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Questions tagged [affine-geometry]

For questions about algebraic geometry that focus on affine space. For affine mappings in linear algebra (i.e. linear mappings plus translations), please use the linear-algebra tag or another appropriate tag.

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Prove continuity of the affine extension mapping between geometric simplicial complexes

Let $\Delta_1$ and $\Delta_2$ be geometric simplicial complexes. Let $K_1$ and $K_2$ be their associated abstract simplicial complexes. Let $f: V(K_1) \to V(K_2)$ be a simplicial mapping. We define ...
Nikolas's user avatar
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1 answer
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Is there a function whose maximizers remain the same after any affine transformations?

Let $f: \mathbb{R_+}^n\to \mathbb{R_+}$ be a function that is strictly increasing in each of its arguments. Let $M_f$ be the set of its maximizers on some fixed compact subset $D\subseteq \mathbb{R_+}^...
Erel Segal-Halevi's user avatar
1 vote
0 answers
38 views

an isometry preserving an equilateral triangle will preserve its vertices

Prove that an isometry preserving an equilateral triangle will preserve its vertices, i.e if $\Delta ABC$ is an equilateral triangle and $f$ is an isometry s.t $f(\Delta ABC)=\Delta ABC$ then $\{A,B,C\...
Alex Nguyen's user avatar
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What affine transformation does the projective transformation correspond to?

In the projective space $P^3(\mathbb{K})$ with the frame $(S_0,S_1,S_2,S_3;E)$, consider a projective transformation as follows: $$\begin{cases} tx_0'=x_0\\ tx_1'=-x_1\\ tx_2'=-x_2\\ tx_3'=x_3.\end{...
Alex Nguyen's user avatar
-2 votes
1 answer
64 views

A diophantine equation with no solution in positive integers $x,y$ i.e $(y(y+1)+1)^2+1\neq 100x$

Hi I ask separately a question regarding the question where I sktech a special case of the Brocard-Ramanujan problem : Problem : Let $x,y$ be positive integers shows that : $$(y(y+1)+1)^2+1=100x\...
Ranger-of-trente-deux-glands's user avatar
2 votes
2 answers
51 views

Affine Map as a Morphism of Affine Vector Spaces

I've recently took interest in morphism and category theory and I'm amazed how it offers a very general notion. However, I'm struggling to apply this for the affine vector spaces. I've seen that a ...
Lyders's user avatar
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-1 votes
2 answers
75 views

Jensen's inequality with affine combination

From page 217 of 'Convex functions' by Arthur Wayne Roberts, Dale Varberg (exercise F) i want proof that: $f:\mathbb{R}\longrightarrow\mathbb{R}$ is affine iff $$ f\biggl(\displaystyle\sum\limits_{i=1}...
user791759's user avatar
0 votes
2 answers
60 views

Projective varieties contained in dense open subsets

Let $X$ be a smooth irreducible projective variety over the complex numbers. Let $U$ be a nontrivial dense open subset of $X$. Does there exist a projective curve $C$ inside $U$? My attempt: Let's ...
cupoftea's user avatar
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One place at infinity

I am currently reading a book titled "Numerical Semigroups," which can be found here. I have a question regarding a definition provided by the author. To begin, let's define $F=y^n+a_1(x)y^{...
Mousa hamieh's user avatar
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Weight of a monomial

I have a question for the mathematicians in affine algebraic geometry: Given an algebraically closed field $k$, we define the projective $n$-space as the quotient space $\mathbb{P}^n = (k^{n+1} - \{0_{...
Mousa hamieh's user avatar
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1 answer
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Midpoint polygon for odd $n$

Given $n$ distinct points $M_1, M_2, M_3, \dots, M_n$ in an affine space $U$; in case $n$ is odd, it's possible to find $A_1,\dots,A_n \in U$ such that $M_1$ is the midpoint of $\overline{A_1A_2}$, $...
J P's user avatar
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1 answer
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An ellipse $\mathcal{E}$ touches a fixed ellipse $\mathcal{C}$ at $A$, prove the length of semi-major axis of $\mathcal{E}$ is constant

$\mathcal{C}$ is an ellipse with center $O$ and semi-major axis length $=a$, semi-minor axis length $|OB|=b$. $A$ is a point moving on $\mathcal{C}$. $E$ is a point on $\mathcal{C}$ such that $OE$ is ...
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1 answer
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How to recognize affinely dependent or independent?

