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
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 ...
2
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0answers
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> ⊂ ...
1
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0answers
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&...
1
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0answers
26 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 ...
1
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0answers
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 ...
1
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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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0answers
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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0answers
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
70 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
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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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 ...
1
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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:
$$ax + b = a^{-1}(ax + ...
2
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0answers
43 views
Definition of a Product
For notational convenience I want to introduce a short-hand notation: Let $A\in\mathbb R^{n\times(k+1)}$ and $x\in\mathbb R^k$. My "product" should represent the affine-transformation $A\bullet x = Sx ...
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0answers
14 views
Proof regarding affine spaces over fields
Consider an affine space $\mathcal{A} = \operatorname{AG}(n,\mathbb{K})$, with $\mathbb{K}$ a field. Show that $\mathcal{A}$ contains $\operatorname{AG}(2,2)$ if and only if $\operatorname{char}(\...
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0answers
31 views
Is this axiomatization of affine plane categorical?
First I'll give some definitions.
Hilbert's plane axioms of incidence: We consider a set $P$ (plane) and a family $\mathcal{L}$ (a family of lines) with axioms:
For any two distinct points $a,b$ ...
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0answers
15 views
Classifying Radon partitions in $\mathbb{R}^n$ whose affine hull is $\mathbb{R}^n$
Specifically, I want to determine all distinct "types" of Radon partitions of $n+2$ points in $\mathbb{R}^n$ for which the affine hull is all of $\mathbb{R}^n$. This is a homework question, so I'm ...
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0answers
5 views
Proving statements regarding bases of affine spaces
! Definitions:
Basis of an affine subspace;
Consider a $d$-dimensional affine subset $D$ of an affine space, and a set of points $S$ in $D$.
If $S$ is a generating set for $D$, then $S$ ...
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votes
1answer
34 views
Checking if a certain transform is affine and computing its inverse
$$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)$...
1
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1answer
30 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,
$\...
1
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0answers
16 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
22 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
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}+\...
0
votes
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
vote
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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votes
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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0answers
30 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
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 ...
1
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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 ...
2
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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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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
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 ...
1
vote
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 ...
3
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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 \...
2
votes
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$ ...
1
vote
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)}, \...
2
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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 ...
1
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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 ...
1
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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 ...
3
votes
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 ...
1
vote
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 ...
2
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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\...
2
votes
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 ...
2
votes
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 ...
