({{, (RefLink[TransformationMatrix, paclet:ref/TransformationMatrix])[tfun] 
gives the homogeneous matrix associated with a (RefLink[TransformationFunction, paclet:ref/TransformationFunction]) object.}})

OpenersNotesSection(Spacer1)Details

For transformations in n dimensions, (RefLink[TransformationMatrix, paclet:ref/TransformationMatrix]) normally gives an (n+1)×(n+1) matrix.

mat[[1;;n,1;;n]] gives the linear part of the transformation; mat[[1;;n,-1]] gives the displacement vector.

OpenersPrimaryExamplesSection(Spacer1)Examples  (3)

OpenersExampleSection0(Spacer1)Basic Examples  (1)

Here t is defined to be a rotation around the z axis:

t=RotationTransform[θ,{0,0,1}]

Get the transformation matrix:

m=TransformationMatrix[t]

The linear part:

m[[1;;3,1;;3]]

The displacement vector:

m[[1;;3,-1]]

OpenersExampleSection1(Spacer1)Scope  (1)

Translation matrix in four dimensions:

(m=TransformationMatrix@TranslationTransform[{x0,y0,z0,w0}])//MatrixForm

Transformation of homogeneous coordinates:

m.{x,y,z,w,1}

Points at infinity do not change under translation:

m.{x,y,z,w,0}

OpenersExampleSection3(Spacer1)Properties & Relations  (1)

The matrix of a general 2D affine transform:

TransformationMatrix@AffineTransform[{Array[a,{2,2}],Array[b,2]}]//MatrixForm

Composition of linear fractional transformations corresponds to the product of their matrices:

{t1,t2}={LinearFractionalTransform[Array[a1,{3,3}]],LinearFractionalTransform[Array[a2,{3,3}]]};

TransformationMatrix[Composition[t1,t2]]==TransformationMatrix[t1].TransformationMatrix[t2]

({{DocumentationFooterIconsRelatedFunction.png, ({{See Also}, {(RefLink[GeometricTransformation, paclet:ref/GeometricTransformation])  ▪  (RefLink[AffineTransform, paclet:ref/AffineTransform])  ▪  (RefLink[LinearFractionalTransform, paclet:ref/LinearFractionalTransform])}})}})

({{DocumentationFooterIconsRelatedGuide.png, ({{Related Guides}, {▪(RefLinkPlain[Geometric Transforms, paclet:guide/GeometricTransforms])}, {▪(RefLinkPlain[Computational Geometry, paclet:guide/ComputationalGeometry])}})}})