Wolfram: Geometric Transformation Functions

Geometric Transformation

When programing graphics, especially when doing things like tiling, or group symmetries, often you need to apply a geometric transformation to the graphics. (e.g. rotation, scaling, shear, reflection, translation)

there are two main things you want to do:

• Apply transformation to points inside graphics.
• Compose transformations as in group theory.

Symbolic Representation of Geometric Transform Functions

the following, each represents a transform function symbolically:

• RotationTransform
• TranslationTransform
• ScalingTransform
• ShearingTransform
• ReflectionTransform
• RescalingTransform
• AffineTransform
• LinearFractionalTransform

they all return a TransformationFunction[data] that represent the transformation symbolically.

These functions are especially suitable if you are doing geometric transformations in the context of group theory. Namely, wallpaper group, space group. It allows you to compose group operations, and finally apply it to graphics.

TransformationFunction

TransformationFunction is the generalized symbolic representation of geometric transformation.

Apply a Transform to Points

• TransformationFunction[data][vector] apply f to the vector.
• TransformationFunction[data][vectorList] apply f to a list of vectors.

TransformationFunction, Other Use

Apply a Transform to Graphics Primitives

use GeometricTransformation

• Wolfram: Apply Transform to Graphics

Compose Transformations

• Wolfram: Compose Transformations

Convert to Matrix Representation

use TransformationMatrix

• Wolfram: Transformation Matrix