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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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1answer
43 views

a circle and a parabola have 3 intersection points

Is it possible that a circle and a parabola on a euclidean plane have 3 intersection points and the center of the circle does not lie on the axis of parabola?
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1answer
32 views

Euclidean Geometry versus Analytic Geometry versus Affine Geometry?

What are the relationships (connections) among: Euclidean (or Plane) geometry Analytic geometry Affine geometry How do these things relate? I know that this is a very general question, so I'm ...
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35 views

What is the name of a linear combination that has coefficients that sum up to zero?

Is there a special name for a linear combination where the coefficient add up to zero?
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1answer
29 views

Geometric interpretation of ranks of matrices gathering coefficients of 3 affine and associated vectorial planes

Given three planes $\pi_{1}$, $\pi_{2}$ and $\pi_{3}$ $\in \mathbb{R}^{3}$ with their respect cartesian equation of the form $A_{i}x + B_{i}y + C_{i}z + D_{i} = 0$, we can determine their relative ...
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2answers
197 views

Given four points, determine a condition on a fifth point such that the conic containing all of them is an ellipse

The image of the question if you don't see all the symbols The given points $p_1,p_2,p_3,p_4$ are located at the vertices of a convex quadrilateral on the real affine plane. I am looking for an ...
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0answers
23 views

Understanding the connection between $PGL(3, K)$ and $AGL(2, K)$

Sorry if this is a trivial question, but I'm having trouble wrapping my head around the connection between these two groups. It seems intuitively clear to me that the group of invertible affine ...
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1answer
23 views

Explicit affine transformation between simplex and subsimplex

Take a simplex $\mathcal P_{n}$ with corner points $[00\dots00]$, $[10\dots00]$,$\dots$,$[11\dots11]$ in $\mathbb R^{n+1}$. Slicing by a hyperplane $\sum_{i=1}^{n+1}x_i=t$ where $0\leq t\leq n$ gives ...
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0answers
23 views

Uniform Bound on fibres of a finite morphism between affine varieties.

Let $X,Y$ be two affine varieties (irreducible zero loci of polynomials over algebraically closed field). Let $\phi : X\rightarrow Y$ be a finite morphism (i.e. K[X] is finitely generated $\phi ^*(K[Y]...
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0answers
36 views

What is an affine $0$ space?

I have encountered $\mathbb{A}^0$, where $\mathbb{A}^n$ is the affine $n$ space over some algebraically closed field. What exactly is $\mathbb{A}^0$? Does this only consist of $1$ element perhaps?
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2answers
13 views

How to convert half plane from parametric to general form?

I have: $ x=2+5t $ $y=5-t, t\geq 0 $ and I am supposed to convert it to general form. I know how to do it when I have plane, but that is half plane and I have no idea what will change. Can anyone ...
2
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1answer
51 views

The complement of a point in $\Bbb{A}^n$ is compact in the Zariski topology.

The book "Invitation to Algebraic Geometry" says the following: The complement of a point in $\Bbb{A}^n$ is compact in the Zariski topology. Why is this this the case? This is thing that is asked ...
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2answers
37 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
32 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 ...
4
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1answer
100 views

Sheaf of a 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
30 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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0answers
44 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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0answers
30 views

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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0answers
31 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
21 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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0answers
22 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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0answers
11 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
76 views

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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0answers
17 views

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
32 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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0answers
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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0answers
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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0answers
14 views

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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0answers
34 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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0answers
7 views

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
38 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
45 views

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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0answers
17 views

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
23 views

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
40 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
17 views

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{...
1
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1answer
53 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
45 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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0answers
32 views

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
55 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
15 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
41 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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0answers
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 ...
1
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1answer
38 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 ...
1
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1answer
59 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
100 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
43 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
34 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)}, \...