The Wolfram Language's symbolic architecture and sophisticated mathematical capabilities allow it to take a uniquely high-level approach to geometric transformations—supporting complete geometric, matrix, and functional representations in any number of dimensions, whether for mechanical systems, computer graphics, or pure mathematics.

Geometric Object Transformations

Rotate  ▪  Translate  ▪  Scale

GeometricTransformation — apply any transformation function to geometric objects

Rotation Matrices in 3D

EulerMatrix — axis-oriented rotations in the current frame (intrinsic rotations)

RollPitchYawMatrix — axis-oriented rotations in the global frame (extrinsic rotations)

EulerAngles, RollPitchYawAngles — gives angles from rotation matrices

Transformation Matrices in nD

RotationMatrix — rotations in any number of dimensions

ScalingMatrix — stretch or shrink along any direction

ShearingMatrix  ▪  ReflectionMatrix  ▪  IdentityMatrix

TransformationMatrix — the matrix representation of a transformation function

Symbolic Transformation Functions in nD

RotationTransform  ▪  TranslationTransform  ▪  ScalingTransform

ShearingTransform  ▪  ReflectionTransform  ▪  RescalingTransform

AffineTransform  ▪  LinearFractionalTransform

TransformationFunction — general symbolic representation of transformations

Composition — symbolic composition of transformation functions

InverseFunction — inverse of transformation functions

Transformation Functions from Points

FindGeometricTransform — find affine or linear fractional transform between point sets

Algebraic Structure

FiniteGroupData — properties of finite groups