# TransformationFunction

TransformationFunction[data]

represents a transformation function that applies geometric and other transformations.

# Details  • objects are generated by constructors such as TranslationTransform, RotationTransform, etc.
• TransformationFunction[][x] applies the transformation function to a vector x, returning a transformed vector.
• TransformationFunction[][{x1,x2,}] for a list of vectors applies the transformation to each vector xi, producing a list of transformed vectors.
• TransformationFunction works with both numerical and symbolic vectors and represents a linear fractional transformation , where AMatrices[{m,n}], bMatrices[{m,1}], cMatrices[{1,n}] and dMatrices[{1,1}].
• For smaller dimensions, it is typically displayed as an transformation matrix . TransformationMatrix can be used to extract the transformation matrix.
• Composition[t1,t2] where ti has transformation matrix gives a new TransformationFunction object with transformation matrix .
• where t has transformation matrix gives a new TransformationFunction object with transformation matrix where is the matrix inverse.
• GeometricTransformation can be used to represent the effect of applying a TransformationFunction object to geometrical or graphics objects when restricted to affine transformations.
• TransformationFunction[][prop] gives the transformation property prop. For a transformation function with transformation matrix , properties include:
•  "AffineQ" whether the transformation is affine or not, it gives True if both c and d are zero "AffineMatrix" the matrix A "AffineVector" the vector b "FractionalVector" the vector c "FractionalConstant" the constant d "ArgumentLength" the length n of the vector x "ResultLength" the length m of the result vector "TransformationMatrix" # Examples

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## Basic Examples(1)

Create a rotation transform:

This rotates the vector {x,y} by angle θ:

## Scope(15)

### Constructing TransformationFunction(10)

A translation by the vector {qx,qy,qz}:

A rotation around the axis:

Scaling along the coordinate axes:

Shearing in the direction by an angle θ:

Reflecting in the plane:

Rescaling the box [xmin, xmax][ymin, ymax] to the unit square:

A general TransformationFunction:

A linear transformation:

An affine transformation:

A linear fractional transformation:

### Working with TransformationFunction as a Function(4)

Here is a rotation of around the axis:

This transforms the axis:

This transforms a list of vectors:

Composing two transformations:

Computing the inverse:

This shows they are inverses:

Computing the partial derivative :

### Working with TransformationFunction as a Formula(1)

This defines a general transform:

This is the corresponding formula:

A derivative:

A limit:

An integral:

A plot:

## Applications(2)

TransformationFunction can be used as an argument to GeometricTransformation:

Integrate a function over a rhombic region: defines a change of variables that maps the unit square to the integration region:

The integrand in the new coordinates:

The Jacobian:

## Properties & Relations(1)

Find the  power of a transformation:

Apply t five times:

Apply tt:

Find the  iteration using RSolve: