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.

learn more… | top users | synonyms

2
votes
2answers
109 views

Finding the defining equations for a simple quotient variety

First of all let me note that I have no experience at all with modern algebraic geometry so if at all possible I would appreciate an answer not involving the concept of a scheme. I have however some ...
0
votes
1answer
38 views

Flexible affine variety

Let X be an affine variety. Aut(X) is automorphisms group of X. SAut(X) is a subgroup spanned by all unipotent(image of additive group of field K) subgruops. X is called flexible if SAut(X) act ...
0
votes
0answers
17 views

How do projections in 3D with homogeneous coordinates work?

Affine 3D transformations can be expressed in homogeneous coordinates by a matrix $M \in \mathbb{R}^{4 \times 4}$. This means we have 16 parameters to calculate. The first thing I asked myself is how ...
1
vote
5answers
75 views

Prove 3 vectors are collinear

I am asked to prove A(2,4), B(8,6), C(11,7) are collinear using vectors. I can work AB by subtracting A from B and BC by subtracting B from C in vector form. I can say that BC = 2AB. But I don't ...
1
vote
1answer
37 views

Show that a set is an affine subset

Let $V$ be a vector space over $\mathbb{F}$ and $S \subseteq V$ a nonempty set. Theorem: $S$ is an affine subset of $V$ if and only if $$ \forall u,v \in S, \forall \lambda \in \mathbb{F}, \...
0
votes
0answers
17 views

Dimesion of an affine variety- solution verification

I have affine variety $V=\{(t,t^2,t^3)|t\in\mathbb{Q}\}$ and if I'm right I have $I(V)=(y-x^2,z-x^3)$ and coordinate ring is $\mathbb{Q}[V]\cong \mathbb{Q[t]}$ (I set $x=t$). I have this definiton : ...
1
vote
1answer
41 views

What's the intuition behind the definition of the tangent space of $\Bbb R^2$?

I'm reading a book on differential forms and on page one it defines the tangent space to $\Bbb R^n$. In what follows I've translated the statements into two dimensions for simplicity. Let $p$ be a ...
2
votes
0answers
33 views

affine transformations, strategy for finding invariant straight lines

At first lets introduce some notation. $\mathcal{A}^n$ is a $n-$dimensional affine space and $V$ is its associated vector space. For any affine subspace of $\mathcal{M}$, its associated vector space ...
0
votes
0answers
8 views

Transforming a vector configuration to an affine point configuration

I am reading the first chapter of Oriented Matroids by Bjorner et. al, in which they consider the set of vectors $\mathbf{v}_1,\mathbf{v}_2,\dots,\mathbf{v}_6$ given as the columns of the following ...
0
votes
1answer
29 views

Affine variety of geometric progressions.

Let $M$ be affine variety $M \subset \mathbb A^n$. $z\in M \Leftrightarrow z = (x, xy, xy^2, ... , xy^{n-1}),where\hspace{2mm} x,y \in \mathbb C, x \neq 0$. I need to find the equations $f_1, ..., ...
3
votes
3answers
95 views

How to gain an intuition of the affine function's definition?

Here is the definition of Affine Functions according to Stephan Boyd (EE263 Stanford) : 1- I believe linearity is more restrictive property of a function than being affine since it requires $f(0) = ...
0
votes
1answer
33 views

How to determine if an affine transformation would cause reflection?

I have a list of affine transformation matrices and I want to write a code to delete the transformation matrices that applying them on an image would cause reflection. after seeing this image in ...
0
votes
0answers
44 views

Why is half space not affine?

I've read that half spaces are convex but not affine. I'm trying to understand this geometrically. Does it mean that if I connect any 2 points in the half space, it may result in a line that extends ...
0
votes
0answers
8 views

There are exactly 8 isometries $F$ with $F(l_{1})=l_{2}$, $F(l_{2})=l_{3}$, $F(l_{3})=l_{1}$ and $l_{1} \cap l_{2} \cap l_{3} $ is fixed point.

$l_{1}, l_{2}, l_{3} $ are 3 pairwise orthogonal lines in $\mathbb{E_3}$ Prove that there are exactly 8 isometries $F$ with $F(l_{1})=l_{2}$, $F(l_{2})=l_{3}$, $F(l_{3})=l_{1}$ and $l_{1} \cap l_{2} \...
0
votes
0answers
26 views

Theorem of Pappus (analytical proof)

Let $x$, $y$, $z$, be points on a line $L$, and let $x'$, $y'$, $z'$ be points on a line $L'$. Assume $xy'\parallel x'y$ and $y'z\parallel yz'$. Show that $xz'\parallel x'z$. I want to prove this in ...
0
votes
1answer
28 views

Find a plane equation from a line and a point

Consider the point $B(1,0,1)$ and the line $R=(1,1,1) + \alpha[1,1,-1]$. Find the equation of a plane that passes through $B$ and contains $R$. What I tried doing was simply setting two different ...
0
votes
1answer
25 views

