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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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 ...
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1answer
20 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 ...
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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
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 ...
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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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102 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 ...
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1answer
44 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 ...
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1answer
29 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 ...
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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 ...
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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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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 ...
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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 $\varphi_n$ ...
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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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46 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 ...
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39 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 ...
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1answer
16 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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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 ...
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52 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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50 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 ...
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1answer
46 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 ...
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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 ...
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1answer
98 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 ...
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1answer
41 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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31 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 ...
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1answer
56 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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77 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 ...
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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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26 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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31 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 ...
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77 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 ...
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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 ...
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2answers
144 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 ...
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1answer
32 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 .
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40 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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21 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 ...
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1answer
33 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 ...
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27 views

Example of affine varieties with a restriction on the dimension of an irreducible component of their intersection

On Hartshorne's book of algebraic geometry, exercise 2.11 (c) page 13 it's ask to prove that for any two linear varieties $Y,Z$ in $P^n$, with $dim Y=r$, $dim Z=s$, if $r+s-n\geq 0$, then $Y\cap Z\neq ...
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1answer
61 views

Show that halfspace is not affine.

Let us define half-space as $$ C = \{x\mid a^Tx\leq b\} $$ Intuitively (or geometrically), I understand why halfspace is not affine. But while I prove that half-space is convex, it seems to hold for ...
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1answer
43 views

Why is this function an affine function?

Why is the following function an affine function? $$f(x)=(P^{1/2}x,c^Tx)$$ I learnt that affine functions have the pattern like $f(x)=Ax+b$, is there any relation between the two function? Maybe I ...
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2answers
118 views

Any continuous function with the mean value property is affine

A function $f(t)$ on an interval $I=(a,b)$ has the mean value property if $$f\left(\frac{s+t}{2}\right)=\frac{f(s)+f(t)}{2} \quad s,t\in I$$ Show that any affine function $f(t)=At+B$ has the mean ...
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1answer
26 views

Finding an invertible element that is not invertible in a subring of a coordinate ring

First, let $K$ be an algebraically closed field. Now let $\phi: X \rightarrow Y$ be a dominant map of affine varieties, so namely $\phi^\star: K[Y]\rightarrow K[X]$ is an injection, and so we have a ...
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1answer
35 views

Is it possible to define the vertex coordinates of a dodecahedron as integer multiples of basis vectors in an affine space?

I am trying to think of an affine space defined by cells (whose edges may differ in length), such that the coordinates of all the vertices in a regular dodecahedron can be expressed as integer ...
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1answer
45 views

Ideals of affine variety

Let $X=V(y^2+x^2y-x^2)$ be an affine variety of affine space of 2 variables. What is the ideals of affine variety, $I(X)$. We know that $X$ consists of the curve $y^2-x^2y=x^2$. So how do we determine ...
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1answer
69 views

basic question on varieties (algebraic geometry)

I study basic algebraic geometry and I saw this exercise: V is the complement of the twisted cubic in $$ A_c^3. $$ i.e. $$ V = A_c^3 - \{(t^3, t^4, t^5) \mid t\in c\}. $$ 1. How can I proove that V is ...
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2answers
45 views

Non-isomorphic two-variable varieties in characteristic 2

Let $K$ be an algebraically closed field of characteristic $2$, and we will consider affine varieties of $\mathbb{A}^2$. Let $X = Z(y-x^2)$ and $Y = Z(xy-1)$. I have shown through some exhausting case ...
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1answer
103 views

Do any books or articles develop basic Euclidean geometry from the perspective of “inner product affine spaces”?

Definitions. By a vector space, I simply mean an $\mathbb{R}$-module. By an affine space, I mean a vector space $X$ (the "translation space") together with a set $P$ (of "points"), together with an ...
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64 views

Some “facts” on oriented angles in the Euclidean affine space of dimension 2

Let $\mathcal{E}^2$ be an Euclidean affine space of dimension $2$ oriented by $R=(O,B=\{u_1,u_2\}$. Let $$\mathcal{r}=\{P \in \mathcal{E}_2 \text{ such that } \overrightarrow{AP} \in \langle u ...
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1answer
42 views

Rigorous definition of “oriented line” in an Euclidean affine space

Let $\mathcal{A}^n$ be an affine space of dimension $n$. For example, let's take $n=3$. A line $\mathcal{s}$ of $\mathcal{A}^3$ is an affine subspace of dimension $1$, that is: $\mathcal{s}=\{P ...
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1answer
201 views

Decompose affine transformation (including shear in x and y)

How can I fully decompose an affine transformation matrix that includes tx, ty, rotation (theta), scale-x (sx), scale-y (sy), shear-x and shear-y? Using this matrix as example: $$A = ...