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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3
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0answers
12 views

Calculate the singular points of affine curve

I want to calculate the singular points of the affine curve $$f(X,Y)=(1+X^2)^2-XY^2 \in \mathbb{C}[X,Y]$$ The point $P=(x,y)$ is singular $\Leftrightarrow$ If $x=0$ we find $y=0$ and then from the ...
0
votes
0answers
27 views

$Ax = b$ & $Ax + b$

Ask a dumb question but confuse me long time. The following is what I know: 1st case $Ax = b$ is an affine set in $x$,i.e. $\{x | Ax = b\}$, and it is linear in $x$. 2nd case $ f(x) = Ax + b$ ...
0
votes
0answers
12 views

Rational representation of conics

Currently I'm beginning my study of rational curves (Rational Bezier and NURBS) all books that I've read tell me that is "well known" that conics can't be represented by Bezier or even a B-Spline. ...
0
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0answers
10 views

Explain how lines and points in the 2D plane form an affine plane?

I think I understand the affine transformation, but I just have trouble describing how lines and points in the 2D plane form an affine plane.
0
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0answers
12 views

Decomposition of 4x4 or larger affine transformation matrix to individual variables per degree of freedom.

There are a couple of problems and solutions where affine matrices are decomposed into their seperate tranformations. However they are all for the 2D case and I`m finding it difficult to generalise it ...
0
votes
1answer
19 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 ...
0
votes
1answer
29 views

Showing affinity of a function - proof help

Let $V$ be the set of sequences whose terms are contained in $\mathbb{R}^n . V$ is the set of functions $x(·) : N → \mathbb{R}^n $ which we denote as $\{x_n\}_n \subset \mathbb{R}^n$. $V$ is a vector ...
-2
votes
0answers
15 views

Is this transformation affine? [closed]

When transforming coordinates from $P(x,y)$ to $P'(u,v)$ using equations $u=x*y\mid a$ and $v=x/y\mid b$, where $a\mid b$ means that $a$ divides $b$, i.e. $b$ is a multiple of $a$, is this ...
0
votes
1answer
53 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 ...
1
vote
1answer
26 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 ...
0
votes
1answer
22 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 ...
1
vote
0answers
22 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, ...
0
votes
2answers
40 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 ...
1
vote
3answers
53 views

Every reflection is an isometry proof

The theorem is that every reflection $R_{S}$ in an affine subspace $S$ of $\mathbb{E}^{n}$ is an isometry: $R_S:\ \mathbb{E}^{n} \rightarrow \mathbb{E}^{n}:\ x \mapsto R_{S}(x) = x + 2 ...
0
votes
1answer
25 views

transformation of a ball to an other by homothety

While reading a proof in which they have defined the following homothety $$\begin{align*} h \colon C &\to C\\ x &\mapsto a+t(x-a) \end{align*}$$ where $a\in C$, C is a convex set of a ...
2
votes
1answer
41 views

What does it mean to fix a point in an affine space?

In their book Metric Affine Geometry, Snapper and Troyer state on page 59: It cannot be stressed enough that the affine space $X$ is not a vector space. Its points cannot be added and there is no ...
2
votes
1answer
63 views

What is an affine space?

I am having trouble understanding what an affine space is. I am reading Metric Affine Geometry by Snapper and Troyer. On page 5, they say: "The upshot is that, even in the affine plane, one can ...
2
votes
1answer
53 views

Is there a characterization of contractible hypersurfaces in $\mathbb{C}^2$.

Let $V$ be an irreducible, algebraic hypersurface in $\mathbb{C}^2$ which is contractible as a topological space. I would like to know the algebraic characterization of such objects. For example, ...
0
votes
1answer
27 views

Question on Logic in translation

let $P,P'$ two affine subspace of $R^{3}$ have we equality between this two statement $$\exists\ u_{0}\in R^{3}\ \mbox{such that } t_{u_0}(P)=P'$$ $$\exists B,A\in PP' \mbox{such that } ...
0
votes
0answers
16 views

help me to check the following argument on a linear transformation

all, Just want to confirm the following argument: Assumption: we know that $S:=\{(x_1,x_2,x_3) : \|(x_1,x_2)\|_2 \leq (1+k)x_3\}$, for a given value $k$. By linear transformation: ...
0
votes
1answer
61 views

Dimension of the intersection of affine subspaces

Let $\alpha,\beta,a,b,c \in \mathbb{R}.$ Consider three affine planes in the affine space $\mathbb{R}^{3}$: $P_{1}$ of equation $x+2y+\beta z=a$, $P_2$ of equation $2x+y=b$, $P_{3}$ of equation ...
4
votes
0answers
52 views

Is the cone over Grassmannian manifold $Gr_2(\mathbb{C}^n)$ an open set of a determinantal variety?