Relative postion of a plane and a hyperplane in $\mathbb{R^4}$

I know what happens in $\mathbb{R^2}$ and $\mathbb R^3$. In $\mathbb{R^2}$ , two lines either intersect in a point or they are parallel. In $\mathbb{R^3}$, two lines (or a line and a plane) can ...
0
votes
2answers
27 views

Find a plane perpendicular to the intersection of two affine subspaces

Consider the two following affine subspaces of $\mathbb{R^3} $: $$S=\{x,y,z)\mid 2x+y+z=1\}$$ and $$T=\{(x,y,z)\mid x-y+2z=0\} $$ Find the plane $H$ perpendicular to the intersección of $S$ and $T$, ...
0
votes
1answer
57 views

Find the dimension of an affine subspace

Find the dimension of the affine subspace of $\mathbb{R^5}$ generated by the points $$p=(-1,2,-1,0,4)$$ $$q=(0,-1,3,5,1)$$ $$r=(4,-2,0,0,3)$$ $$s=(3,-1,2,5,2)$$ Is it as trivial as simply finding $\...
0
votes
0answers
33 views

References on composition of affine transformations

I'm doing this problem, but I don't know how to start: Classify the affine transformation of the affine plane obtained composing a special homology and a homotecy. I know that, in a reference $R$...
0
votes
1answer
22 views

Let $f$ be an affine transformation. The fixed points are inside invariant lines?

I'm studying geometry. We're classifying affinities, and my professor wrote: This affine transformation is an hyperbolic transformation. It has a fixed point and two invariant lines. The fixed point ...
2
votes
0answers
27 views

Alternative to affine space

I've been reading up on affine geometry. An affine space (correct me if I'm wrong) is a set of "points" along with a set of translations on those points such that for any two points $P, Q$ there ...
0
votes
1answer
14 views

Finding composed affine tranformation

How do you find the composition $F_1 \circ F_2 \circ F_3$ of three affine transformations if $F_1$ is the reflection about the $yz$-plane, $F_3$ is the rotation over $\pi /2$ around the x-axis and $...
1
vote
0answers
63 views

Show that the singular locus $\Sigma$ of an affine variety $V$ contains no irreducible component of $V$.

This is a problem from Ideals, Varieties, and Algorithms by Cox et. al. Let $V=V_1\cup \cdots \cup V_r$ be a decomposition of variety into its irreducible components. Let $\Sigma$ be the ...
0
votes
0answers
150 views

Estimate transformation between two sets of vectors in different coordinate systems

My question is similar to this, except in this case there is reflection instead of free translation. I have two sets of N vectors in ...
0
votes
1answer
49 views

Building Euclidean space

What's the minimum amount of extra "structure" do we need to add to the general concept of an affine space to get Euclidean space? That includes the concepts of angle and distance, in which we can ...
2
votes
1answer
30 views

Coordinate-free expression of a rotation

I'm interested in coordinate free (non-matrix based) approaches to geometry. What I'd like to do is to show that every Galilean transformation can be written uniquely as the composition of a rotation,...
4
votes
1answer
64 views

Change the variables in $Q(x,y,z)=(x-y+z-1)^2-2z+4$ to have $Q(f(u,v,w))=u^2+v$

I have a problem with this exercise. Initially, they gave me this polynom, and I had to complete the squares: $$Q(x,y,z)=x^2-2xy+2xz+y^2-2yz+z^2-2x+2y-4z+5.$$ I've done it, and I've checked with ...
2
votes
1answer
15 views

Invariant affine subspaces: It's possible that $\dim(f(V))\neq\dim(V)$?

I'm studying geometry right now. I saw that an affine subspace $V$ is invariant under $\ f\ $ if $\ f(V)\subset V$. After reading that, I wondered this: Is it possible that $\dim(f(V))\neq\dim(V)$? ...
4
votes
0answers
67 views

Affine geometry book for physicist

I'm looking for a textbook to help me with understanding the geometry of Galilean relativity and the Galilean group. The reason is that I tried going through V.I. Arnold's Mathematical Methods, but ...
2
votes
2answers
83 views

Map from $\mathbb{A}^1 \rightarrow \mathbb{A}^2$

Let the map $\varphi_n:\mathbb{A}^1 \rightarrow \mathbb{A}^2$ be defined by $t\rightarrow(t^2,t^n)$. -Show that if n is even, the image of $\varphi_n$ is isomorphic to $\mathbb{A}^1$ and $\varphi_n$ ...
1
vote
0answers
100 views

What is the affine space and what is it for?

These two topics already exist: (preface: got in contact with affine space through computer graphics subject in university) What are affine spaces for? What are differences between affine space and ...
0
votes
0answers
47 views

Rational maps between affine varieties

If I want to check that the map $$\phi:C_1\rightarrow C_1,\hspace{0.5cm}\phi(x,y)=(\phi_1(x,y),\phi_2(x,y))$$ between two affine plane curves is rational I just should check that $\phi_1$ and $\...
0
votes
0answers
40 views

Does a bijection that preserves collinearity have to be affine?

