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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.

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
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 ...
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?
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.
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&...
17
votes
2answers
2k views

What *is* affine space?

In my recent reading of various books and notes on algebraic geometry and scheme theory, I have come across three definitions of affine $n$-space over a field $k$: $\mathbb{A}_k^n$ is $k^n$ 'without ...
17
votes
4answers
1k views

When is a vector “glued” to the origin?

Let $V$ be a real finite-dimensional vector space (I guess this forces $V$ to be $\mathbb{R}^n$). My intuition is that a vector $v\in V$ must be "glued" to the origin, since the origin is the only ...
14
votes
4answers
3k views

Why is the affine hull of the unit circle $\mathbb R^2$?

In Boyd's "Convex Optimization" it defines the affine hull of a subset $C$ of $\mathbb R^n$ as $$\text{aff} C = \left\{\theta_1 x_1 + \ldots +\theta_k x_k \mid x_1, \ldots x_k \in C, \theta_1 + \...
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 ...
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 ...
13
votes
3answers
867 views

Hole in the axioms of Hartshorne's “Foundations of Projective Geometry”?

I'm currently working my way through Foundations of Projective Geometry by Hartshorne, and he states the axioms characterizing an affine plane as: An affine plane is a set $\mathbb{X}$ together ...
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 ...
11
votes
3answers
670 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?
11
votes
1answer
204 views

Looking for elementary proof that irreducible/smooth curve in $\mathbb C^2$ is connected in Euclidean topology of $\mathbb C^2$

Let $f(X,Y)\in \mathbb C[X,Y]$ be an irreducible polynomial. I know that the zero set of $f$ , $V(f):=\{(a,b)\in \mathbb C^2 : f(a,b)=0\}$ is connected in the usual Euclidean topology of $\mathbb C^2$ ...
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 $\lambda \in [0,1]$. ...
11
votes
0answers
307 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 ...
10
votes
2answers
995 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 ...
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 ...
9
votes
2answers
4k views

Definition of Affine Independence in Brondsted's Convex Polytopes?

At one point in the book (An Introduction to Convex Polytopes, by Arne Brondsted) a definition of affine independence is given as follows, An n-family $(x_{1},...,x_{n})$ of points from $\mathbb{R}^...
8
votes
1answer
2k views

What is an affine space?

I am having trouble understanding what an affine space is. I am reading Metric Affine Geometry by Snapper and Troyer. On page 5, they say: "The upshot is that, even in the affine plane, one can ...
8
votes
4answers
545 views

Show that every rotation in $\mathbb{R^3}$ can be written as the product of two rotations of order 2.

Show that every rotation in $\mathbb{R^3}$ can be written as the product of two rotations of order 2. Here's my attempt at a solution: We know that any rotation in $\mathbb{R^3}$ can be ...
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 ...
7
votes
1answer
118 views

Zariski topology and product topology on $\mathbb A_k^m \times \mathbb A_k^n$

Let $k$ be an algebraically closed field and $\mathbb A_k^n$ be the affine $n$-space. We can give the Zariski topology on it. Then consider $X=\mathbb A_k^m \times \mathbb A_k^n$ under product ...
7
votes
1answer
3k views

Affine Independence $\iff$ Linearly Independent

I guess I'm having some trouble getting my head around the notion of affine independence. As I've been taught, a set of vectors $\{\vec{x_1},\ldots,\vec{x_n}\}\subset \mathbb{R}^d$ is affinely ...
7
votes
1answer
684 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 ...
7
votes
1answer
379 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 ...
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
3answers
5k views

Origin in vector space?

In the wikipedia article about vector space I do not understand this sentence Roughly, affine spaces are vector spaces whose origin is not specified. A vector space does not need an origin. When ...
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) \...
6
votes
2answers
197 views

Given four points, determine a condition on a fifth point such that the conic containing all of them is an ellipse

The image of the question if you don't see all the symbols The given points $p_1,p_2,p_3,p_4$ are located at the vertices of a convex quadrilateral on the real affine plane. I am looking for an ...
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-...
6
votes
2answers
731 views

About the ramification locus of a morphism with zero dimensional fibers

This question arises from my somewhat frustrating attempts to understand what etale means (in the world of algebraic varieties for now) and marry the more advanced algebraic geometry references and ...
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 ...
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 ...
6
votes
0answers
221 views

Affine plane curves classification

Define an affine plane conic as $\operatorname{Spec}A$ where $A=k[x,y]/(f)$ and $f$ a quadratic polynomial with no multiple factors. Define an equivalence relation on the set of alline plane conics ...
6
votes
0answers
236 views

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

Let $G$ be an affine group scheme, and $\mathrm{Dist}(G)$ its hyperalgebra. I am wondering what is the relationship between $\mathrm{Dist}$(G) and $G$ interms of Cohomology? Is there a cohomology ...
6
votes
1answer
206 views

Existence of a special set of q+2 points in the finite affine plane over $\mathbb F_q$

I am working in the finite affine plane over $\mathbb F_q$ with $q=2^n$. Such a plane has $q^2$ points, $q^2+q$ lines, each line has $q$ points, and by a point is passing $q+1$ lines. There are $q+1$ ...
6
votes
4answers
885 views

Term for similarity transformation which is not a translation

What's the best (i.e. most concise) term to refer to an orientation-preserving similarity transformation which is not a translation? Here are some descriptions I could think of, but all of them feel ...
5
votes
2answers
5k views

affine vs projective tranformation

I'm trying to grasp the difference between the affine and projective transformations...I got the point of the line at infinity but their matrix representation is not yet clear enough: ...
5
votes
1answer
1k views

Intersection of affine varieties is affine

Let $M,N\subset\mathbb{P}^n$ quasiprojective varieties such that there exist isomorphisms $i\colon M\rightarrow Z(a)\subset \mathbb{A}^m$ and $j\colon N\rightarrow Z(b)\subset \mathbb{A}^m$ for ideals ...
5
votes
1answer
322 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 $\...
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 ...
5
votes
3answers
2k views

Showing that if $fg=gf$ and $fh=hf$, then $gh=hg$, where $f$, $g$, and $h$ are affine functions

Given real numbers $a$ and $b$ ($a \ne 0$), let $f_{a,b}$ be the function $\mathbb{R} \to \mathbb{R}$ defined by $x \mapsto ax+b$. The set of such functions is a permutation group on $\mathbb{R}$, ...
5
votes
2answers
68 views

Is there an algebraic structure for affine spaces

So in normal Vector spaces, you have vectors and scalars, where the vectors form a field, and scalars form an abelian group. But in affine spaces you have the notion of point as opposed to vector, ...
5
votes
2answers
538 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
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 ...
5
votes
1answer
213 views

Find the tangent space of $\mathrm{Aff}(n)$

Find the tangent space of $\mathrm{Aff}(n)$. see Proof: Tangent space of the general linear group is the set of all squared matrices $\mathrm{Aff}(n)$ is the set of all matrices of the form $$ ...
5
votes
1answer
546 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 ...
5
votes
1answer
185 views

Affine map $f : P_1 \to P_2$ between two planes

I'm learning affine geometry, specifically affine maps, and need help with the following problem : We give the affine planes $$P_1 = \{(x, y, z) \in \mathbb R^3 : 3x + 2y + z = 6\} \quad \...