I have been trying to understand example of affinely dependent (AD) or affinely independent(AI). But from the above images how are leftmost two figures AI and is rightmost AD? I am not able to ...
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3 answers
79 views

Why affinely independent points in $\mathbb{R^d}$ is $d+1$?

Below text quoted from Jiff Matousek book: Affine dependence of $a_1 ,\dots, a_n$ is equivalent to linear dependence of the $n-1$ vectors $a_1 - a_n, a_2 - a_n, \dots, a_{n-1}-a_n$ . Therefore, ...
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Find all points M such that MA<MB.

In the Euclidean affine plane $\mathbb{E}$, let $A$ and $B$ be two distinct points. Prove that the set {$M \in \mathbb{E} \mid MA < MB$} is a half-plane determined by the perpendicular bisector of ...
Thanh Thanh's user avatar
1 vote
1 answer
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Are there affine rings that are properly affine?

Let $A$ be an abelian group. If $B$ is a nonempty set equipped with a transitive free action of $A$ (written additively), say that $B$ is an affine abelian group (equivalently, $B$ is an $A$-principal ...
Smiley1000's user avatar
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1 vote
1 answer
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Is it possible to extract the translation of an affine transformation matrix independent of rotation center and angle?

I have $2$ images rotated by $60^\circ$ to each other with different center of rotation. The here presented matrices are affine transformation matrices derived from OpenCV: https://docs.opencv.org/4.x/...
TMul's user avatar
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$AG(2,2)$ and $AG(3,2)$ and their resp. triangular (Fano plane-like) and tetrahedral ($PG(3,2)$-like) representations. Are my assumptions correct?

From the "Finite affine planes" section of the Wikipedia article "Affine plane (incidence geometry)", in a (finite) affine plane of order $n$: each line contains $n$ points, each ...
Kevin M. Lamoreau's user avatar
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1 answer
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If $M_1 \cup M_2 $ is a hyperplane in $\mathbb{R}^n$ then $M_1 = M_2$.

Let $M_1$ and $M_2$ be affine subspaces of $\mathbb{R}^n$ such that they are not proper subsets of each other. If $M_1 \cup M_2 $ is a hyperplane in $\mathbb{R}^n$ then $M_1 = M_2$. What I did is as ...
bruno's user avatar
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2 answers
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Let $M_1$ and $M_2$ be affine subspaces of $\Bbb R^n$ with $M_1 \cap M_2 \neq \emptyset$ and $\dim(M_1 \cap M_2)=\dim(M_1)$. Then $M_1 \subset M_2$.

Let $M_1$ and $M_2$ be affine subspaces of $\mathbb{R}^n$ with $M_1 \cap M_2 \neq \emptyset$ and $\dim(M_1 \cap M_2)=\dim(M_1)$. Then $M_1 \subset M_2$. I tried to prove by contradiction and took an ...
bruno's user avatar
  • 425
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1 answer
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Canonical isomorphism between tangent space and translation vector space of affine space

I've a question about the following. Take an affine space $(E,V)$ and consider the tangent space $T_aE$ at a point $a$. From John Lee book "Introduction to smooth manifolds" chapter 3 there ...
Carlo C's user avatar
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3 votes
2 answers
253 views

Why is maximum number of joints of 6 lines is 4?

The following is considered in Lary Guth's Polynomial Methods in Combinatorics, page 14. Let $L$ be a set of lines in $\mathbb R^3$. A point $x$ which lies in some set of three non-co-planar lines of $...
frelg's user avatar
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Rotate and Scale Around Different Origins using Local Coordinates

How can I chain rotate and scale operations (each with different origins) while keeping each operation origin in relation to the original local component space? Using matrix denotation (i.e. T for ...
Cole Perschon's user avatar
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0 answers
44 views

Dominant rational maps and dimension of affine varieties

Say we have $f : X \dashrightarrow Y$ as a dominant rational map between two affine varieties. Is it necessarily true that $\dim Y \leq \dim X$? $f$ being dominant means that we must have that $f(\...
Jeff's user avatar
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2 votes
1 answer
117 views

Shortest distance between two affine subspaces through orthogonal projection

I'm trying to show the following: Let $V$ be a finite dimensional euclidean vector space, with two vector subspaces $S_1 \subset V$ and $S_2 \subset V$. Suppose that $X, Y$ are affine subspaces with $...
MaChaeHa's user avatar
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0 answers
51 views

Strong Seperation between hypreplane and an affine subspace

Let $h \in \mathbb{R}^n\backslash \{0\}$ and $r \in \mathbb{R}$. A sure that $M$ is an affine subspace of $\mathbb{R}^n$ with $H(h, r) \cap M=\phi$. I tried to use Banach separation theorem to show ...
bruno's user avatar
  • 425
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0 answers
35 views

Show that an affine subspace of $\mathbb{R}^d$ is a Borel set.

