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.
10
votes
2answers
307 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 ...
10
votes
3answers
129 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?
8
votes
3answers
784 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?
7
votes
0answers
81 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 ...
6
votes
2answers
265 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 ...
6
votes
1answer
128 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 ...
6
votes
3answers
324 views
Why is the affine hull of the unit circle R^2?
In Boyd's "Convex Optimization" it defines the affine hull of a subset $C$ of $R^n$ as
$$\text{aff} C = \left\{\theta_1 x_1 + \ldots +\theta_k x_k \mid x_1, \ldots x_k \in C, \theta_1 + \ldots ...
5
votes
3answers
150 views
Showing that if $fg=gf$ and $fh=hf$, then $gh=hg$, where $f$, $g$, and $h$ are affine functions
Given real numbers $a$ and $b$ ($a \ne 0$), let $f_{a,b}$ be the function $\mathbb{R} \to \mathbb{R}$ defined by $x \mapsto ax+b$. The set of such functions is a permutation group on $\mathbb{R}$, ...
4
votes
2answers
945 views
Decomposition of a nonsquare affine matrix
I have a $2\times 3$ affine matrix
$$
M = \pmatrix{a &b &c\\ d &e &f}
$$
which transforms a point $(x,y)$ into $x' = a x + by + c, y' = d x + e y + f$
Is there a way to ...
4
votes
3answers
160 views
The vanishing ideal $I_{K[x,y]}(A\!\times\!B)$ is generated by $I_{K[x]}(A) \cup I_{K[y]}(B)$?
Let $K$ be a field, $x=(x_1,\ldots,x_m)$, $y=(y_1,\ldots,y_n)$, $A\!\subseteq\!\mathbb{A}^m_K$, $B\!\subseteq\!\mathbb{A}^n_K$. Does there hold $$I_{K[x,y]}(A\!\times\!B)=\langle\langle I_{K[x]}(A) ...
4
votes
1answer
131 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 ...
4
votes
0answers
97 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 $q+1$ ...
3
votes
4answers
94 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 ...
3
votes
2answers
123 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 ...
3
votes
3answers
77 views
Closed subset of an affine variety… is it affine?
Preliminaries
So, first of all let me give you the definitions I'm dealing with. Let $k$ be an algebraically closed field, and $\mathbb{A}^n = k^n$.
An affine variety is a closed and irreducible ...
3
votes
1answer
135 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 ...
3
votes
1answer
149 views
Affine Spaces and Affine transformations
Can anyone please recommend a book that describes Affine Spaces and Affine Transformations? Many books i saw described it very briefly. Can anyone please suggest a book that deals with it in detail?
...
3
votes
1answer
183 views
Further questions on barycentric coordinates
Following on from my previous post...
I'm going through this PDF file describing barycentric coordinates and trying to make sure I understand everything fully as I need to implement and support these ...
3
votes
1answer
64 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 ...
3
votes
1answer
89 views
Fitting Shape in Circle for Shape Classification
I need to classify arbitrary 2D shapes. The classification should be invariant to at least affine transform. To achieve this invariance, I decided to "normalize" each shape by fitting it to a unit ...
3
votes
0answers
75 views
Measuring Similarity of Affine Transformations
I am currently working on a problem where a calibration Algorithm provides me with an Affine Transformation that transforms a 2D Image to it's assumed Position in a 3D Volume. To evaluate the accuracy ...
3
votes
1answer
154 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 ...
3
votes
0answers
315 views
Meaning of affine transformation
From Wikipedia, I learned that an affine transformation between two vector spaces is a linear mapping followed by a translation.
But in a book Multiple view geometry in computer vision by Hartley ...
3
votes
0answers
258 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 $\lambda \in [0,1]$.
...
2
votes
3answers
700 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.
2
votes
3answers
96 views
Count points and lines in $\mathbb{A}^2(\mathbb{F}_p)$
Let $p$ be a prime, then $\mathbb{F}_p$ is a finite field.
$\mathbb{A}^2(\mathbb{F}_p)$ is an affine plane.
Number of points in $\mathbb{A}^2(\mathbb{F}_p)$ is $p^2$.
I look at a line equation ...
2
votes
1answer
33 views
Projectivities $\pi:\mathbb KP^1\rightarrow\mathbb KP^1$
I am a little bit confused conerning the following example of projectivities $\pi:\mathbb KP^1\rightarrow\mathbb KP^1$.
On the affine part $\mathbb K\subseteq \mathbb KP^1$ they are exactly the ...
2
votes
1answer
64 views
Projective geometry well defined bijection
I consider the sphere $\mathbb S^n:=\{x\in\mathbb R^{n+1}: \|x\|=1 \}$ and the equivalence relation $x\sim y:\Leftrightarrow x=\pm y$.
How can it be shown that the inclusion $\mathbb ...
2
votes
1answer
104 views
What does affinity means? (In categorical terms)
I know that traditionally, an affine space is "what is left from a vector space, after removing the origin". Given any set $X$ and a ring $K$, we can consider the set of all formal affine linear ...
2
votes
1answer
71 views
Equation of the line in an affine plane over a polynomial field
What are some examples of this? Say for $F_{4}$. I know this is a very simple question, but I can't find any info on it.
Edit: Yes, I was thinking of $F_{2}[x]/(x^2+x+1)$. I was confused.
