Projectivities with fixed points on every line of the plane
Projectivities with fixed points on every line of the plane
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平面每条线上有固定点的投影率
DOI：
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发表时间：
1946
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通讯作者：
R. Baer
中科院分区：
文献类型：
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作者：
R. Baer
射影的固定元素系统包含任意两点的连线和任意两条连线的交点。因此，它在结构上非常像所考虑的平面的一个子平面;因此，人们可以预期投射性的结构将由其固定元素系统的结构所支配，只要这个系统不是“太小”。“为了证实这一点，我们建议在本说明中研究一类投射性，我们称之为准透视性。它们的特点是每一条线都带有一个固定点，或者等价地，每一个点都在某条固定线上。每一个透视性都是一个拟透视性，而一个拟透视性不是透视性当且仅当它的固定元素系是一个射影子平面。对合也是准透视性，如果帕普斯定理在所考虑的平面上有效，则每个准透视性都是透视性或对合。但是在真实的四元数域上的射影平面中已经存在既不是透视也不是对合的准透视，并且我们给出了Despermesian射影平面中的准透视的一个完整的综述。我们的结果变得特别引人注目的情况下，有限的射影平面。如果这样一个平面上的每条直线都有n + 1个点，那么我们可以证明不存在恰好有n个不动点的射影，一个射影是拟透视性当且仅当它的不动点的个数至少是w + 1，并且它是透视性当且仅当它的不动点的个数是w+1或n+2。如果一个拟透视性不是透视性，则n=i，其中i+1是一条固定直线上的不动点的个数。以下符号将在整个过程中使用。我们考虑一个射影平面II，其中笛沙格定理可能成立，也可能不成立。如果P和Q是II中的两个不同点，则P+Q是通过P和Q的唯一确定的直线;如果h和k是两条不同的直线，则hk是它们相交的唯一确定的点。投射性<f>是物体之间的1：1和穷举对应。
The system of fixed elements of a projectivity contains with any two points the line connecting them and with any two lines their intersection. It is, therefore, in its structure very much like a subplane of the plane under consideration ; and thus one may expect the structure of the projectivity to be dominated by the structure of the system of its fixed elements, provided this system is not "too small." To substantiate this we propose to investigate in this note a class of projectivities which we term quasi-perspectivities. They are characterized by the property that every line carries a fixed point or, equivalently, that every point is on some fixed line. Every perspectivity is a quasi-perspectivity, and a quasi-perspectivity is not a perspectivity if, and only if, the system of its fixed elements is a projective subplane. Involutions are quasi-perspectivities too, and if the Theorem of Pappus is valid in the plane under consideration, then every quasi-perspectivity is a perspectivity or an involution. But already in the projective plane over the field of real quaternions there exist quasi-perspectivities which are neither perspectivities nor involutions, and we give a complete survey of the quasi-perspectivities in Desarguesian projective planes. Our results become particularly striking in the case of finite projective planes. If every line in such a plane carries n + 1 points, then we may show that there do not exist projectivities possessing exactly n fixed points, that a projectivity is a quasi-perspectivity if, and only if, the number of its fixed points is at least w + 1, and that it is a perspectivity if, and only if, the number of its fixed points is w+1 or n+2. If a quasi-perspectivity is not a perspectivity, then n=i where i+1 is the number of fixed points on a fixed line. The following notations will be used throughout. We consider a projective plane II in which the Theorem of Desargues may or may not hold. If P and Q are two different points in II, then P+Q is the uniquely determined line passing through P and Q; if h and k are two different lines, then hk is the uniquely determined point in which they meet. A projectivity <f> is a 1:1 and exhaustive correspondence between the