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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8
votes
0answers
197 views

Transformations that map points inside the sphere to points inside the sphere

I am trying to figure out what is the most general linear transformation that maps points inside the unit sphere to points inside the unit sphere. I am slightly abusing the word linear here by ...
6
votes
0answers
587 views

Is every convex-linear map an affine map?

Let's say that a map $f: V \rightarrow W$ between finite-dimensional real vector spaces is convex-linear if $f(\lambda x + (1-\lambda)y) = \lambda f(x) + (1-\lambda)f(y)$ for all $\lambda \in [0,1]$. ...
4
votes
0answers
112 views

Differences and similarities between Euclidean and Minkowski geometry

I am trying to get my head around the differences and similarities between Euclidean and Minkowski plane geometry. AS far as I understand it they are both affine geometries meaning the parallel ...
4
votes
0answers
37 views

What does affine invariance mean in the context of the Newton's method?

The textbook Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (by Ascher, Mattheij, and Russell) states on page 329: [W]e observe that Newton's method is affine ...
4
votes
0answers
76 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 ...
4
votes
0answers
117 views

Is it true that blowing up a quasi-affine variety at a nonsingular point never introduces new singularities?

If we let $M$ be a quasi-affine variety, is it true in general that the blowup of $M$ at a non-singular point $p$ does not introduce new singularities? I came across this statement in my reading, but ...
4
votes
0answers
95 views

Describing $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as gluing of affine charts

How can we describe $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as a gluing of affine charts? I'm having trouble with this problem, perhaps because I don't understand ...
3
votes
0answers
87 views

When does a homogeneous morphism have only finite fibers?

Suppose that we have a map ${\bf f}:=(f_1,f_2,\cdots ,f_n):\mathbb{C}^n\rightarrow \mathbb{C}^n$ given by $$ \mathbb{C}^n\ni {\bf z}:=(z_1,z_1,\cdots,z_n)\rightarrow \big(f_1({\bf z}),f_2({\bf ...
3
votes
0answers
41 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 ...
3
votes
0answers
331 views

Good textbook on geometries

I am looking for a good textbook that thoroughly covers euclidean, affine, projective and non-euclidean geometries. I will be starting graduate school in algebraic geometry next year and I would like ...
3
votes
0answers
114 views

Smallest $\sigma$-algebra of $\Bbb A^n$ containing all affine algebraic subsets.

Let $k=\overline k$. What is the smallest $\sigma$-algebra $\Sigma$ containing all affine algebraic subsets? I am interested in the analogous question for $\operatorname{Spec} k[x_1,\dots,x_n]$, but ...
3
votes
0answers
74 views

Cardinality of quasiaffine variety

The excercise 1.4.8(a) of Hartshorne's Algebraic Geometry says Show that any variety of positive dimension over $k$ has the same cardinality as $k$. Using Hartshorne's notation, we define a ...
3
votes
0answers
140 views

Measuring Similarity of Affine Transformations

I am currently working on a problem where a calibration Algorithm provides me with an Affine Transformation that transforms a 2D Image to it's assumed Position in a 3D Volume. To evaluate the accuracy ...
3
votes
0answers
620 views

Meaning of affine transformation

From Wikipedia, I learned that an affine transformation between two vector spaces is a linear mapping followed by a translation. But in a book Multiple view geometry in computer vision by Hartley ...
2
votes
0answers
79 views

Questions about Affine algebraic group scheme over an infinite field K

For an easily comprehension of my questions I write some definitions: An affine algebraic group scheme over $K$ is a representable group-functor from $K$-algebras category, with a finitely generated ...
2
votes
0answers
44 views

Automorphism of $\mathbb{A}^2$ which maps the finite set of points to the finite set of points

Let $\mathrm{k}$ be infinite field. $P_1,\dots,P_n, Q_1,\dots,Q_n \in \mathbb{A}^2$ and $P_i \neq P_j, Q_i \neq Q_j$. I want to find automorphism(in a.g. sense) which maps $P_i$ to $Q_i$. I have tried ...
2
votes
0answers
30 views

Dependence of linear algebra theorems of the commutativity of the field.

In the linear algebra course I took vector spaces where introduced with a (commutative) field. The classical theorems are proven under this assumption. However, I was wondering what implications it ...
2
votes
0answers
15 views

Does $\dim (A_1\otimes A_2)=\dim(V_1\otimes V_2)$ for all affine spaces $A_{1,2}$, their vector spaces $V_{1,2}$ and the operations $\cap,+$?

Let $A_1=P_1+V_1,A_2=P_2+V_2$ be affine spaces. My teacher uses $\dim$ on affine spaces and the embedded vector spaces interchangeably, which is correct by definition for $\dim A_1=\dim V_1$, but ...
2
votes
0answers
77 views

What if segments are not infinitely divisible?

