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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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18 views

Simple Affine Stretch of Triangle

I am having a little trouble with the following problem: Use explicit affine transformation, i.e., written down in matrix form, to stretch triangle having vertices $(\frac{1}{2},0), (\frac{1}{2},\...
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1answer
25 views

Dimension of an affine plane

In geometry, an affine plane is defined as a system of points which fullfill: 1) Any two distinct points lie on a unique line. 2) Given any line and any point not on that line there is a unique line ...
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32 views

Sheaf of Closed Subset

I’ve been given the following definition: Let $(X,\mathcal{O}_X)$ be a ringed space which is locally isomorphic to an affine algebraic variety, and $Y\subseteq X$ be closed. Then for an open $V\...
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1answer
28 views

Understanding the symbol <A,B> in affine spaces

I'm trying to solve this exercise: A subset F of an affine space is an affine subspace if and only if for all points A and B of F, the inclusion <A, B> ⊂ ...
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38 views

Lines in projective and affine space

I'm trying to understand lines in affine and projective space in order to solve problems 2.15 and 4.13 in Algebraic Curves by William Fulton: https://www.google.com/url?sa=t&source=web&rct=j&...
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Why is $\mathbb{R}^2$ endowed with the taxicab metric isomorphic to the infinity distance model of the cartesian real plane? (Hartshorne exercise 8.9) [duplicate]

My question deals with the following exercise from Hartshorne's Euclid: Geometry and Beyond: Following our general principles, we say that two models $M,M'$ of our geometry are isomorphic if there ...
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30 views

Find a set of polynomials whose common zero set is $\{(1, 2), (0, 5)\}$.

Find a set of polynomials $\{P_1, \dots, P_n\}$, all of whose coefficients are real numbers, whose common zero set is the given set. I know what a zero set is, but I think my confusion comes from ...
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0answers
20 views

$W=\mathbb{P}(S)$ for some (k+1)-dimensional subspace $S$ of $\mathbb{k}^{n+1}$

Let $W$ be a subset of $\mathbb{P}_n(\mathbb{k})$. It is known that for every affine map $A$ (with which $W$ intersects) $A \cap W$ is a k-dimensional affine space. Is it true that $W=\mathbb{P}(S)$ ...
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19 views

3D Affine Rotation Matrix from Orthogonal Vectors

How does one define an affine rotation matrix in order to rotate a 3D volume to align with a new coordinate system? The current coordinate system is $\mathbf{x}, \mathbf{y}, \mathbf{z}$ and I want to ...
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10 views

Why is the stabilizer of a line segment a cylinder?

There is a passage in a paper I'm reading discussing the stabilizer of an edge. For an edge (passing through the origin), its stabilizer (in $\operatorname{GL}_2(\mathbb{R})$) must fix the ...
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2answers
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How is this a coherent definition of “hyperplane”?

My textbook says the following: A hyperplane is a set of the form $$\{ x \mid a^T x = b \},$$ where $a \in \mathbb{R}^n$, $a \not= 0$, and $b \in \mathbb{R}$. Analytically it is the ...
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Proving that $\mathbb{K}$ is a skew field (see post for detailed description of $\mathbb{K}$).

Consider a Desarguesian affine space $\mathcal{A}$. Choose a fixed point $o$ and a fixed line $\mathbb{K}$ through $o$. Select another, arbitrary point $e$ on $\mathbb{K}$. For each point $x \in \...
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1answer
30 views

There cannot be a concept of parallelism in a homogeneous projective space?

Page 4 of my computer vision textbook, Multiple View Geometry in Computer Vision (Second Edition), by Hartley and Zisserman, states the following: Affine geometry. We will take the point of view ...
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3 views

Homotheties in an affine Desarguesian space: conjugation and factors

Definition: a class of conjugated homotheties (conjugated by a translation) is called a factor. I really don't understand this definition of a factor. I believe it has something to do with the ...
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1answer
24 views

Understanding affine cypher / Euclidean math

$$e(x) = (ax + b)$$ $$d(y) = a^{-1}(y-b)$$ I need to prove that $$e(x)=d(y)$$ iff $$(a^2)=1$$ and $$b(a+1)=0,$$ so I tried working with $d(ex)$ to show they can be equivalent: $$ax + b = a^{-1}(ax + ...
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43 views

Definition of a Product

For notational convenience I want to introduce a short-hand notation: Let $A\in\mathbb R^{n\times(k+1)}$ and $x\in\mathbb R^k$. My "product" should represent the affine-transformation $A\bullet x = Sx ...
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Proof regarding affine spaces over fields

Consider an affine space $\mathcal{A} = \operatorname{AG}(n,\mathbb{K})$, with $\mathbb{K}$ a field. Show that $\mathcal{A}$ contains $\operatorname{AG}(2,2)$ if and only if $\operatorname{char}(\...
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31 views

Is this axiomatization of affine plane categorical?

