Monday, December 5, 2022

# Fundamental Theorem Of Affine Geometry

## Title: The Fundamental Theorem Of Affine Geometry In $^n$

Fundamental Theorem of Affine Geometry (Problems) | 5th BSC maths | Geometry | Calicut University

Abstract: Let $L^0$ be the algebra of equivalence classes of real valued randomvariables on a given probability space, and $^n$ the $n$-ary Cartesianpower of $L^0$ for each integer $n\geq 2$. We consider $^n$ as a freemodule over $L^0$ and study affine geometry in $^n$. One of our mainresults states that: an injective mapping $T: ^n\to ^n$ which islocal and maps each $L^0$-line onto an $L^0$-line must be an $L^0$-affinelinear mapping. The other main result states that: a bijective mapping $T:^n\to ^n$ which is local and maps each $L^0$-line segment onto an$L^0$-line segment must be an $L^0$-affine linear mapping. These results extendthe fundamental theorem of affine geometry from $\mathbb R^n$ to $^n$.

## Title: The Fundamental Theorems Of Affine And Projective Geometry Revisited

Abstract: The fundamental theorem of affine geometry is a classical and useful result.For finite-dimensional real vector spaces, the theorem roughly states that abijective self-mapping which maps lines to lines is affine. In this note weprove several generalizations of this result and of its classical projectivecounterpart. We show that under a significant geometric relaxation of thehypotheses, namely that only lines parallel to one of a fixed set of finitelymany directions are mapped to lines, an injective mapping of the space must beof a very restricted polynomial form. We also prove that under mild additionalconditions the mapping is forced to be affine-additive or affine-linear. Forexample, we show that five directions in three dimensional real space sufficeto conclude affine-additivity. In the projective setting, we show that n+2fixed projective points in real n-dimensional projective space , through whichall projective lines that pass are mapped to projective lines, suffice toconclude projective-linearity.

Subjects:

## Applications Of The Fundamental Theorems Of Affine And Projective Geometry

The fundamental theorem of affine/projective geometry says that a bijection between two finite dimensional spaces that preserves the relation of collinearity is a affine/projective isomorphism.

Fundamental Theorem of Affine geometry let $X,X’$ be two finite dimensional affine spaces over two fields $K,K’$ of same dimension $d\geq 2$, and let $f:X\to X’$ be a bijection that sends collinear points to collinear points, i.e. such that for all $a,b,c\in X$ that are collinear, $f,f,f$ are collinear too. Then $f$ is a semi-affine isomorphism.

This means that there is a field isomorphism $\sigma:K\to K’$ such that for any point $a\in X$ the map induced by $f_a: X_a\to X’_$ is a $\sigma$-semi-linear isomorphism.

Fundamental Theorem of Projective geometry let $P,P$ be two finite dimensional projective spaces over two fields $K,K’$ of same dimension $d\geq 2$, and let $f:P\to P$ be a bijection that sends collinear points to collinear points. Then $f$ is a semi-linear isomorphism.

What are some applications of this? I remember that you can use the projective verion of this to prove that any automorphism of $SO$ is given by conjugation with some orthogonal matrix. Are there some other beautiful applications?

In the time since I asked this question I have tried to find some applications of this. If one wishes to apply this theorem one needs to be in a situation where there naturally arise lines. I thought of two such situations:

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## The Fundamental Theorem Of Affine Geometry In Regular L0

be a probability space and L the algebra of equivalence classes of real-valued random variables defined on is said to be regular if x = y for any given two elements x and y in M such that there exists a countable partition { is the characteristic function of A A n its equivalence class. The purpose of this paper is to establish the fundamental theorem of affine geometry in regular L -modules such that V contains a free L -submodule of rank 2, if T is stable and invertible and maps each L -line segment, then T must be L

## Theorem \ Fundamental Theorem Of Affine Geometry

Suppose an affine transformation maps a nondegenerate triangle \ to a triangle \. Then \ is nondegenerate, and

Proof

Since an affine transformation maps lines to lines, the triangle \ is nondegenerate.

