Involutory

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

[画像: lambda^'=(alphalambda+beta)/(gammalambda+delta), ]

(1)

applying the transformation a second time gives

[画像: 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) =

(4)

delta^2-alpha^2 =

(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

[画像: lambda^'=(alphalambda+beta)/(gammalambda-alpha), ]

(7)

which is the general form of a 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.

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Involutory

Cite this as:

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

Involutory

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