2
votes
1answer
55 views
non-affine functions
it is obvious that if $f$ is an affine function, then $f$ has this property: there exist two function $g$ and $h$ such that $f(t+s)=g(t)+h(s)$ for all $t,s \in\mathbb{R}$. My question is: is there ...
2
votes
1answer
79 views
Finding a coset
I'm given $V$ a vector space over a field $\mathbb{F}$. Letting $v_1$ and $v_2$ be distinct elements of $V$, define the set $L\subseteq V$: $L=\{rv_1+sv_2 | r,s\in \mathbb{F}, r+s=1\}$. (This is the ...
2
votes
1answer
178 views
Turning affine planes into projective planes
How can we show that an affine plane of order $n$ can always be turned into a projective plane of order $n$?
Say I start with an affine plane, and split it into $n+1$ parallel classes, add a point ...
2
votes
1answer
241 views
Projective Geometry: Why is multiplication defined this way?
I am trying to understand this new way of multiplying in projective geometry.
Why is it defined like this? Also does this have anything to do with multiplication using a slide ruler? (The picture ...
2
votes
0answers
44 views
$f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$
Suppose, that $f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$
Please help me prove, that $f$ is a composition of rotation about an axis and moving along this axis.
I don't ...
2
votes
1answer
97 views
fixed point projective geometry
I am thinking about the following:
Let $\sigma:\mathbb C P^n\rightarrow\mathbb C P^n$ be a projectivity with $\sigma\circ\sigma=id_{\mathbb C P^n}$. I define the set of all fix points by ...
2
votes
1answer
75 views
Affine Transformation ― correct direction of scale
due to the fact that I am not mathematician I hope the question wont be ejected cause of triviality. But here we go:
what is given:
In an svg graphic, I have an element on which several are ...
2
votes
0answers
35 views
Set of all affine maps between two polytopes
Let $V$ be a finite-dimensional real vector space, let $P, Q \subseteq V$ be polytopes with $P \subseteq Q$. (Let a polytope be defined as the convex hull of finitely many points.) I'm interested in ...
2
votes
0answers
215 views
Applications of the fundamental theorems of affine and projective geometry.
The fundamental theorem of affine/projective geometry says that a bijection between two finite dimensional spaces that preserves the relation of collinearity is a (semi-) affine/projective ...
2
votes
1answer
112 views
Finding the singularities of affine and projective varieties
I'm having trouble calculating singularities of varieties: when my professor covered it it was the last lecture of the course and a shade rushed.
I'm not sure if the definition I've been given is ...
2
votes
1answer
84 views
Domain rotations in the Mellin integral transform
Let's consider the following form of the Mellin integral transform:
$$m_{pq} =\iint\limits_{D_R} \! x^p y^q f(x,y) \, dx\; dy, \, D_R={\{(x,y)\,|\,x^2 + y^2 \le R^2\}}$$
If we scale the domain of the ...
1
vote
2answers
181 views
n-dimensional affine space that is not isomorphic with the n-dimensional real space
I'm looking for an example of $n$-dimensional affine space that is isomorphic with $\mathbb{R}^n$ as affine space but not with respect to other properties (for example it has different ordering etc.)
...
1
vote
4answers
49 views
Find the equation of plane containing line described by
Please help me in this really easy task
Find the equation of plane containing line described by
$x+3y-2z=1$, $2x-y+2z=3$, containing point $(1,1,3)$
1
vote
1answer
29 views
Skew planes in $\mathbb{A}^4$
Can there be two skew planes in $\mathbb{A}^4$?
By this I mean two disjoint planes $\pi_1,\pi_2\subset\mathbb{A}^4$ such that their underlying direction vector spaces only intersect at zero.
1
vote
2answers
53 views
Find the line passing thought the point $p=(1,2,0)$, paralel to the plane…
Find the line passing thought the point $p=(1,2,0)$, paralel to the plane $P=\{x,y,z \mid x+2y-z=-4\}$ and crossing the line $L=\{(x,y,z):x+2y=2, y+z=4\}$
So I've tried to put the equation of plane ...
1
vote
1answer
26 views
How to find equation system describing affine space, having base of linear space and a vector
How to find equation system describing affine space, having base of linear 'overspace' and a vector?
Suppose that I've vectors $\alpha$ and $\beta$, so that $W=\text{lin}(\alpha, \beta)$, and a ...
1
vote
2answers
61 views
Equation of plane — point/vector pedagogy
Suppose we have a point $\mathbf P$ and a vector $\mathbf n$ in plain ordinary 3D space. Here I am deliberately using upper-case letters for points, and lower-case points for vectors, since they are ...
1
vote
1answer
54 views
Motion in affine geometry
$V$ is a finite euclidean vectorspace and $\sigma:V->V$ is a motion, this means that $d(\sigma(a_i),\sigma(a_j))=d(a_i,a_j)$ for an affine coordinate system $a_0,...,a_n$
I know the following two ...
1
vote
5answers
136 views
How to precisely distinguish vectors and points? [duplicate]
Possible Duplicate:
Distinction between vectors and points
I have a doubt about the distinction between points and vectors. I know there's already a topic about that here in the web site, ...
1
vote
1answer
185 views
What is the relation between complex numbers and transformation matrices?
I read addition and multiplication with complex numbers can be represented as translation and rotation in a 2D plane.
I am using this to move around objects on the screen. I have an offset number, ...