If \ or \, then the second statement follows directly from the proposition. Otherwise consider points \ and \ defined by

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## Who First Proved The Fundamental Theorem Of Projective Geometry

The following theorem is often called the fundamental theorem of projective geometry:

Let $k$ be a field and let $n \geq 3$. Let $X$ be the partially ordered set of nonzero proper subspaces of $k^n$. Then every poset automorphism of $X$ is induced by a semi-linear automorphism of $k^n$, i.e. a set map $f:k^n\rightarrow k^n$ for which there exists a field automorphism $\tau:k \rightarrow k$ such that$$f = \tau f + \tau f$$for all $c_1,c_2 \in k$ and $\vec_1,\vec_2 \in k^n$.

Question: who was the first person to prove this, and where does their proof appear? I know it has its origins in 19th century work of von Staudt, but I don’t think that the above theorem appears in his work. On page 52 of Baer’s book “Linear Algebra and Projective Geometry”, he says that the first proof was due to Kamke, but he does not give a reference.

The version you state is definitely a 20th century development, only marginally related to Von Staudt’s theorem. Here is a translation of the relevant section of Karzel & Kroll’s Geschichte der Geometrie seit Hilbert, p. 51 :

One might add that Darboux already states and attributes the theorem:

It is easy for example to recognize with v. Staudt that a projective or homographic correspondence in the plane or in space can be defined by the sole condition that aligned points in one figure correspond to aligned points in the other.

He only sketches a proof, and Schur comments:

## Local Fundamental Theorem Of Affine Geometry

The fundamental theorem of affine geometry implies that a bijection $f \colon \mathbb R^n \to \mathbb R^n$ is affine if $n \geq 2$ and $f$ maps lines to lines. I was wondering, does this hold locally? Suppose $\Delta$ is a simplex of dimension $n$, $f \colon \Delta \to \Delta$ a bijection mapping each vertex to itself, and mapping segments to segments, can we readily say that $f = \textrm_\Delta$?

• 2$\begingroup$It seems it suffices to prove it for $n=2$, as you can easily proceed by induction from there.$\endgroup$ This site has become a dump.Aug 27, 2021 at 21:58
• $\begingroup$Correct, if it’s true for some $n$, then each “face” is mapped identically, any inside point is at the intersection of the image of the segments whose edges are on the boundary, passing through the considered point, hence the proposition holds for $n+1$.$\endgroup$

No, this is not true. Let us say $\Delta=\:\sum_i x_i=1\}$. Let $a_0,\dots,a_n> 0$ be any sequence of positive constants and define $g=$. Then there is a bijection $f:\Delta\to\Delta$ given by $f=g/s)$ where $s)$ denotes the sum of the entries of $g$. This bijection fixes the vertices of $\Delta$ and is not the identity unless all the $a_i$ are the same. However, I claim that it sends segments to segments.

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## How To Show That The Only Affine Automorphism Of $\mathbb$ Is The Identity Using Fundamental Thorem Of Affine Geometry

Let the set of real numbers $\mathbb$ be endowed with the natural affine structure . The question is how to prove that the only affine automorphism in affine space $\mathbb$ is the identity, using the fundamental theorem of affine geometry. The affine automorphism is an affine map which is an automorphism. The fundamental theorem of affine geometry states as follows:

Let $X,X’$ be affine spaces of the same affine dimension $\ge2$. Let $f:X\to X’$ be a set-theoretical bijection which takes any three collinear points in $X$ into collinear points in $X’$. Then $f$ is semiaffine.

I have no idea know how to relate the fundamental theorem of affine geometry to the problem in question. In particular, the affine space $\mathbb$ has affine dimension $1$, not $\ge2$ as stipulated in the theorem. What contradiction can I get if I assume there is a non-identity automorphism in affine space $\mathbb$? I would appreciate your help greatly if you can direct me how to use this powerful fundamental theorem.

PS, for reference, the screenshot of the proposition and the fundamental theorem is copied below:

The statements about $\mathbb$ and $\mathbb$ in the Proposition are about field automorphisms. The point is that

• It is well known that the only field automorphism of $\mathbb$ is the identity .
• We can view $\mathbb$ as a 2-dimensional affine geometry over $\mathbb$.