x3), plot::Scene2d(plot::Transform2d(A, x1, x2, x3)), Scaling = Constrained, Layout = Horizontal):```

`delete x1, x2, x3, A:`

## Parameters

 `b2d` The 2D shift vector: a list with 2 entries. Also vectors generated by `matrix` and arrays are accepted. The entries must be numerical values or arithmetical expressions of the animation paramater `a`. `b2d` is equivalent to the attribute `Shift`. `A2d` The 2D transformation matrix: a 2×2 matrix, a 2×2 array, a list of 2 lists, or a plain list with 4 entries. The entries must be numerical values or arithmetical expressions of the animation paramater `a`. `A2d` is equivalent to the attribute `Matrix2d`. `obj1, obj2, …` Plot objects `a` Animation parameter, specified as `a```` = amin..amax```, where `amin` is the initial parameter value, and `amax` is the final parameter value.