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.
1,202
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What is the difference between Procrustes analysis and the Linear Transformation in terms of Shape Analysis?
Let's say you have two objects, each described by some 2D corresponding points.
In order to compare these two shapes, you can multiple algorithms:
Procrustes analysis
Search the Linear / Affine ...
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2
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651
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Example of an affine space that is not euclidean
Wikipedia says that Euclidean space is affine (obvious), but that not all affine spaces are Euclidean.
I understand that Euclidean space has extra structure defined on it, namely metrics of distance ...
4
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2
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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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1
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Convexity of affine function.
Can someone help me with a proof that affine function preserves convexity?
Given that $f$ is convex, $A$ is in $\mathbb{R}^{M\times N}$ and $b$ is in $\mathbb{R}^m$ then show that $g(x) = f(Ax+b)$ is ...
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1
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How to prove the triangle inequality becomes equality in a space with minkowski distance metric
Assume we have a vector space, $S$, with a distance function $D: S\times S \rightarrow \mathbb{R}^+$ with the following four assumptions about $D$:
(1) $D(x,y) = 0$ if and only if $x=y$
(2) $D(x,y) = ...
4
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2
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How to parametrise a hyperplane without making assumptions on the coefficients of its normal vector?
Consider a vector $w\in\mathbb R^n$, and its orthogonal complement:
$$w^\perp\equiv\left\{v\in\mathbb R^n: \langle v,w\rangle\equiv\sum_{i=1}^n v_i w_i=0\right\}.$$
I'm looking into ways to ...
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2
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Euclidean Geometry versus Analytic Geometry versus Affine Geometry?
What are the relationships (connections) among:
Euclidean (or Plane) geometry
Analytic geometry
Affine geometry
How do these things relate?
I know that this is a very general question, so I'm ...
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1
answer
386
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Sheaf of a Closed Subset
I’ve been given the following definition:
Let $(X,\mathcal{O}_X)$ be a ringed space which is locally isomorphic to an affine algebraic variety, and $Y\subseteq X$ be closed. Then for an open $V\...
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1
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Coefficients in affine hull vs. coefficients in affine independence
The definition of the affine hull of a set of vectors $\{a_1,\dots,a_n\}$ is
$$ \bigg\{x=\sum_{i=1}^n\lambda_ia_i\ \bigg|\ \sum_{i=1}^n\lambda_i=1\bigg\}. $$
On the other hand, a set of points $\{...
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1
answer
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Image of a map $f : k^3 \longrightarrow k^3$
Let $k$ be an algebraically closed field and define $f : k^3 \longrightarrow k^3$ by $$f(x,y,z) = (x, xy, xyz).$$
I would like to verify that the image of this map is $$f(k^3) = \{ (0,0,0) \} \cup k^...
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2
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Sphere to ellipsoid affine transformation matrix
I am trying to find the minimum bounding box of an ellipsoid. In my search, I found this answer and also some other nice descriptions like this one to the problem. I am not a mathematician (I need ...
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1
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Not understanding the concept of "irreducibility" for quasi-projective varieties
I am self-studying Shafarevich's "Basic Algebraic Geometry" and I am having some trouble understanding the decomposition of quasi-projective varieties into irreducible components. I work over an ...
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How is it possible to add a point to a vector?
I'm reading Linear Algebra book, stating that points are elements in the affine space $\mathcal{A}$, associated to a certain linear space $\mathbb{A}$. The elements in $\mathbb{A}$ are free vectors. ...
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2
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Affine transformation with translation
Basically, the issue I am trying to solve is that of "zooming round a point in a graphic, so that this point stays put on the display".
First, here is how I display the curve within a rectangle on ...
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Affine transformations - the meaning of contractivity
An affine transformation $\omega \colon \mathbb{R}^2 \to \mathbb{R}^2$ is a linear mapping followed by a translation, in other words
$$
\omega(x) = Ax+t =
\begin{pmatrix}
a & b \\ c & d
\end{...
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Why is for an algebraically closed field $k$, $\mathbb{A}^1_k \setminus \{0\} \simeq \operatorname{Spec} k[x, x^{-1}]$
Let $k$ be an algebraically closed field and $\mathbb{A}^1_k:=\operatorname{Spec}(k[x])$ be the affine 1-space. Why is then $\mathbb{A}^1_k \setminus \{0\} \simeq \operatorname{Spec}(k[x,x^{-1}])$ ?
...
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Relation Between Definitions of Geometric Independence [duplicate]
I'm struggling to see how these two definitions of geometric independence are related.
In Elements of Algebraic Topology by J. Munkres the following definition is given:
Given a set $\{a_0,a_1,\ldots,...
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3
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What is the parametric form of the unique line which crosses these other three lines?
In the real affine space $\Bbb A^4$ let $A=(1,2,1,0), A'=(1,2,2,-1), B=(1,0,0,0), B'=(2,0,0,0), C=(2,1,1,0), C'=(-2,1,-1,0).$
Now let $a$ be the line that passes through $A$ and $A'$, $b$ through $B$ ...
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How many $k$-dimensional affine subspaces?
I'm trying to determine the number of $k$-dimensional affine subspaces of a $n$-dimensional affine space ($k \leq n$) $X$ over $\mathbb{F}_q$, where $q$ is the power of a prime number.
Let $V$ be ...
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1
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Rigorous definition and relations between point/vector/affine space/vector space/basis/frame/coordinate system
I am trying to understand the exact relation between all these things:
point
vector
affine space
vector space
basis
frame
coordinate system
Can you explain me rigorously (in the mathematical sense) ...
4
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1
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657
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Does convexity of all projections imply convexity in higher dimensions?
