# Rational functions and Mobius transformations

## Mobius and his transformations

The answer leads us to an interesting family of rational functions introduced by the prominent German geometer August Ferdinant Mobius. Mobius also was an architect of the revival of projective geometry in the 19th century.He introduced a family of functions or transformations that are quotients of one linear polynomial over another linear polynomial. An example of such a*Mobius transformation*, also called a

*fractional linear transformation*, is \[\Large{y=\frac{5x-1}{3x+2}}.\]Both the numerator and denominator are linear, or constant, polynomials.We see that this graph indeed looks something like \(\normalsize y=1/x\), or more precisely \(\normalsize y=-1/x\), which in addition has been translated in the plane.Why do we use the word transformation here and not function? Mobius, being a geometer more than an analyst, was interested in thinking about functions dynamically and geometrically, as moving points around. Hence the word transformation.

## A connection with linear algebra

The general case of a Mobius transformation is \[\Large{y=\frac{ax+b}{cx+d}}.\]Such a function can be specified by a*matrix*\[\Large{\begin{pmatrix} a & b\\ c & d \end{pmatrix}}\]and there is a natural connection with elementary linear algebra, as we shall see when we investigate composition of Mobius transformations.Mobius transformations are special cases of

*rational functions*, which are quotients of polynomials.

#### Maths for Humans: Linear, Quadratic & Inverse Relations

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