Consider a bijection between two affine spaces of the same dimension $n$ (let's assume $n\ge 1$ to avoid trivialities) which sends any three collinear points into collinear points. Must such a ...
0
votes
1answer
17 views

Rotation of a hyperbola in affine geometry

Given the hyperbola $x^2 - 3xy + y^2 + 4x - 5y + 2 =0$ I have translated this by $x+\frac{7}{5}$ and $y-\frac{2}{5}$ and got $x^2 - 3xy + y^2 = \frac{9}{5}$ Now, the bit where I'm stuck; I have ...
1
vote
0answers
21 views

Relationship between affine functions and affine sets?

A function $f: \mathbf{R}^n \to \mathbf{R}^m$ is affine if it is a sum of a linear function and a constant ($f(x) = Ax + b$). A set $S \subseteq \mathbf{R}^n$ is affine if for any $x_1,x_2 \in S$ and ...
1
vote
0answers
57 views

Why affine variety not vector space variety?

I am new to algebraic geometry. A basic question baffles me: why is the setting the affine space not the vector space?
2
votes
0answers
82 views

Geometry book for the university with solved exercises (affine space, euclidean space, etc…)

I'm looking for a book with solved exercises of affine space, affine transformations, etc... I found a lot of books and pdf's with theory, but none of them contained solved exercises, and I'm having ...
3
votes
1answer
53 views

Axioms of Affine Space

In every definition of an affine space I see, the affine space is defined as a set $A$ with an associated vector space $V$ with a group action of $V$ on $A$. But I also see that vector spaces are ...
1
vote
1answer
37 views

counter example - affine space

In the affine-n-space $\mathbb A^n_k$ (where $k$ is algebraic closed) you can define for an algebraic set $X$: $I(X)=\left \{ f\in k[x_1,x_2,...,x_n] | \forall a \in X \,\,\,f(a)=0))\right \}$ I ...
4
votes
1answer
100 views

Affine transformations - the meaning of contractivity

An affine transformation $\omega \colon \mathbb{R}^2 \to \mathbb{R}^2$ is a linear mapping followed by a translation, in other words $$ \omega(x) = Ax+t = \begin{pmatrix} a & b \\ c & d \end{...
2
votes
1answer
43 views

Why only one $\infty $ point for each parallel class of lines in $\mathbb{R}^2$?

I heard of projective geometry since high school. But I never managed to understand it in a systematic way. It is said that the projective plane $\mathbb{P^2}(\mathbb{R})$ over the real numbers $\...
1
vote
0answers
32 views

The hexagon in Pappus' theorem and its relation to the more usual form

I am reading about geometry. Pappus' affine great (compared to a weaker small) theorem is introduced as follows: If the angles of a hexagon lie alternatingly on two intersecting straight lines and ...
0
votes
1answer
63 views

Can all affine transformations be just expressed as a combination of the common transformations we are taught?

(At the time I was writing these questions, I forgot about Projection, and was focusing on isomorphic transformations, so I suspect I may have made some mistake with my presumption in 1. — please ...
2
votes
0answers
97 views

Three lines are concurrent (or parallel) $\iff$ the determinant of its coordinates vanishes.

I'm trying to prove the concurrency condition for three lines lying on a plane. This condition says that: Let \begin{cases} ax + by + cz=0 \\ a'x – b'y + c'z=0 \\ a''x + b''y + c''z=0 \end{...
2
votes
0answers
50 views

Three points of an affine space are collinear $\iff \det(A)=0$, with $A$ the matrix of the barycentric coordinates.

I'm doing this exercise: Let $3$ different points of an affine plane, with barycentric coordinates $X=(x_0,x_1,x_2), Y=(y_0,y_1,y_2), Z=(z_0,z_1,z_2)$ respect to a fixed reference frame. Prove ...
0
votes
0answers
28 views

Proof that dimension of set is n

I have a Discrete Geometry question and I would really appreciate if someone could help me out with this. I have a set $K\subset \mathbb{R}^n$ s.t. $int(B_{n}) \subseteq K \subseteq B_{n}$ where $int(...
0
votes
0answers
33 views

Complete affine variety is a finite set

When I read Newstead's book, "Introduction to Moduli Problems", I found on page 5 the following sentence: "It is easy to see that if $X$ is a complete irreducible variety, then $A(X)=k$; it follows ...
4
votes
0answers
86 views

Affine-regular hexagon in convex body

An affine-regular $n$-gon is a non-degenerate affine image of the regular $n$-gon. It seems to be a standard fact in combinatorial geometry that inside every convex compact set $K\subseteq \mathbb R^2$...
3
votes
0answers
49 views

Could we talk about affine spaces before vector spaces?

I was recommended the book Geometry by Michele Audin by a professor when I asked about learning more about affine geometry. I like the book, but it's raised a question. To me, it seems that it would ...