Let $Gr_2(\mathbb{C}^n)$ the Grassmann manifold of the planes in $\mathbb{C}^n$. It is, via Plucker embedding, a projective variety. If we consider the cone $C$ over $Gr_2(\mathbb{C}^n)$, is it ...
3
votes
1answer
37 views

Is the $n$-sphere $x_1^2+\cdots+x_n^2-1=0$ a rational variety in $\mathbb{A}^n$?

I asked a question a few days ago about where the function field $k(x,\sqrt{1-x^2})$ was purely transcendental over $k$, for $k$ algebraically closed. It turned out to be true, so I know this proves ...
0
votes
0answers
28 views

$P+Q:=\varphi_{O}^{-1}\left(\varphi_{O}(P)+\varphi_{O}(Q)\right)$

let X is affine space and $\overrightarrow{X}$ is vector space associted to X $$\begin{array}{ccccc} & \varphi_{O} : & X & \longrightarrow & \overrightarrow{X}\\ & & ...
0
votes
2answers
34 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 ...
1
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1answer
52 views

The affine line with two points removed

To which affine variety $V$ is $\mathbb{A}^1 \setminus \{0, 1\}$ isomorphic to? What would be the isomorphism in this case? Any help would be appreciated.
0
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0answers
25 views

Determine the expression for a continuous affine transformation

In this problem I'm doing, I'm being asked To determine the affine transformation matrix which maps triangle V to triangle W. I'm also being asked to determine this matrix's continuous ...
0
votes
0answers
47 views

Let $T(x)=Ax+v$, where $A\in GL (n,\mathbb{R})$ and $v \in \mathbb{R}^n$. Show that T sends affine subspaces of $\mathbb{A}^n$ to affine subspaces.

I'm not too familiar with Affine geometry so I feel as though I'm missing something from my solution: Let $E$ be an affine subspace defined by $E=\{P+ \textbf{v}\, |\, \textbf{v} \in U\}$ where $U$ ...
2
votes
1answer
38 views

Domain of definition of $(1-y)/x$ on $x^2+y^2=1$?

I'm self-teaching myself some basic algebraic geometry, and I wanted to double check something that seems too easy. An exercise sheet I found asks to compute the domain of definition of the rational ...
0
votes
0answers
15 views

Two affine subspaces parallel

Theorem: Two affine subspaces $V,V'$ of $(X,\overrightarrow{X})$ are said to be parallel $(V\parallel V')$ if there is a translation such that $t_{\vec{u}}(V)=V'$ $V\parallel V' ...
2
votes
1answer
28 views

Are the quasi-affine subsets of $\mathbb{A}^1_F$ necessarily open or closed?

For what follows, my definition of a quasi-affine subset of $\mathbb{A}^n_F$ is one which can be written as $Z_1\setminus Z_2$, where $Z_1$ and $Z_2$ are closed subsets of $\mathbb{A}^n_F$. (I think ...
1
vote
4answers
61 views

A function is convex and concave, show that it has the form $f(x)=ax+b$

A function is convex and concave, it is called affine function. That is the function: $$f(tx+(1-t)y)=tf(x)+(1-t)f(y),\, \, t\in (0,1) $$ Force $y=0$(suppose $0$ is in the domain of $f(x)$), we ...
2
votes
2answers
31 views

If we delete two points $x,y$ from $\mathbb{A}^1$, can we without loss of generality assume $x=0, y=1$?

My intuition is that we can assume this. More precisely, what I mean is, suppose $\mathbb{A}^1_k$ is the affine space over an algebraically closed field $k$. If $x,y$ are any two distinct points in ...
0
votes
0answers
27 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 ...
3
votes
2answers
87 views

Why is $\text{Mor}_{\mathrm{reg}}(*,W)\to \text{Hom}_{k-\mathrm{alg}}(k[W],k[*])$ not surjective when $W=\mathbb{A}^2\setminus\{(0,0)\}$.

