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哦, 我明白了:
因为整个$\mathbb CP^1$就是一射影直线, 无论怎么变换都“保持共线性”
$z\mapsto1/\bar z$保持共线性(collineation)且不是射影变换(homography)
The operation of taking the complex conjugate in the complex plane amounts to a reflection in the real line.
With the notation $z^∗$ for the conjugate of $z$, an anti-homography is given by
$$ f(z)={\frac {az^{*}+b}{cz^{*}+d}}. $$
Thus an anti-homography is the composition of conjugation with a homography, and so is an example of a collineation which is not an homography. For example, geometrically, the mapping $ f(z)=1/z^{*} $ amounts to circle inversion. The transformations of inversive geometry of the plane are frequently described as the collection of all homographies and anti-homographies of the complex plane.
As mentioned by Blaschke and Klein, Michel Chasles preferred the term homography to collineation. A distinction between the terms arose when the distinction was clarified between the real projective plane and the complex projective line. Since there are no non-trivial field automorphisms of the real number field, all the collineations are homographies in the real projective plane, however due to the field automorphism of complex conjugation, not all collineations of the complex projective line are homographies. In applications such as computer vision where the underlying field is the real number field, homography and collineation can be used interchangeably.
Line-preserving transformations
A transformation of a projective plane which preserves lines is called a collineation. In every projective plane, a projective transformation will be a collineation. In the common Euclidean plane with the real numbers as the underlying field, the converse can be shown as well: every collineation is a projective transformation. If the underlying field has a non-trivial automorphism, the way the complex numbers do, this is no longer true. For example, one can compute the complex conjugate of every coordinate, thus preserving lines although that operation is not a projective transformation. |
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Czhang271828
发表于 2023-5-22 22:57