First I'll give some definitions. Hilbert's plane axioms of incidence: We consider a set $P$ (plane) and a family $\mathcal{L}$ (a family of lines) with axioms: For any two distinct points $a,b$ ...
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15 views

Classifying Radon partitions in $\mathbb{R}^n$ whose affine hull is $\mathbb{R}^n$

Specifically, I want to determine all distinct "types" of Radon partitions of $n+2$ points in $\mathbb{R}^n$ for which the affine hull is all of $\mathbb{R}^n$. This is a homework question, so I'm ...
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Proving statements regarding bases of affine spaces

! Definitions: Basis of an affine subspace; Consider a $d$-dimensional affine subset $D$ of an affine space, and a set of points $S$ in $D$. If $S$ is a generating set for $D$, then $S$ ...
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1answer
34 views

Checking if a certain transform is affine and computing its inverse

$$T(x,y)=(2x+y+5,3x+2y+2)$$ Check if $T$ is affine and compute $T^{-1}$. I'm not sure if I'm using the right methods here or if I am missing something, but I went about proving that $T(A)+T(B)=T(A+B)$...
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1answer
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Point Addition in Affine Space

The only operations defined on points in an affine space are point-vector addition. this yields a new point. point-point subtraction. this yields a vector. This can be extended to an affine sum, $\...
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Is there any incompatibility between affine spaces and Hilbert spaces? [closed]

I was wondering if there is such a thing like a Hilbertian affine space. I've seen the definition of an Euclidian affine space, which is: An affine space (A, V, φ) is an Euclidean affine space if ...
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2answers
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An automorphism of an affine space preserves parallelism: question on proof

Definitions: An automorphism of an affine space is a permutation of the set $\mathcal{P}$ of points that preserves lines and planes (if the dimension it at least 3). This is the proof given in my ...
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2answers
33 views

Prove that General Affine Transformations preserve ratios of lengths

Let $A$ be a matrix with determinant 1. Then we call a general affine transformation, a transformation of the form \begin{align*} \begin{bmatrix}x'\\y'\end{bmatrix}=A\begin{bmatrix}x\\y\end{bmatrix}+\...
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1answer
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Affine space and convex sets in the context of Euclidean space

I am a bit confused as to the relationship between the ideas of vector space, affine space, and convex sets in the context of Euclidean space $\mathbb{R}^d$. As of now, this is how I see it. $\mathbb{...
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1answer
31 views

Basis of an affine subspace

Consider an affine subspace $D$ of an affine space or affine plane $\mathcal{A}$. Every set of points that are not elements of a proper affine subspace of $D$ is called a generating set of $D$. If ...
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3answers
41 views

Name for ratio of three co-linear points in affine geometry

Given three points $a_1$, $a_2$, $a_3$ in an affine space such that $\dim \langle a_1, a_2, a_3\rangle = 1$ (i.e. they are co-linear) and $a_1\ne a_2$, my professor defined the "simple ratio" (making ...
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1answer
28 views

Derivative being linear

Very straight forward question, so I'm studying differentiation between (infinite) normed vector spaces and when considering the very basic example of $f(x)=x^2+2x$ from reals to real we have the ...
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Why is an affine hull of a set of three non-collinear vectors of $\mathbb{R}^2$ equal to $\mathbb{R}^2$?

Three vectors are $(1,0)$, $(-1,2)$, and $(3,1)$. I tried to relate the hull with an affine set, and so a flat. However, I don't find their relation. From what I learn about affine combinations, ...
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1answer
51 views

How to find a 2D coordinate field's corners in a 3D Coordinate field if I have 3x 3D points with 3x2D Points?

In order to solve "this" problem, i have to transform my corner-points from a 2D Space to my 3D Space. But my two coordinate fields are only defined by their relation to each other. They have the ...
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0answers
14 views

Notion of line on a complex affine space

$\newcommand{C}{\mathbb{C}} \newcommand{\A}{\mathbb{A}} \newcommand{\i}{\hspace{0.1em}\mathrm{i}} \newcommand{\R}{\mathbb{R}}$ Let $V = (\C, \C, +, \cdot)$ be the one-dimensional complex vector space ...
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1answer
34 views

definition of affine polynomial [closed]

I'm reading a paper and it is written inside it "affine polynomial" but I don't know this definition, and couldn't find it on the web. Could you please help me if you know it?
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214 views

Euclidean incidence spaces

Structure $\left<X,B\right>$ is an $n$-dimensional Euclidean incidence space iff $B$ is ternary betweenness relation associated with $n$-dimensional Euclidean geometry. That is, Euclidean ...
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1answer
35 views

“Affine” vector spaces with groups?

