({{, (RefLink[LinearFractionalTransform, paclet:ref/LinearFractionalTransform])[m] 
gives a (RefLink[TransformationFunction, paclet:ref/TransformationFunction]) that represents a linear fractional transformation defined by the homogeneous matrix m. }, {, (RefLink[LinearFractionalTransform, paclet:ref/LinearFractionalTransform])[{a,b,c,d}]
represents a linear fractional transformation that maps r to (a.r+b)/(c.r+d). }})

OpenersNotesSection(Spacer1)Details

(RefLink[LinearFractionalTransform, paclet:ref/LinearFractionalTransform]) gives a (RefLink[TransformationFunction, paclet:ref/TransformationFunction]) that can be applied to vectors.

For ordinary linear fractional transforms in n dimensions, m is an (n+1)×(n+1) matrix.

(RefLink[LinearFractionalTransform, paclet:ref/LinearFractionalTransform]) in general supports s×(n+1) matrices for transformations in n dimensions.

In (RefLink[LinearFractionalTransform, paclet:ref/LinearFractionalTransform])[{a,b,c,d}], a is a matrix, b and c are vectors, and d is a scalar.

OpenersPrimaryExamplesSection(Spacer1)Examples  (4)

OpenersExampleSection0(Spacer1)Basic Examples  (1)

This creates the linear fractional transform (a.x+b)/(c.x+d):

a={{1,2},{3,4}};
b={5,6};
c={7,8};
d=17;

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

This is the corresponding formula:

t[{x,y}]

OpenersExampleSection1(Spacer1)Scope  (3)

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

LinearFractionalTransform[{{{1,2},{3,4}}, {5,6},{7,8}}]

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

LinearFractionalTransform[{{1,2,5},{3,4,6},{7,8,1}}]

Suppose you have a linear fractional transform t:

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

The inverse is computed by applying (RefLink[InverseFunction, paclet:ref/InverseFunction]):

s=InverseFunction[t]

This shows that s and t are inverses:

Composition[s,t]

This shows the same thing using formulas:

f=t[{x,y}]

g=s[{u,v}]

f/. Thread[{x,y}->g]//Simplify

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

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