If I have $n$ 2-dimensional convex "regions" that are the projections along $n$ (independent) dimensions of a n-dimensional compact subspace:
Does that imply that the latter is convex? Is the ...
4
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1
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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 ...
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1
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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 $\sigma\circ\sigma=\mathrm{id}_{\mathbb C P^n}$. I define the set of all fix points by $\...
4
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1
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Finding the missing centre or vertex of a projected rectangle.
In the attached image BCDE is a rectangular object that is projected through focal point J to produce a perspective image with vertices B'C'D'E' on the vertical blue plane. The plane of the rectangle ...
4
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1
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The intersection of the facets on the polyhedron
Given a point set $E=\{\alpha_j\}_{j=1}^m\subset \mathbb{N}^{n}$ ($1\leq m< \infty$).
Define the polyhedron $\mathcal{N}(E)$ to be the convex hull of the set
\begin{equation*}
\bigcup_{j=1}^m \...
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1
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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 \...
4
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1
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Orthogonality in affine subspaces
(Note: edited after the first comment)
Let $A, B$ be nonempty subspaces of an affine space $E$ such that $A \cap B = \emptyset$.
I am asked to prove that there are $a \in A$ and $b \in B$ such that $...
4
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1
answer
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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
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1
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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 in ...
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Are there affine planes with no abelian group of translations?
I'm reading the classic book "Geometric algebra" by Emil Artin and I have a question/reference request about a remark in it.
In order to make this question self-contained, let me start by ...
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0
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108
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Ellipsoid in $L^p([0,1],\lambda)$ spaces?
Let us consider $L^p([0,1],\lambda)$ spaces, were $\lambda$ is simply the lebesgue measure.
These are Banach spaces for $p\ge1$ (of course). It is well known (see for example here: Embedding of Lp ...
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Incidence geometry with curves instead of lines
Let $K$ be an algebraically closed field. Say that a subset $C$ of the affine plane $K^2$ is an irreducible curve if there is an irreducible polynomial $f\in K[X,Y]$ (where $X$ and $Y$ are ...
4
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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 ...
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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 $\...
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0
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Automorphisms of $\mathbb{A}^1_R$
When $R$ is an integral domain, the automorphisms of the affine line are all of the form $X \mapsto aX + b$ with $a \in R^\times$ and $b \in R$; the proof is the same as in the case of $R$ a field, ...
4
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0
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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 ...
4
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1
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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.$$
I've done it, and I've checked with ...
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0
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Geometry book for the university with solved exercises (affine space, euclidean space, etc...)
I'm looking for a book with solved exercises of affine space, affine transformations, etc... I found a lot of books and pdf's with theory, but none of them contained solved exercises, and I'm having ...
4
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0
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Affine-regular hexagon in convex body
An affine-regular $n$-gon is a non-degenerate affine image of the regular $n$-gon. It seems to be a standard fact in combinatorial geometry that inside every convex compact set $K\subseteq \mathbb R^2$...
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Questions about Affine algebraic group scheme over an infinite field K
For an easily comprehension of my questions I write some definitions:
An affine algebraic group scheme over $K$ is a representable group-functor from $K$-algebras category, with a finitely generated ...
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What does affine invariance mean in the context of the Newton's method?
The textbook Numerical Solution of Boundary Value Problems for Ordinary Differential Equations (by Ascher, Mattheij, and Russell) states on page 329:
[W]e observe that Newton's method is affine ...
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0
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When does a homogeneous morphism have only finite fibers?
Suppose that we have a map ${\bf f}:=(f_1,f_2,\cdots ,f_n):\mathbb{C}^n\rightarrow \mathbb{C}^n$ given by $$ \mathbb{C}^n\ni {\bf z}:=(z_1,z_1,\cdots,z_n)\rightarrow \big(f_1({\bf z}),f_2({\bf z}),\...
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Calculate the singular points of affine curve
I want to calculate the singular points of the affine curve $$f(X,Y)=(1+X^2)^2-XY^2 \in \mathbb{C}[X,Y]$$
The point $P=(x,y)$ is singular $\Leftrightarrow$
If $x=0$ we find $y=0$ and then from the ...
4
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0
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Is the cone over Grassmannian manifold $Gr_2(\mathbb{C}^n)$ an open set of a determinantal variety?
Let $Gr_2(\mathbb{C}^n)$ the Grassmann manifold of the planes in $\mathbb{C}^n$. It is, via Plucker embedding, a projective variety. If we consider the cone $C$ over $Gr_2(\mathbb{C}^n)$, is it ...
4
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0
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What are sliding vectors mathematically?
What is the mathematical definition of sliding vectors and their operations, as used in mechanics? What kind of mathematical structure do they form? Does the operation of constructing the "space" of ...
4
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0
answers
1k
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Good textbook on geometries
I am looking for a good textbook that thoroughly covers euclidean, affine, projective and non-euclidean geometries. I will be starting graduate school in algebraic geometry next year and I would like ...
4
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0
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311
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Is it true that blowing up a quasi-affine variety at a nonsingular point never introduces new singularities?
If we let $M$ be a quasi-affine variety, is it true in general that the blowup of $M$ at a non-singular point $p$ does not introduce new singularities? I came across this statement in my reading, but ...
4
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0
answers
176
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Describing $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as gluing of affine charts
How can we describe $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(m))$ as a gluing of affine charts? I'm having trouble with this problem, perhaps because I don't understand ...
4
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0
answers
346
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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 ...
4
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0
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248
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Cardinality of quasiaffine variety
The excercise 1.4.8(a) of Hartshorne's Algebraic Geometry says
Show that any variety of positive dimension over $k$ has the same cardinality as $k$.
Using Hartshorne's notation, we define a quasi-...