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Questions tagged [affine-geometry]

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.

30
votes
3answers
7k views

$\mathbb{A}^{2}$ not isomorphic to affine space minus the origin

Why is the affine space $\mathbb{A}^{2}$ not isomorphic to $\mathbb{A}^{2}$ minus the origin?
117
votes
4answers
86k views

What is the difference between linear and affine function

I am a bit confused. What is the difference between a linear and affine function? Any suggestions will be appreciated
6
votes
3answers
1k views

The vanishing ideal $I_{K[x,y]}(A\!\times\!B)$ is generated by $I_{K[x]}(A) \cup I_{K[y]}(B)$?

Let $K$ be a field, $x=(x_1,\ldots,x_m)$, $y=(y_1,\ldots,y_n)$, $A\!\subseteq\!\mathbb{A}^m_K$, $B\!\subseteq\!\mathbb{A}^n_K$. Does there hold $$I_{K[x,y]}(A\!\times\!B)=\langle\langle I_{K[x]}(A) \...
63
votes
7answers
28k views

What are differences between affine space and vector space?

I know smilar questions have been asked and I have looked at them but none of them seems to have satisfactory answer. I am reading the book a course in mathematics for student of physics vol. 1 by ...
6
votes
1answer
382 views

Cross-ratio relations

The way I define the cross-ratio in projectve geometry: Let $P_0,P_1,P_2,P_3$ being four points on a projective line G, such that $P_0,P_1,P_2$ are pairwise distinct. Let $\pi:\mathbb KP^1\rightarrow ...
14
votes
3answers
9k views

Difference between Projective Geometry and Affine Geometry

I recently started reading the book Multiple View Geometry by Hartley and Zisserman. In the first chapter I came across the following concepts.. Projective geometry is an extension of Euclidean ...
2
votes
1answer
240 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 ...
23
votes
5answers
4k views

What are affine spaces for?

I'm studying affine spaces but I can't understand what they are for. Could you explain them to me? Why are they important, and when are they used? Thanks a lot.
9
votes
2answers
12k views

Decomposition of a nonsquare affine matrix

I have a $2\times 3$ affine matrix $$ M = \pmatrix{a &b &c\\ d &e &f} $$ which transforms a point $(x,y)$ into $x' = a x + by + c, y' = d x + e y + f$ Is there a way to ...
12
votes
2answers
2k views

Proof $\mathbb{A}^n$ is irreducible, without Nullstellensatz

As the title suggests, could anyone either provide me with or direct me to a proof that affine n-space $\mathbb{A}^n$ is irreducible, without using the Nullstellensatz? This is an exercise in a ...
2
votes
1answer
148 views

All polynomial parametric curves in $k^2$ are contained in affine algebraic varieties

I have started working through the textbook Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea and I am stuck on one part of an introductory question. The question begins by getting one to ...
5
votes
2answers
539 views

Difference between Euclidean space and $\mathbb R^3$

What is the difference between Euclidean space and $\mathbb R^3$? I have found in some books that they are the same, but in other references like Wikipedia, it says that a vector in $\mathbb R^3$ is ...
5
votes
1answer
323 views

Bijection $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ preserves collinearity $\iff \ \ f(x)=Ax+b$

I don't know how to prove the following: Bijection $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ preserves collinearity $\iff \ \ f(x)=Ax+b$, where $A \in GL_2(\mathbb{R})$, $b$ is a fixed vector in $\...
3
votes
1answer
1k views

Affine Plane of Order 4 Picture?

I am unable to construct an Affine Plane of Order 4, I can construct an Affine plane of Order 3, and 2. But am unable to find the construction of four anywhere, It would be greatly appreciated if ...
5
votes
1answer
1k views

Prove, in this figure, that $EFGH$ is a parallelogram

In the following figure, $ABCD$ is a parallelogram, and $O$ is any point. Parallelograms $OAEB, OBFC, OCGD, ODHA$ are completed. Prove that $EFGH$ is a parallelogram. We can obtain a fairly trivial ...
0
votes
1answer
197 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 ...
20
votes
3answers
20k views

Rotation Matrix of rotation around a point other than the origin

In homogeneous coordinates, a rotation matrix around the origin can be described as $R = \begin{bmatrix}\cos(\theta) & -\sin(\theta) & 0\\\sin(\theta) & \cos(\theta) & 0 \\ 0&0&...
14
votes
1answer
161 views

fitting points into partitions of a square

A friend of mine came up with the following problem: Let $\{X_1, X_2, ..., X_n\}$ be an arbitrarily finite partition of the unit square $[0, 1]^2$. Let $\{P_1, P_2, ..., P_m\}$ be a finite set of ...
11
votes
2answers
1k views

Since the Curvature tensor depends on a connection(not metric), is it the relevant quantity to characterize the curvature of Riemannian manifolds?

