14. Certain linear fractional transformations

If f(x) is given by the linear fractional transformation

${\displaystyle f(x)={\frac {(\cos \alpha )x-\sin \alpha }{(\sin \alpha )x+\cos \alpha }},}$

and similarly

${\displaystyle g(x)={\frac {(\cos \beta )x-\sin \beta }{(\sin \beta )x+\cos \beta }},}$

then

${\displaystyle f{\big (}g(x){\big )}=g{\big (}f(x){\big )}={\frac {{\big (}\cos(\alpha +\beta ){\big )}x-\sin(\alpha +\beta )}{{\big (}\sin(\alpha +\beta ){\big )}x+\cos(\alpha +\beta )}}.}$

More tersely stated, if for all α we let f_α be what we called f above, then

${\displaystyle f_{\alpha }\circ f_{\beta }=f_{\alpha +\beta }.\,}$

If x is the slope of a line, then f(x) is the slope of its rotation through an angle of −α.