Suppose we're working over an algebraically closed field $k$. If $V\subseteq\mathbb{A}^n$ and $W\subseteq\mathbb{A}^m$ are affine algebraic sets, there is a well known bijective correspondence $$ ...
2
votes
1answer
55 views

Why is $f(X)$ open or closed if $f:X\to\mathbb{A}^1(k)$ is regular?

I have a question about a certain property of regular maps into $\mathbb{A}^1(k)$. This is my notation for the affine space over $k$, algebraically closed. Suppose $f:X\to \mathbb{A}^n(k)$ is a ...
0
votes
0answers
22 views

Computing the dimension of the affine hull of a set of vectors

I have a few hundred vectors that live in $R^d$ with $d$ larger than 100. I'm interested in finding the dimension of the affine hull of these points. Computing the dimension of the linear subspace ...
1
vote
1answer
40 views

If $n\geq 2$, why is $k[\mathbb{A}^n\setminus\{p\}]=k[\mathbb{A}^n]$?

Assuming $n\geq 2$, why is the coordinate ring of affine $n$-space over an algebraically closed field $k$ unchanged if we delete a point? That is, if $p\in \mathbb{A}^n$, why does ...
0
votes
1answer
30 views

how to obtain transformation matrix A in y = Ax + b notation?

I'm trying to obtain original transform matrix A and its translation vector b From y=Ax+b equation. I have original values of vectors before transform and translation (x) and vectors after transform ...
4
votes
3answers
90 views

Why are $xy=0$ and $x^2-x=0$ not isomorphic?

I'm not sure how to rigourously back up my intuition that the curves given by $x^2-x=0$ and $xy=0$ in $\mathbb{A}^2_k$ are not isomorphic. If I denote the varieties as $V_1=V(xy)$ and ...
1
vote
1answer
39 views

The number of $(d-1)$-faces in a $d$-polytope is at least $(d+1)$

I have to try to prove the following: Let $V = {v_1,...,v_n} \subset \mathbb{R}^d$ be a point configuration affinely spanning $\mathbb{R}$ (i.e., $\operatorname{aff}(V) = \mathbb{R}^d)$. Let $H$ be ...
2
votes
1answer
34 views

Why is every open in $\mathbb{A}^1$ necessarily principal?

Let $U\subseteq\mathbb{A}^1$ be an open set in affine $1$-space. Why is $U$ necessarily a principal open set? Since $U$ is the complement of a closed set, I write $U=\mathbb{A}^1\setminus V(S)$ for ...
0
votes
1answer
39 views

Express a point as an affine combination of another two points(3D collinearity)

So, given the points A(1,2,2), B(2,4,2) and C(3,6,2) I have to show that they are collinear. ...
0
votes
1answer
35 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 $(x,y,z)\mapsto(x,y)$. ...
0
votes
1answer
35 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 ...
0
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0answers
116 views

What are sliding vectors mathematically?

What is the mathematical definition of sliding vectors and their operations, as used in mechanics? What kind of mathematical structure do they form? Does the operation of constructing the "space" of ...
1
vote
1answer
28 views

A set $S\subseteq\mathbb{A}^n$ is quasi-affine iff $S=Z\setminus V$ for closed $Z$ and $U$?

I'm confused by a remark in note I'm reading. It essentially says, Let $S\subseteq\mathbb{A}^n$ be a subset of affine $n$-space over an algebraically closed field. It's clear that $S$ is ...
1
vote
1answer
53 views

Distinction between point and vector outside of US ( particularly Germany and Eastern Europe )

There was a long discussion in a forum I visit in where a calculus teacher was being critical of Stewarts Calculous for making a distinction between points and vectors. He argued that no such ...
0
votes
1answer
36 views

Making a matrix full rank through affine transformations

If I have (finite) $k$ vectors, $u_1,...,u_k\in\mathbb{R}^N$ that are in general linearly dependent is it possible to take positive affine transformations of the form: $$u'_i=\alpha_i u_i ...
2
votes
1answer
41 views

Surface area of transformed sphere

So if I have a sphere with center C and radius R and then apply one or more affine transformations (so any combination of rotating, scaling and translating), how would I go about finding the surface ...