Such a transformation is the most general form of conformal mapping of a domain. This group can be given the structure of a complex manifold in such a way that composition and inversion are holomorphic maps.
Note that the fixed points are counted here with multiplicity ; the parabolic transformations are those where the fixed points coincide. Either or both of these fixed points may be the point at infinity. The roots are. Parabolic transforms have coincidental fixed points due to zero discriminant. For c nonzero and nonzero discriminant the transform is elliptic or hyperbolic. This corresponds to the situation that one of the fixed points is the point at infinity.
In this case the transformation will be a simple transformation composed of translations , rotations , and dilations :. Thus any map that fixes at least 3 points is the identity.
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Indeed, any member of the general linear group can be reduced to the identity map by Gauss-Jordan elimination, this shows that the projective linear group is path-connected as well, providing a homotopy to the identity map. The Lefschetz—Hopf theorem states that the sum of the indices in this context, multiplicity of the fixed points of a map with finitely many fixed points equals the Lefschetz number of the map, which in this case is the trace of the identity map on homology groups, which is simply the Euler characteristic.
This corresponds to the fact that the Euler characteristic of the circle real projective line is 0, and thus the Lefschetz fixed-point theorem says only that it must fix at least 0 points, but possibly more. We first treat the non-parabolic case, for which there are two distinct fixed points. If one of them is already at infinity then g can be modified so as to fix infinity and send the other point to 0. The fixed point equation for the transformation f can then be written. Observe that, given an ordering of the fixed points, we can distinguish one of the multipliers k of f as the characteristic constant of f.
Reversing the order of the fixed points is equivalent to taking the inverse multiplier for the characteristic constant:. The fixed point formula for a parabolic transformation is then. From the above expressions one can calculate:. The point midway between the two poles is always the same as the point midway between the two fixed points:.
These four points are the vertices of a parallelogram which is sometimes called the characteristic parallelogram of the transformation. Its characteristic polynomial is equal to. Then these functions can be composed , giving. That is, define functions g 1 , g 2 , g 3 , g 4 such that each g i is the inverse of f i. Then the composition.
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For example, the preservation of angles is reduced to proving that circle inversion preserves angles since the other types of transformations are dilation and isometries translation, reflection, rotation , which trivially preserve angles. A generalized circle is either a circle or a line, the latter being considered as a circle through the point at infinity. Even if it maps a circle to another circle, it does not necessarily map the first circle's center to the second circle's center.
The cross ratio of four different points is real if and only if there is a line or a circle passing through them. Two points z 1 and z 2 are conjugate with respect to a generalized circle C , if, given a generalized circle D passing through z 1 and z 2 and cutting C in two points a and b , z 1 , z 2 ; a , b are in harmonic cross-ratio i. This property does not depend on the choice of the circle D.
This property is also sometimes referred to as being symmetric with respect to a line or circle. Two points are conjugate with respect to a circle if they are exchanged by the inversion with respect to this circle. With every invertible complex 2-by-2 matrix. In other words, the map.
John Stillwell's book Naive Lie Theory motivates the subject in the first chapter using an exploration of quaternions acting by conjugation on the unit sphere. I think his explanation is great for someone at any point in their undergraduate career, however the subsequent chapters may be extremely slow for anyone already acquainted with lie theory, group theory or general topology. There is a much more in depth coverage of the details of this in another book which I am trying to recall the name of. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered.
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