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LinearFractionalTransform[m]

gives a TransformationFunction that represents a linear fractional transformation defined by the homogeneous matrix m.

LinearFractionalTransform[{a,b,c,d}]

represents a linear fractional transformation that maps to .

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LinearFractionalTransform

LinearFractionalTransform[m]

gives a TransformationFunction that represents a linear fractional transformation defined by the homogeneous matrix m.

LinearFractionalTransform[{a,b,c,d}]

represents a linear fractional transformation that maps to .

Details

Examples

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Basic Examples  (1)

This creates the linear fractional transform :

Wolfram Language code: a = {{1, 2}, {3, 4}}; b = {5, 6}; c = {7, 8}; d = 17;
Wolfram Language code: t = LinearFractionalTransform[{a, b, c, d}]

This is the corresponding formula:

Wolfram Language code: t[{x, y}]

Scope  (3)

If the scalar d is omitted, it is taken to be 1:

Wolfram Language code: LinearFractionalTransform[{{{1, 2}, {3, 4}}, {5, 6}, {7, 8}}]

A single matrix is taken to be the homogeneous representation of the transform:

Wolfram Language code: LinearFractionalTransform[{{1, 2, 5}, {3, 4, 6}, {7, 8, 1}}]

Suppose you have a linear fractional transform t:

Wolfram Language code: t = LinearFractionalTransform[{{1, 2, 5}, {3, 4, 6}, {7, 8, 1}}]

The inverse is computed by applying InverseFunction:

Wolfram Language code: s = InverseFunction[t]

This shows that s and t are inverses:

Wolfram Language code: Composition[s, t]

This shows the same thing using formulas:

Wolfram Language code: f = t[{x, y}]
Wolfram Language code: g = s[{u, v}]
Wolfram Language code: f /. Thread[{x, y} -> g]//Simplify
Wolfram Research (2007), LinearFractionalTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/LinearFractionalTransform.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_linearfractionaltransform, author="Wolfram Research", title="{LinearFractionalTransform}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/LinearFractionalTransform.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_linearfractionaltransform, organization={Wolfram Research}, title={LinearFractionalTransform}, year={2007}, url={https://reference.wolfram.com/language/ref/LinearFractionalTransform.html}, note=[Accessed: 12-June-2026]}