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A Linear Transformation of period two. Since a Linear Transformation has the form,

\lambda'={\alpha \lambda+\beta\over\gamma\lambda+\delta},
\end{displaymath} (1)

applying the transformation a second time gives
\end{displaymath} (2)

For an involuntary, $\lambda''=\lambda$, so
\end{displaymath} (3)

Since each Coefficient must vanish separately,
$\displaystyle \alpha\gamma+\gamma\delta$ $\textstyle =$ $\displaystyle 0$ (4)
$\displaystyle \delta^2-\alpha^2$ $\textstyle =$ $\displaystyle 0$ (5)
$\displaystyle \alpha\beta+\beta\delta$ $\textstyle =$ $\displaystyle 0.$ (6)

The first equation gives $\delta=\pm\alpha$. Taking $\delta=\alpha$ would require $\gamma=\beta=0$, giving $\lambda=\lambda'$, the identity transformation. Taking $\delta=-\alpha$ gives $\delta=-\alpha$, so
\end{displaymath} (7)

the general form of an Involution.

See also Cross-Ratio, Involution (Line)


Woods, F. S. Higher Geometry: An Introduction to Advanced Methods in Analytic Geometry. New York: Dover, pp. 14-15, 1961.

© 1996-9 Eric W. Weisstein