# plot::Transform2d

Affine linear transformation of 2D objects

### Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

## 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 to 2D objects.

The transformation matrix `A` can be specified by a list of lists

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

with the sublists representing the rows.

A plain list

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

represents the matrix row by row in 2D.

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`. Cf. the help page of `ViewingBox`. In such a case, you should 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

We visualize a linear transformation 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:`

### Example 2

For some applications, it is very popular to plot a function in 3D together with a projection of its contour lines onto the lower or upper bounding plane. MuPAD® has no direct option for this, but with `plot::Transform3d`, it is possible to achieve the same effect. Assume you have the function under consideration in a `plot::Function3d` object:

```f := plot::Function3d(sin(x*y)+cos(x^2-y), x=-3..3, y=-3..3, Submesh=[1,1]):```

To plot contour lines at all, we use the attribute `ZContours`. Since we don't want to change our `f`, we create a modified copy using `plot::modify`:

`plot(plot::modify(f, ZContours = [Automatic, 10]))`

To only get contour lines, we have to change a few more parameters: We need to switch off the surface and the parameter lines. Then, we add height coloring to our lines and use `plot::Transform3d` to project them onto the plane z = - 2.5. Finally, we plot these lines together with the original function:

```plot(f, plot::Transform3d([0, 0, -2.5], // shift vector [1, 0, 0, // transformation matrix 0, 1, 0, 0, 0, 0], plot::modify(f, Filled = FALSE, XLinesVisible = FALSE, YLinesVisible = FALSE, ZContours = [Automatic, 10], LineColorFunction = // height coloring ((x, y, z) -> [(z+2)/4, 0, (2-z)/4]))))```

## 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 of the appropriate dimension `a` Animation parameter, specified as `a```` = amin..amax```, where `amin` is the initial parameter value, and `amax` is the final parameter value.