I almost got myself mixed up I a philosophical discussion again. Somebody was talking about the Planck time and length which are, according to him, the minimal possible time and distance, and how ...
2
votes
0answers
51 views

A question for epigraph and affine function

I'm working with a problem the epigraph of a real-valued function $f$ is a halfspace $\iff$ $f$ is a real-valued affine fuction. First, I quickly recall some definitions: A (closed) halfspace is a ...
2
votes
0answers
57 views

How to describe the variety for lines through the origin

Let $I(x)$ be an ideal generated by the affine variety $X$. $I(X)=\langle xy,yz,zx\rangle$ gives the three coordinate axes through the origin (because it is the intersection of the $3$ planes $xy=0$, ...
2
votes
0answers
41 views

Semisimple part of a nilpotent connected affine algebraic group

These notes on affine algebraic groups mention the following theorem. Let $G$ be a connected nilpotent affine algebraic group (over an algebraically closed field $k$), and denote $G_s$ and $G_u$ ...
2
votes
0answers
38 views

Monotononically Increasing Water Filling Solution?

$\mathbf{I}$ is the $K\times K$ identity matrix. $\mathbf{h}_i\in\mathbb{C}^{M\times1}\quad\forall1\leq i\leq K$ are column vectors. Consider the solution of the convex optimisation problem over ...
2
votes
0answers
172 views

categorical description of the Minkowski sum of polytopes

Consider the category $\textbf{Poly}$ of polytopes, where the objects are convex hulls of finite subsets of $\mathbb{R}^d$ for arbitrary $d \in \mathbb{N}$ and where the morphisms are affine maps ...
2
votes
0answers
101 views

Relationship between hyperalgebra (algebra of distributions) of an affine group scheme to its cohomology

Let G be an affine group scheme, and Dist(G) its hyperalgebra. I am wondering what is the relationship between Dist(G) and G interms of Cohomology? Is there a cohomology theory for Dist(G), if so ...
2
votes
0answers
57 views

$\varphi$ affine if with $V = \{V_{i}\}_{i=1, .., m}$ affine covering such that $\varphi^{-1}(V_{i})$ affine

I found an exercise of mine, that I solved, but now I am not sure anymore about the details, and some parts of what I wanted to say there. I have the affine open sets $V_{1}, ..., V_{m}$ and ...
2
votes
0answers
52 views

$f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$

Suppose, that $f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$ Please help me prove, that $f$ is a composition of rotation about an axis and moving along this axis. I don't ...
2
votes
0answers
98 views

Affine set and linear equation

Prove or disprove the following statement. For any affine set C in R^n, there exists a solution set of linear equation that express C.
1
vote
0answers
13 views

why should add one column using Moore-Penrose pseudoinverse

I have a code from someone that I dont understand: This code is written in matlab and the function is to estimate linear geometric transformation [1] of a matrix using pinv. The size of first matrix ...
1
vote
0answers
22 views

Octonions - affine space

I'm writing a project on Cayley's algebra. I have some topics which I have to follow and I've managed to solve most of them,except 2. I have written about their rule of multiplication,together with ...
1
vote
0answers
32 views

transform orthonormal coordinate system to another

I have one orthonormal coordinate system ABC that it's origin is the point p0. I would like to transform it to another orthonormal coordinate system A'B'C', that it's origin is p1. I know how to ...
1
vote
0answers
18 views

Proving Pappus' theorem in a finite Affine Geometry

Let $\mathcal{A}$ be an affine plane with a finite amount of points on each line. Suppose that Desargues' theorem holds in $\mathcal{A}$. Then it is known that we can associate a division ring ...
1
vote
0answers
14 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 ...
1
vote
0answers
16 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 ...
1
vote
0answers
26 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 ...
1
vote
0answers
26 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 ...
1
vote
0answers
24 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 ...
1
vote
0answers
74 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 ...
1
vote
0answers
40 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, ...
1
vote
0answers
40 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 ...
1
vote
0answers
35 views

Linearizable reductive group action.

Let $k$ be a 0 characteristic field, and $G$ a reductive group in $GA_2(k)$ (the group of automorphisms of k[x,y] as k-algebra). How is it possible to deduce that $G$ is conjugated to a subgroup of ...
1
vote
0answers
53 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 ...
1
vote
0answers
58 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 ...
1
vote
0answers
51 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 ...
1
vote
0answers
139 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: ...
1
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0answers
189 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 + ...
1
vote
0answers
99 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 ...
1
vote
0answers
101 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.
1
vote
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 $\bigcup\mathcal{H}$, ...
1
vote
0answers
680 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 ...