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
230 views

Line-preserving transformations

Is there a name for the class of transformations on the Euclidean plane (or projective plane) that preserves lines? They are not all affine transformations; consider a perspective projection $p$ in ...
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1answer
33 views

affine variety of infinitely many polynomials can be represented as an affine variety of its finite subset

Let $f_1,f_2,\cdots$ be an infinite sequence of polynomials in $k[x_1,\cdots,x_n]$ and let $V(f_1,f_2,\cdots)=\{(a_1,\cdots,a_n)\in k^n:f_i(a_1,\cdots,a_n)=0$ for $i=0,1,\cdots\}$. Show that there is ...
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1answer
430 views

Intersection of affine subspaces is affine

So if I have two affine subspaces, each is a translate ( or coset) of some linear subspace. I want to show that the intersection of such affine subspaces is again affine, particularly in ...
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1answer
211 views

Lines projective space

I have a question concerning the answer of Georges Elencwajg in Lines in projective space There he states that the line $\overline {AB}=\mathbb P(\Lambda)\subset \mathbb P^n$ has its points of the ...
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1answer
111 views

projective geometry hyperplane

For $j=0,\ldots,n$ consider the affine hyperplane $A_j:=e_j+\langle e_0,\ldots,e_{j-1},e_{j+1},\ldots,e_n\rangle$ in $\mathbb K^{n+1}$ and the associated embedding $\tau_j:\mathbb ...
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1answer
43 views

$\alpha$ affine iff graph is affine subspace

I am just checking different analogous of $\alpha:V \longrightarrow W$ being affine. I have problems with this one: $\alpha:V \longrightarrow W$ affine $\iff$ $G_\alpha=\{(v,\alpha(v): v\in V)\}$ is ...
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1answer
123 views

How to get around non-commutativity of matrix multiplication?

I have a problem with a matrix equation/transformation problem which I need solving. I have two transformations $A_1$ and $A_2$, both of which can be expressed as $A_i = R_i \times B_i$, $R_i$ ...
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1answer
118 views

Affine Subspace Confusion

I'm having some trouble deciphering the wording of a problem. I'm given $V$ a vector space over a field $\mathbb{F}$. Letting $v_1$ and $v_2$ be distinct elements of $V$, define the set $L\subseteq ...
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1answer
252 views

use homography to rotate around x/y axes

I need to construct a homography out of a 3x3 rotation matrix. I am fundamentally misunderstanding some part of how homographies are constructed. I have been assuming that a homography is ...
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1answer
58 views

Confusion regarding convex and affine set

I am a bit confused regarding convex and affine set. When they mention set, does it mean the set consisting of all the points belonging to the line or shape respectively?
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1answer
334 views

Minimising a matrix equation to find 'best fit' affine matrix

Here is my problem: I have an image divided into segments. Each segment consists of pixels with coordinates (x,y) called vector $v$, each pixel has a length 3 vector RGB called $I(v)$. I want to ...
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0answers
10 views

Representation of Affine Maps

I'm just looking for a reference or the proof that every affine map $f:V\rightarrow W$ between two possible different linear spaces $V$ and $W$: $$ f[\lambda x+ (1-\lambda) y]=\lambda ...
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0answers
11 views

Non-affinely parametrized geodesics

Consider a non-affinely parameterised geodesic, i.e., a geodesic whose tangent vector field obeys $\nabla_X X = fX$ for some function $f$. Prove that one may reparameterise the geodesic so the tangent ...
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0answers
18 views

Vanishing points from three collinear points

I would like to find the 2D vanishing point from a three collinear points as is shown in "Multiple View Geometry in Computer Vision" Example 2.19 (see here). What I did so far: 1 - I've extracted ...
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0answers
20 views

Do affine spaces have coordinate transformations?

I asked a question on Physics SE and there seemed to be some confusion as to whether affine spaces could have coordinate transformations. Specifically, the particular space I was working with was ...
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0answers
23 views

Determining if a set is projective or not

In $\mathbb{P}^3$ define the following sets: $$X=\{w_0w_1^2=w_2^2w_3-w_3^3\}\\Y_1=\{y_3=0\}\\Y_2=\{\sum_{i=0}^3 w_i=0\}\\Y_3=\{w_0+w_1+w_2+2w_3=0\}$$ Does the set $Z=X\cap Y_3\setminus((X\cap ...
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1answer
37 views

Affine transformation invariants and lie groups

Is it possible to generate geometric properties which are invariant under affine transformations? I'm trying to learn about lie groups and lie algebras with the example of the lie group of affine ...
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0answers
25 views

Intersections of convex hulls

Given a set of $n$ points $\{A_1, \ldots , A_n\}$ of the plane and every possible triangle formed with $3$ points $A$, I would like to describe the intersections fo theses triangles. By intersection, ...
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0answers
36 views

What are the prerequisites to understand Affine Invariant Fourier Descriptors?