Everything is in the title. I was thinking of using the fact that an affine subspace of $\mathbb{R}^d$ is the set of solutions to a system of linear equations, but I can't figure out how to describe ...
Alex's user avatar
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0 answers
34 views

Having Trouble Formalizing a Combinatorial Argument

Let $\{X_i\}_{i = 1} ^d$ be symmetric linearly independent i.i.d random vectors in $\mathbb{R}^n$. I wish to show $$ \mathbb{P}\{ 0 \in \mathrm{conv}\{ X_i \}_{i \in I}| 0 \in \mathrm{aff}\{ X_i \}_{...
Partial T's user avatar
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-1 votes
1 answer
97 views

A quadric has the origin as centre if and only if its equation has no terms of degree $1$ [closed]

I'd like to know how to prove that if a(n affine) quadric of A^n (regarded as the affine space canonically associated to K^n, being a field) has the origin as centre, then its equation has no terms of ...
Amanda Wealth's user avatar
1 vote
0 answers
94 views

Does unique convex combination imply affine independence?

Why we need affine independent to ensure the unique representation of a vector from convex hull. As far as I understood, the converse of the theorem is not true. In other words, if any vector 𝑣 is in ...
bruno's user avatar
  • 425
1 vote
1 answer
61 views

Hausdorff-ness of irreducible affine algebraic sets [closed]

In what case is $X$, an irreducible affine algebraic set not a Hausdorff topological space with respect to the Zariski topology? My understanding is that this would only occur if $X$ is infinite, ...
Jeff's user avatar
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0 votes
1 answer
63 views

Prove $S = \{ (a,b) \in \mathbb{A}^2 \ | \ a \overline{a} + b \overline{b} = 1\}$ is not Zariski closed

I'm working on a problem where I have to prove that Consider the subset $S$ of $\mathbb{A}^2$, the affine space over $\mathbb{C}$, defined by $$ S = \{ (a,b) \in \mathbb{A}^2 \mid a \overline{a} + ...
Oopsilon's user avatar
  • 129
0 votes
1 answer
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Projective closure as a closure operator

I am studying basic projective and affine geometry and I've come across the notion of projective closure (by now, only in spaces obtained from K^n by projectivisation). It has been defined (somewhat ...
Amanda Wealth's user avatar
-1 votes
1 answer
94 views

Proposition about the union of affine subspaces

NOTE: I will use the symbol $"Y<X"$ to denote $Y$ is an affine subspace of $X$ I have this proposition: Let $X$ be an affine space. If $M<X$ and $N<X$ such that $M \cup N < X$ $\...
Juanma's user avatar
  • 89
2 votes
1 answer
119 views

Affine map maps simplex onto a simplex

I am reading John Lee's Introduction to Topological Manifolds and I am trying to prove Theorem 5.39 (Simplicial Maps are determined by vertex maps. Let $K$ and $L$ be simplicial complexes. Suppose $...
nomadicmathematician's user avatar
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0 answers
25 views

Affine subspace parallel proof (wanting a second opinion)

Let $X$ an affine space and $S,T$ two affine subspace of $X$. I want your opinion on my proof about the statement If $S \cap T = \emptyset$ its okay. If $S \cap T \neq \emptyset$, we have $P \in S \...
Tohiea's user avatar
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0 answers
54 views

Only one $(m+1)$-plane containing $m$-plane and a given point

In the Affine space $A$, given $m-$plane $\alpha$ and a point $P\notin\alpha$. Prove that there is only one $(m+1)-$plane containing $\alpha$ and $P$. I can prove through $m+1$ independent points, ...
user avatar
2 votes
1 answer
105 views

Affine vs projective from a higher standpoint

While studying geometry, I came to know that a projective space (of dimension n associated to a vector space over a field K) can be seen as an affine space (of the same dimension and on the same field)...
Amanda Wealth's user avatar
1 vote
0 answers
67 views

