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.
1
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1answer
42 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?
2
votes
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 ...
0
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0answers
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?
0
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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 ...
6
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
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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]...
1
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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?
0
votes
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
votes
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 ...
0
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2answers
36 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},\...
0
votes
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
votes
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\...
0
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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> ⊂ ...
1
vote
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&...
2
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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 ...
1
vote
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 ...
1
vote
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 $W=\mathbb{P}(S)$ ...
0
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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 ...
0
votes
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 ...
0
votes
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 ...
1
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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 \...
0
votes
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 ...
0
votes
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
vote
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:
$$ax + b = a^{-1}(ax + ...
2
votes
0answers
43 views
Definition of a Product
For notational convenience I want to introduce a short-hand notation: Let $A\in\mathbb R^{n\times(k+1)}$ and $x\in\mathbb R^k$. My "product" should represent the affine-transformation $A\bullet x = Sx ...
0
votes
0answers
14 views
Proof regarding affine spaces over fields
Consider an affine space $\mathcal{A} = \operatorname{AG}(n,\mathbb{K})$, with $\mathbb{K}$ a field. Show that $\mathcal{A}$ contains $\operatorname{AG}(2,2)$ if and only if $\operatorname{char}(\...
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0answers
34 views
Is this axiomatization of affine plane categorical?
First I'll give some definitions.
Hilbert's plane axioms of incidence: We consider a set $P$ (plane) and a family $\mathcal{L}$ (a family of lines) with axioms:
For any two distinct points $a,b$ ...
0
votes
0answers
7 views
Proving statements regarding bases of affine spaces
! Definitions:
Basis of an affine subspace;
Consider a $d$-dimensional affine subset $D$ of an affine space, and a set of points $S$ in $D$.
If $S$ is a generating set for $D$, then $S$ ...
0
votes
1answer
38 views
Checking if a certain transform is affine and computing its inverse
$$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)$...
1
vote
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,
$\...
1
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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 ...
0
votes
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 ...
1
vote
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}+\...
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
51 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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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, ...
0
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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 ...
1
vote
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 ...
2
votes
1answer
40 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?
0
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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
vote
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
vote
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 ...
3
votes
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 \...
2
votes
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$ ...
1
vote
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=...
0
votes
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)}, \...
