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TransformationMatrix[tfun]

gives the homogeneous matrix associated with a TransformationFunction object.

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TransformationMatrix

TransformationMatrix[tfun]

gives the homogeneous matrix associated with a TransformationFunction object.

Details

  • For transformations in n dimensions, TransformationMatrix normally gives an × matrix.
  • mat[[1;;n,1;;n]] gives the linear part of the transformation; mat[[1;;n,-1]] gives the displacement vector.

Examples

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Basic Examples  (1)

Here is defined to be a rotation around the axis:

Wolfram Language code: t = RotationTransform[θ, {0, 0, 1}]

Get the transformation matrix:

Wolfram Language code: m = TransformationMatrix[t]

The linear part:

Wolfram Language code: m[[1 ;; 3, 1 ;; 3]]

The displacement vector:

Wolfram Language code: m[[1 ;; 3, -1]]

Scope  (1)

Translation matrix in four dimensions:

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

Transformation of homogeneous coordinates:

Wolfram Language code: m.{x, y, z, w, 1}

Points at infinity do not change under translation:

Wolfram Language code: m.{x, y, z, w, 0}

Properties & Relations  (1)

The matrix of a general 2D affine transform:

Wolfram Language code: TransformationMatrix@AffineTransform[{Array[a, {2, 2}], Array[b, 2]}]//MatrixForm

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

Wolfram Language code: {t1, t2} = {LinearFractionalTransform[Array[a1, {3, 3}]], LinearFractionalTransform[Array[a2, {3, 3}]]};
Wolfram Language code: TransformationMatrix[Composition[t1, t2]] == TransformationMatrix[t1].TransformationMatrix[t2]
Wolfram Research (2007), TransformationMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/TransformationMatrix.html.

Text

Wolfram Research (2007), TransformationMatrix, Wolfram Language function, https://reference.wolfram.com/language/ref/TransformationMatrix.html.

CMS

Wolfram Language. 2007. "TransformationMatrix." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/TransformationMatrix.html.

APA

Wolfram Language. (2007). TransformationMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TransformationMatrix.html

BibTeX

@misc{reference.wolfram_2026_transformationmatrix, author="Wolfram Research", title="{TransformationMatrix}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/TransformationMatrix.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_transformationmatrix, organization={Wolfram Research}, title={TransformationMatrix}, year={2007}, url={https://reference.wolfram.com/language/ref/TransformationMatrix.html}, note=[Accessed: 12-June-2026]}