Affine transformations. Reflection

I'm trying to develop a programme, which transforms a 2d image. I'm trying to create a simple reflection. Based on a theory affine transformation is determined as follows: Given a transformation ...
Jack Havis's user avatar
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0 answers
23 views

Affiness, $U_{2,4}$ and $M(K_4).$

I do not know why $M(K_4)$ is not affine over $GF(2)$ or $GF(3)$ but it is affine over all fields with more than 3 elements. I proved that $U_{2,4}$ is $\mathbb F$-representable iff $|\mathbb F| \geq ...
Intuition's user avatar
  • 3,181
1 vote
0 answers
23 views

Characteristic polynomial and bounded regions.

I know that the number of bounded regions of a homogeneous hyperplane arrangement $\mathcal{A}$(a collection of n hyperplanes) in $\mathbb R^d$is ${ n - 1 \choose d}$ but how can this be expressed in ...
Intuition's user avatar
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0 votes
1 answer
51 views

Understanding contraction in hyperplane arrangements.

Here are two figures that shows a hyperplane arrangement's contraction (this is from McNulty book, "Matroids, a geometric introduction"): I am not sure why a became a line in the right ...
Intuition's user avatar
  • 3,181
0 votes
1 answer
48 views

When are projective, affine geometries uniform matroids?

I am trying to understand the following corollary in James Oxley book: A simple rank-r matroid M that is representable over $GF(q)$ has at most $\frac{q^r - 1}{q - 1}$ elements. Moreover, if $|E(M)| = ...
Intuition's user avatar
  • 3,181
2 votes
0 answers
56 views

Some properties in Projective Geometry

I am trying to understand the following two Propositions in James Oxley's book "Matroid Theory" Prop. 6.1.3 Let $M$ be a simple rank-r matroid and $\mathbb F$ be a field. The following ...
Hope's user avatar
  • 95
1 vote
1 answer
26 views

Vandermonde hyperplanes: affine general position?

let $K$ be a field (e.g., a finite field). Fix a dimension $n$, and, for $m > n$, $m$ distinct elements $v_1, \ldots , v_m$ of $K$. consider the following Vandermonde-like matrix: \begin{equation} \...
BD107's user avatar
  • 611
1 vote
1 answer
56 views

affine geometries that are self-dual matroids.

I want to know which of the affine geometries $AG(n,q)$ are self-dual matroids? I have proved before that uniform matroids $U_{n,m}$ that are self duals are those who satisfies that $m = 2n.$ I am ...
Intuition's user avatar
  • 3,181
2 votes
1 answer
160 views

a full proof for the Fundamental Theorem of affine geometry

Let $n\geq 2$ and a bijection $T:\ \mathbb R^n\longrightarrow\mathbb R^n$ satisfies $\ T(0)=0\,$ and $\,T$ maps straight lines to straight lines. Then $T$ is a linear map. My proof : Let $(P)\subset\...
PermQi's user avatar
  • 579
12 votes
2 answers
355 views

If $T:\ \mathbb R^n\longrightarrow\mathbb R^n$ is additive then $T$ is linear

Let the bijection $T:\, \mathbb R^n\longrightarrow\mathbb R^n,\ n\geq2$ satisfy \begin{align} T(u+v)=T(u)+T(v),\ \forall u,v\in\mathbb R^n\tag1 \end{align} and \begin{align} T\big(\langle u\rangle\big)...
PermQi's user avatar
  • 579
1 vote
1 answer
517 views

Affine function $\mathbb{R}^n \to \mathbb{R}^n$ preserves angles if it maps a ball to a ball

I'm finding it difficult to formulate my intuition for the following problem: Let $\phi: \mathbb{R}^n \to \mathbb{R}^n$ be an affine function such that $\phi(B_1) = B_2$ for two balls in $\mathbb{R}^...
Anon's user avatar
  • 1,791
0 votes
0 answers
64 views

If $T$ maps the lines to the lines then $T$ is additive

Let $T:\, \mathbb R^n\to\mathbb R^n\ \, (n\geq 2)$ be a bijective map that takes straight lines to straight lines, and $T(0)=0$. I want to show that $T$ is linear. So far, I have proved that $T$ maps ...
PermQi's user avatar
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