I need to implement Affine Invariant Fourier Descriptors on matlab, the objective is to compare two objects one reference and other transformed by affine transformation for recognition, my problem is ...
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0answers
49 views

A confusion regarding Affine spaces

Take an Affine space $\Bbb{A}$ over the field $K$. How would you determine the points satisfying any polynomial $f(x)$? If there is no fixed origin, points can be given names with reference to ANY ...
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1answer
38 views

linearization of $\log(|x|)$

I am trying to convexify $\log(|x|)$. I think its concave. So I am trying to get an affine upper bound through linearization. But the problem is there are two concave functions because of the absolute ...
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1answer
32 views

Angle of two planes in $\mathbb E_4$

I have two planes (given in parametric form) in $\mathbb E_4$: $\alpha$: (7,3,5,1) + t(0,0,1,0) + s(3,3,0,1) and $\beta$: (1,5,4,1) + r(0,0,0,-1) + p(2,0,0,1), and I have to find angle between them. I ...
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1answer
93 views

Why a 2D Affine Transformation matrix is 3 by 3

The matrix which I get for Scaling , Shearing and Rotation are follows: Scale: Shear Rotation Why do we need Homogenous Co-ordinate to get the transformation matrix as listed below? (need a ...
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0answers
57 views

Linear algebra, affine space, and floor function

My question is mostly: is there a name for this kind of things. I am mostly interested by finding book or articles about what follows, but without even a word or a name, it is quite hard to search for ...
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0answers
47 views

affine translation in direction of a vector

Suppose I have a line segment in 3D-space, having end-points $(a,b)$. I want to translate this segment by $w$ units in the direction specified by 3 angles $\alpha,\beta,\gamma$ with respect to ...
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1answer
92 views

The graph of a regular function is an algebraic set, and intersection of hypersurfaces is finite?

i have some problems with these exercises, can you give me a hint? Let $f:\mathbb A^n_k\rightarrow\mathbb A^m_k$ be a regular function. If $X\subset\mathbb A^n_k$ is an algebraic set, show that the ...
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0answers
135 views

Number of vector and affine subspaces of dimension $ k$ of $E$ over $\mathbb{F_q}$

Problem (comments after): Let $\mathbb{F_q}$ be a finite field of cardinal $q$ and $\mathcal{E}$ an affine espace of dimension $n$ directed by the vector space $E$. Show that: ...
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1answer
226 views

Using absolute coordinates in 2D affine transformation matrix

In my 2D animation program I have a sprite which transformation is described by a 2D affine transformation matrix (SVGMatrix): $$ \begin{bmatrix} a & c & e \\ b & ...
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0answers
153 views

Multi-affine function

Suppose I have a three-variable function $f(x_1, x_2, x_3)$, $f : \mathbb{R}^3 \to \mathbb{R}$. If it is linear for $x_1$, $x_2$ and $x_3$ we can say it has the form $f(x_1, x_2, x_3) = c_1x_1 + ...
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0answers
95 views

Distance between two affine lines using determinant of Gramian matrix.

I've a task to find the distance in $E^4$ between: $L = [1,2,-1,4] + \text{lin}((1,2,-1,0))$ and $M = [2,3,1,5] + \text{lin}((2,1,0,2))$ My efforts to find the correct solution: Let ...
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0answers
100 views

Prove something is affine?

For any subspace $K$ and any point $u$, prove $K+u$ is affine. Or if you have an affine set $V$ and point $u$, then prove $V-u$ is a subspace.
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0answers
66 views

Separation of Euclidean Space

Consider a finite collection $\mathcal{H}$ of hyperplanes of $\mathbb{R}^n$ that have a common line. Given some $A \subseteq \mathbb{R}^n$ that is homeomorphic to a subset of $\bigcup\mathcal{H}$, ...
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0answers
570 views

Convexity of affine function.