1) The definition of the Riemann curvature tensor does not include a metric. So, if we have a smooth manifold(not a Riemannian manifold), we can define the Riemannian curvature tensor for it by just ...
11
votes
1answer
1k 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 $\lambda \in [0,1]$. ...
8
votes
1answer
6k views

Is perspective transform affine? If it is, why it's impossible to perspective a square by an affine transform, given by matrix and shift vector?

I'm a bit confused. I want to program a perspective transformation and thought that it is an affine one, but seemingly it is not. As an example, I want to perspective a square into a quadrilateral (as ...
5
votes
1answer
4k views

Affine sets and affine hull

Mathematically an affine hull can be expressed as $ Aff[C] = \{\theta_1x_1 + \theta_2x_2 .... \theta_nx_n| x_i \in C \ \ \sum_{i=1}^{n}\theta_i = 1 \}$ Intuitively can anyone explain what this ...
6
votes
2answers
11k views

Definition of an affine subspace

I am reading this introduction to Mechanics and the definition it gives (just after Proposition 1.1.2) for an affine subspace puzzles me. I cite: A subset $B$ of a $\mathbb{R}$-affine space $A$ ...
6
votes
0answers
81 views

a geometric problem about area [duplicate]

This is a problem which baffled me for years. Many possibly know the solution. See the figure below. $ABCD $ is some arbitrary quadrilateral. Each side is divided evenly into three parts. It is ...
3
votes
4answers
1k views

Polar form of (univariate) polynomials: looking for a proof

Recently I stumbled upon the following theorem — I'd like to read a comprehensible (i.e. understandable for an engineer) proof for it: Given a polynomial $F(t)$ of degree $n$, there exists a unique ...
2
votes
1answer
459 views

alternative definition of Affine map

Let $f:X\longrightarrow Y$ be a function on real vector spaces (note that $X,Y$ have arbitrary dimensions). If $T(x)=f(x)-f(0)$ is linear, $f$ is called an affine map. Prove that $f$ is affine if and ...
0
votes
3answers
4k views

Equation of a line in homogenous coordinates given 2 points in affine coordinates

So if I have 2 points $A$ and $B$ such that $F(A) = (1; a, a^3)$, and $F(B) = (1; b, b^3)$. how do I find the equation of this line in homogeneous coordinates? So I know how to get a line the "...
0
votes
1answer
955 views

How to prove: Every affine set can be expressed as the solution set of a system of linear equations [closed]

How can I prove that every affine set can be expressed as the solution set of a system of linear equations? Please note that set $C \subseteq \textbf{R}^n$ is said to be affine if for any $x_1 , x_2 ...
11
votes
3answers
671 views

Product of two algebraic varieties is affine… are the two varieties affine?

Let $X_1$ and $X_2$ two algebraic varieties such that their product $X_1\times X_2$ is affine. Are $X_1$ and $X_2$ affine then? If this is not true, could you give a counterexample?
7
votes
1answer
685 views

Two affine varieties are not isomorphic

Given the affine variety $A:=Z(y^{2}-P(x)) \subset \mathbb{C} ^{2} $, where $P(x)$ is a polynomial with $\deg P \geq 2$, I need to show that $A$ is not isomorphic to $ \mathbb{C}$. I know it has ...
6
votes
1answer
159 views

Is a similarity map necessarily affine linear?

My text on fractal geometry introduces the following definition: A map $S: \mathbb R^n \to \mathbb R^n$ is called a similarity map if $$\exists c>0 \ \forall x,y \in \mathbb R^n: |S(x)-S(y)|=c|x-...
5
votes
1answer
547 views

Effect the zero vector has on the dimension of affine hulls and linear hulls

I am currently working through "An Introduction to Convex Polytopes" by Arne Brondsted and there is a question in the exercises that I would like a hint, or a nudge in the right direction, please no ...
5
votes
0answers
111 views

Is the complement of a complex affine algebraic set in an irreducible complex affine algebraic set (path) connected in the euclidean topology?

Let $n \ge 2$, and $V$ be an affine algebraic set in $\mathbb C^n$ and $W$ be an irreducible affine algebraic set in $\mathbb C^n$, with $V \subsetneq W$ ; then is it true that $W \setminus V$ is ...
4
votes
2answers
1k views

Sphere to Ellipsoid affine transformation matrix

I am trying to find the minimum bounding box of an ellipsoid. In my search, I found this answer and also some other nice descriptions like this one to the problem. I am not a mathematician (I need ...
3
votes
1answer
221 views

Affine space $A^n$ and definition of difference.