I wish to consider a structure that is like an affine space, but does not use a vector space as the affine structure, rather uses a group. That is, we shall "forget" the scaling structure of the ...
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1answer
56 views

complex coordinate to screen coordinate problem

As an expansion to my fractal software, I've decided to add center plus width as an image location/definition. Which leads me to the question of how do I convert the complex number given to an x,y ...
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0answers
96 views

Projective-invariant differential operator

This question has been cross-posted to MathOverflow. Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that \begin{align*} &T(g) = 0 \...
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1answer
41 views

Ideal of regular functions in $Z(xy-z^2)$ vanishing on $(x,y)$ is not principal

Let $H = Z(xy-z^2) \subset \Bbb A ^ {3}$. And let $L = Z(x,z)$. I need to show that $L \subset H$ , that $dim(L) = dim (H) -1$ and that the ideal of regular functions on $H$ which vanishing on $L$ ...
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1answer
31 views

Affine transformations satisfying conditions

I'm asked to find ALL affine transformations from $\mathbb{R^3}$ to itself which satisfy that the point $(-1,2,2)$ is fixed and that the lines $$\textbf{a}: y-z=y+z-2=0$$ and $$\textbf{b}: z-1=x-z=...
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0answers
13 views

Intersections of hyperboloids and affine maps

Consider the cartesian product $\mathbb{H}_{n}$ of $n$ $m-1$-dimensional forward hyperbolae in $\mathbb{R}^{mn}$ as given by the parametrization: $\mathbb{H}_i: \ \ x_i=\sqrt{(\vec{x}_{i+1}^2+1)}, \...
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0answers
126 views

$GL(n,G)$ General Linear group of a group $G$?

I know you can take the General linear group of some vector space $V$: $GL(V)$. For example, suppose I have a three-sphere, elements of which I might represent as $SU(2)$ since they're diffeomorphic....
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1answer
25 views

Codimension of a subset of $Hom(E,F)$

Let $E$ and $F$ be vector spaces over the field of complex numbers. Consider $$W=\{f\in Hom(E,F)\,:\, \textrm{dim }ker(f)\geq c\}\,.$$ The claim is that $W$ is a closed subvariety of $Hom(E,F)$ of ...
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1answer
54 views

What means ''direction'' in hyperbolic geometry?

We can define the concept of ''direction'' as the equivalence class of parallel lines, but this is a good definition only if the Euclide parallels axiom is assumed, so that there is only one ...
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1answer
50 views

Coxeter, Introduction to Geometry, ordered geometry, parallelism of rays and lines

Since Coxeter’s “Introduction to Geometry“ is a classic, I think I can ask a question referring to it (2.ed, 1969). For me it is important, because it lies at the foundation of how Coxeter defines ...
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2answers
256 views

Identifying the Plane at Infinity in the World Necessitates Determining the Affine Geometry of the World?

Page 18 of my computer vision textbook, Multiple View Geometry in Computer Vision (Second Edition), by Hartley and Zisserman, states the following: 1.8 Auto Calibration $\vdots$ Generally ...
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2answers
80 views

Prove a set $V$ is not algebraic

I need help showing that the set $V = \{(a,b) \in \mathbb{A}^2(\mathbb{C})\ \vert \ \vert a\vert^2 + \vert b\vert^2 = 1\}$ is not an algebraic subset of complex affine 2-space. I believe that I can ...
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2answers
77 views

Showing affine transformations group generated by $2x$ and $x+1$ is the Baumslag-Solitar group.

I want to compute the presentation groups of $\langle f,g\rangle$ the generated group of affine transformations with $f(x)=2x$ and $g(x)=x+1.$ The affirmation is $\langle f,g\rangle=\langle a,b\...
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0answers
85 views

Show that the affine transformation of a polyhedron is a polyhedron.

Let $P=\{x \in \mathbb{R}^n \mid Ax \geq b\}$ be a nonempty polyhedron for a matrix $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^{m}$. Let $F:\mathbb{R}^m \rightarrow \mathbb{R}^n$ be an ...
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1answer
67 views

Intersection of n hyperplanes in $\mathbb{R}^n$

For all unit vector $\nu \in \mathbb{R}^n$ consider an affine hyperplane $A_{\nu}$ orthogonal to the direction $\nu $. Now consider n linearly independent unit vectors $\nu_ 1 , \nu_2, \dots, \nu_n \...
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0answers
23 views

Is Geometric Median Affine Equivariance?

Does the geometric median have the natural property - Affine Equivariance? That is the depth of the geometric median and its relative location to other data points do not change under affine ...