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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23 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 ...
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0answers
12 views

How to go from Affine to a Non-linear transformation

If you were able to take two sets of matrices and transform one to attempt to match the other using an affine method such as Least Squares, how could you replicate this process better using a ...
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0answers
5 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 ...
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1answer
27 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
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3answers
73 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) = ...
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1answer
15 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 ...
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0answers
27 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 ...
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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} ...
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0answers
19 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 ...
0
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1answer
22 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
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1answer
22 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 ...
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2answers
21 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
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1answer
45 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
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0answers
24 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 ...
0
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1answer
18 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
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0answers
23 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 ...
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1answer
12 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 ...
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0answers
51 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 ...
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0answers
37 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
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1answer
39 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
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1answer
28 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 ...
4
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1answer
63 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
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1answer
14 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 $\dim(f(V))\neq\dim(V)$? ...
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50 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
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2answers
81 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 $\varphi_n$ ...
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0answers
47 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 ...
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0answers
42 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
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0answers
37 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 ...
0
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1answer
15 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 ...
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0answers
17 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 ...
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0answers
49 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?
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0answers
31 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
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1answer
36 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
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1answer
34 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
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1answer
96 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 ...
2
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1answer
39 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 ...
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0answers
30 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
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1answer
53 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 ...
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0answers
59 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 ...
2
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0answers
41 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 ...
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0answers
25 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 ...
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0answers
27 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
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0answers
73 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 ...
3
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0answers
48 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 ...
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2answers
116 views

An irreducible quadric hypersurface is rational?

Here quadric hypersurface just means it is generated by a polynomial with degree 2. I can guess the idea is to project the hypersurface from a fixed point P, to some plane by drawing a line through ...
-1
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1answer
24 views

Affine hull of two points in R4 [closed]

I try to describe an affine hull of two points (1,3,2,4) and (1,4,2,3) so i try to make the linear equation which describe it .
2
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0answers
38 views

Affine geometry textbook

What's a good recommendation for a book on affine geometry at the undergrad level? I ask because I skimmed through the first bit of Vladimir Arnold's Mathematical Methods of Classical Mechanics and ...
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0answers
20 views

The points of an affine variety where a given rational map is regular

This is a very general question, because I can't seem to find this in the notes for the course. So let $X$ be an affine algebraic variety and $K[X] = K[\mathbb{A}^n]/I(X) = ...
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1answer
32 views

Let $B=A+t\vec{AC}$ with $t\ne1$, show that $A=B+s\vec{BC}$ for $s=t/(t-1)$

I have to prove the following: Let $B=A+t\vec{AC}$. Let $t:=(A, B, C)=\frac{\vec{AB}}{\vec{AC}}$. Prove that $(B, A, C)=\frac{t}{t-1}$. I've been trying by two different ways but I always obtain ...
0
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1answer
27 views

Every irreducible component of an affine cone contains its vertex

Let $X=V(F_1,...,F_k)\subset \mathbb{P}^n$with $F_i\in k[X_0,...,X_n]$ an projective algebraic set. Let $C(X)\in \mathbb{A}^{n+1}$ the affine cone over $X$, that is $C(X)=\theta^{-1}(X)\cup ...