I'm not sure if this question would be more appropriate in Physics.SE, if so let me know. I need help in understanding this quote from "Arnold - Mathematical Methods in Classical Mechanics" (This is ...
3
votes
0answers
114 views

What is the relationship between an affine space and an affine set?

I just learned what an affine space is and its basic properties. It is formally defined as $\Bbb A=(A,V,f)$ where $A$ is a set of points, $V$ is a vector space, and $f:A^2→V$ is a function st. for any ...
2
votes
0answers
83 views

Does an irreducible real affine algebraic set/ its complement has finitely many connected components in the Euclidean topology?

Let $n\ge 2$ and $V$ be an irreducible affine algebraic set in $\mathbb R^n$ . Then is it true that $V$ has only finitely many connected components in the Euclidean topology of $\mathbb R^n$ ? Does $\...
1
vote
1answer
2k views

Intersection of affine subspaces is affine

So if I have two affine subspaces, each is a translate ( or coset) of some linear subspace. I want to show that the intersection of such affine subspaces is again affine, particularly in $\mathbb{R}^...
1
vote
1answer
54 views

Basis of an affine subspace

Consider an affine subspace $D$ of an affine space or affine plane $\mathcal{A}$. Every set of points that are not elements of a proper affine subspace of $D$ is called a generating set of $D$. If ...
0
votes
1answer
49 views

Is the group generated by $f(x)= x+2^{\nu_2(2x)}$ and $g(x)=2x$ a 2-dimensional affine space on $\Bbb Z[\frac12]\setminus 0$?

Does the free abelian group generated by composition of $f(x)=x+ 2^{\nu_2\left(2x\right)}$ and $g(x)=2x$ define a 2-dimensional affine space on $\Bbb Z\left[\frac12\right]\setminus0$? $2^{\nu_2(x)}$ ...
0
votes
1answer
1k views

Affine plane of order 4?

I cannot seem to construct an affine plane of order 4. I have the construction for order 3- but cannot seem to come up with or find the construction for 4 anywhere. Could someone show me a picture of ...
0
votes
1answer
1k views

dimension of quotient space

I am confused about the following: In Wiki: => dim(vector space) - dim(subspace) = dim(quotient space) In S. Boyd's textbook of cvx (p.22) => dim(subspace) = dim(affine set) Problem: As far ...
4
votes
4answers
473 views

Sufficient condition for a function to be affine

If for a function $f:\mathbb{R}\to\mathbb{R}$, I can prove for any real $x,y$, that $f(\frac{x+y}{2})=\frac{f(x)}{2}+\frac{f(y)}{2}$, can I say for sure that it is affine, as in of the form $f(x)=ax+b$...
4
votes
2answers
607 views

Computing irreducible components of algebraic set

Consider the algebraic set $V(X^2-YZ,X-XZ)$. Find the irreducible components of this set and show that $I(V)=(X^2-YZ,X-XZ)$. I reasoned that $X-XZ=0$ iff $X=0$ or $Z=1$. If $X=0$, we get $Y=0$ or $Z=...
4
votes
0answers
60 views

Translating and inflating a set of $k$-dimensional subspaces of $\mathbb F_p^n$ to form a cover by affine hyperplanes?

Fix a prime number $p$ and consider the affine space $V = \mathbb F_p^n$. Let $k < n/2$. Consider subspaces $V_1, \ldots, V_n \subseteq V$ of dimension $k$, and take $v_i \notin V_i$. Do there ...
4
votes
1answer
198 views

Centre of a quadric

I found the following sentence in my linear linear algebra book (affine and projective geometry): $Q:V \to \mathbb{K}$ is a quadric (quadratic function) and $\alpha\in Aff(V)$. $Aff(V)$ is the set of ...
3
votes
1answer
568 views

Prove that Newton's Method is invariant under invertible linear transformations

As part of an assignment for a course in nonlinear optimization, I have to prove that Newton's Method is invariant under a non-singular (invertible) linear transformation, $x=Uy$. Specifically, I was ...
3
votes
2answers
75 views

Is the notion of “affineness” more general than “linearity”, or vice versa?

This is an incredibly dumb question, so please bear with me. An affine transformation $T$ is equal to a linear transformation $L$ plus a translation $t$. This suggests that affine transformations are ...
3
votes
2answers
275 views

Identifying the Plane at Infinity in the World Necessitates Determining the Affine Geometry of the World?

Page 18 of my computer vision textbook, Multiple View Geometry in Computer Vision (Second Edition), by Hartley and Zisserman, states the following: 1.8 Auto Calibration $\vdots$ Generally ...
3
votes
1answer
151 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 \...