Can someone help me with a proof that affine function preserves convexity? Given that $f$ is convex, $A$ is in $\mathbb{R}^{M\times N}$ and $b$ is in $\mathbb{R}^m$ then show that $g(x) = f(Ax+b)$ is ...
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1answer
483 views

Affine subspace iff statement

I cannot find a proof for the following statement, though the statement itself seems common enough: $X$ is an affine subspace of $\mathbb{R}^n \iff \forall a\in X: X - \{a\}$ is a linear subspace. ...
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2answers
3k views

Definition of an affine subspace

I am reading this introduction to Mechanics and the definition it gives (just after Proposition 1.1.2) for an affine subspace puzzles me. I cite: A subset $B$ of a $\mathbb{R}$-affine space $A$ ...
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1answer
60 views

Is $\mathbb{A}^1\times\mathbb{P}^1\cong\mathbb{P}^1\times\mathbb{P}^1$?

Just curious, is it true that $\mathbb{A}^1\times\mathbb{P}^1\cong\mathbb{P}^1\times\mathbb{P}^1$? Here I'm writing $\mathbb{A}^1$ is affine space, and $\mathbb{P}^1$ projective space, both over an ...
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1answer
291 views

Research Paper and Affine Subspace

I was reading a research paper titled Purity and Reid's Theorem by A.Blass and J.Irwin and i have the problem with the explanation of the proof of the first theorem, that is theorem 1.1. In the proof ...
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1answer
36 views

Finding a line $L\subset V(y-xz)\subset\mathbb A^3_k$

I want to find lines $L\subset V(y-xz)$ and $M\subset\mathbb A_k^2$ such that $$ V(y-xz)\setminus L \simeq \mathbb A_k^2\setminus M\ . $$ Hint suggests that I use the projection $(x,y,z)\mapsto(x,y)$. ...
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1answer
43 views

Are quasiaffine subsets of $\mathbb{A}_F^n$ always necessarily open or closed?

Something I was wondering about lately, suppose $\mathbb{A}_F^n$ is affine space over a field $F$ which is algebraically closed. When I say a quasi-affine set, I mean a set that is locally closed, so ...
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1answer
28 views

Is it possible, to create a affine transformation matrix, from a function?

I have a function, which maps every point in the 3D space, to an other. How is it possible, to find a matrix, which works the same, as the function, if I multiply 3D vectors with it? It's sure, the ...
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2answers
50 views

How to define an affine transformation using 2 triangles?

I have $2$ triangles ($6$ dots) is a $2D$ plane. The points of the triangles are: a, b, c and x, y, z I would like to find a ...
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2answers
46 views

fixed points of an affine transformation is unique iff $1 \notin SP(\vec{f} )$

Let $f$ be Affine transformation from $E$ to $E$ (always we assume it finite dimensional ) and $\overrightarrow{f}$ is the linear mapping associated to $f$. Then the map $f$ has a unique fixed ...
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2answers
32 views

Number of parameters to specify an affine transformation in n dimensions

In general, how many parameters does it take to specify an affine transformation in $n$ dimensions, and how does one go about proving this? For example, in 2 dimensions it takes 6 parameters, and in ...
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1answer
33 views

Fixpoints of affine transformations

I want to find out all the possibilities what fixpoints of an affine transformation can be in 2-dim vector space. If the transformation is identity, then it is trivial - fixpoints describe the ...
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2answers
57 views

Identify a quadric

Could you tell me how to identify a given quadric? Given a conic section, I should find an orthonormal affine frame in $\mathbb{R}^2$ (with standard dot product) in which the equation has a canonical ...
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1answer
46 views

Affine transformation, if $L_1, L_2 - $ skew lines, $f(L_1), \ f(L_2) $ are parallel, then $f$ is not injective

Could you tell me how to prove that if $f$ is affine transformation, $L_1, L_2 $ are skew lines, $f(L_1), \ f(L_2) $ are parallel, then $f$ is not injective?
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1answer
70 views

Variety Affine Space

I have the following question which i'm not sure how to work out... $For\ f=6x^2y-xy^2-2y^3+1\ and\ \ h=3x-2y\ \in \mathbb{C}[x,y]$ Show that V(f,h) is empty. What can you say about the ideal ...
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1answer
384 views

Affine transformation matrixes

I could use some advise with the following problem: Lets say there is a cuboid that has two distinguished points - that is one of its vertexes ($A$) and the other one is somewhere on the surface ...
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1answer
17 views

Distance from affine vector space?

I've got an affine vector space $W$ defined by a collection of vectors $\{v_1, v_2, ... v_n\}$. Each vector in that space could be represented as a sum of the form $\sum_{i=1}^n w_i * v_i$, where ...
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1answer
22 views

Two discrete lines always intersect at a point

In my lecture notes we have the following: $K$ field Extension of the affine space. Relation between points and lines: Two discrete points define an unique line and two discrete lines always ...