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.
127
questions
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$\mathbb{A}^{2}$ 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?
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5
answers
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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.
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8
answers
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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 ...
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3
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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) \...
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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=...
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2
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What does it mean to be "affinely independent", and why is it important to learn?
I was studying linear optimization and i saw the term Affine independence. I came across this http://www.cis.upenn.edu/~cis610/geombchap2.pdf while trying to get a better understanding of the topic.
...
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6
votes
1
answer
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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 ...
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3
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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 ...
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22
votes
3
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What is the 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 ...
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2
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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 ...
- 153
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2
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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 ...
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6
votes
1
answer
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Affine group, semi-direct product and linear transformations
According to wikipedia the Affine group is the semi-direct product of a vector space $V$ and the general linear group $GL(V)$. Here is the definition of the semi-direct product in terms of matrices ...
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3
votes
1
answer
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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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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.
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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 ...
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3
answers
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How to define an affine transformation using 2 triangles?
I have $2$ triangles ($6$ dots) on a $2D$ plane.
The points of the triangles are: a, b, c and x, y, z
I would like to find a ...
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3
votes
1
answer
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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 ...
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3
votes
1
answer
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Connectedness and path connectedness, of irreducible affine algebraic set in $\mathbb C^n$, under usual Euclidean topology
Let $n\ge 2$ and $V$ be an irreducible affine algebraic set in $\mathbb C^n$ . Then is it true that $V$ is path connected (or at least connected ) in the usual Euclidean topology of $\mathbb C^n$ ?
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2
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1
answer
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a full proof for the Fundamental Theorem of affine geometry
Let $n\geq 2$ and a bijection $T:\ \mathbb R^n\longrightarrow\mathbb R^n$ satisfies
$\ T(0)=0\,$ and $\,T$ maps straight lines to straight lines.
Then $T$ is a linear map.
My proof :
Let $(P)\subset\...
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votes
3
answers
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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&...
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2
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Prove that $v_0, v_1,...,v_k$ are affinely independent if and only if $v_1 - v_0,...,v_k - v_0$ are linearly independent
Definition: Let $v_0, v_1.. v_k$ be points in $\mathbb{R}^d$. These points are called affinely independent if there do not exist real numbers $\alpha_0, \alpha_1...\alpha_k$ that are not all zero ...
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2
answers
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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$ ...
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6
votes
1
answer
356
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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-...
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3
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Intersection of affine subspaces is affine
If I have two affine subspaces, each is a translation (or coset) of some linear subspace. I want to show that the intersection of such affine subspaces is also affine, particularly in $\mathbb{R}^d$. ...
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6
votes
2
answers
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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 ...
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votes
1
answer
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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 $\...
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2
answers
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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 ...
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5
votes
1
answer
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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 ...
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3
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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 "...
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3
votes
1
answer
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vector space dimension of linear varieties
Suppose $Y=Z(f_1,\cdots ,f_r)\subseteq \mathbb{A}^n_k$ where each $f_i$'s are linear homogeneous polynomials which are $k$-linear independent. Then $Y$ is also a vector space over $k$. My question: Is ...
user276115
2
votes
1
answer
98
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Shortest distance between two affine subspaces through orthogonal projection
I'm trying to show the following:
Let $V$ be a finite dimensional euclidean vector space, with two vector subspaces $S_1 \subset V$ and $S_2 \subset V$. Suppose that $X, Y$ are affine subspaces with $...
- 33
2
votes
2
answers
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Jacobian ideal of $f$ and normality
Let $k$ be an algebraically closed field of characteristic $0$. Suppose $f\in k[X_1,\ldots,X_n]$ is such that $f\in \mathcal{J}(f)$, where $\mathcal{J}(f)$ is the Jacobian ideal of $f$ (i.e. ideal ...
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1
answer
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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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votes
2
answers
775
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Are non-bijective isometries necessarily affine functions?
Any isometry on $\mathbb{E}^n$, in the sense of a distance-preserving bijective function, is an affine function (see here, here and here). An affine function is defined in the following way:
If $X$ ...
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votes
2
answers
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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 ...
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4
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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 + \...
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3
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Since the Curvature tensor depends on a connection (not metric), is it the relevant quantity to characterize the curvature of Riemannian manifolds?
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 ...
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1
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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 $\lambda \in [0,1]$.
...
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14
votes
1
answer
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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 ...
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votes
3
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What is the difference between linearly and affinely independent vectors?
What is the difference between linearly and affinely independent vectors? Why does affine independence not imply linear independence necessarily? Can someone explain using an example?
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2
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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}^...
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11
votes
1
answer
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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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3
answers
951
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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?
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5
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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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10
votes
1
answer
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If $u_i$ are affinely independent, are they also linearly independent?
I am wondering about affinely independent and just linearly independent. On Wikipedia it is explained that $u_i$ are affinely independent if $u_1 - u_0, ...,u_k -u_0$ are linearly independent. It is ...
- 1,767
10
votes
1
answer
447
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Does ONLY the ellipse have these properties?
Two parallel lines are tangent to an ellipse. Between those two lines, every line parallel to those two intersects the ellipse in two points.
The precise midpoint between those two points lies exactly ...
9
votes
1
answer
10k
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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 ...
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votes
1
answer
1k
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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 ...
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6
votes
3
answers
9k
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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 ...
- 1,425
6
votes
4
answers
2k
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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 ...
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