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Involutory


A linear transformation of period two. Since a linear transformation has the form,

 lambda^'=(alphalambda+beta)/(gammalambda+delta),
(1)

applying the transformation a second time gives

 lambda^('')=(alphalambda^'+beta)/(gammalambda^'+delta)=((alpha^2+betagamma)lambda+beta(alpha+delta))/((alpha+delta)gammalambda+betagamma+delta^2).
(2)

For an involutory, lambda^('')=lambda, so

 gamma(alpha+delta)lambda^2+(delta^2-alpha^2)lambda-(alpha+delta)beta=0.
(3)

Since each coefficient must vanish separately,

gamma(alpha+delta)=0
(4)
delta^2-alpha^2=0
(5)
beta(alpha+delta)=0.
(6)

Equation (5) requires delta=+/-alpha. Taking delta=alpha in turn requires that gamma=beta=0, giving lambda=lambda^', i.e., the identity map, while taking delta=-alpha gives delta=-alpha, so

 lambda^'=(alphalambda+beta)/(gammalambda-alpha),
(7)

which is the general form of a line involution.


See also

Cross Ratio, Line Involution

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References

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

Referenced on Wolfram|Alpha

Involutory

Cite this as:

Weisstein, Eric W. "Involutory." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Involutory.html

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