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.

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1answer
18 views

Express a point as an affine combination of another two points(3D collinearity)

So, given the points A(1,2,2), B(2,4,2) and C(3,6,2) I have to show that they are collinear. ...
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1answer
30 views

Finding a line $L\subset V(y-xz)\subset\mathbb A^3_k$

I want to find lines $L\subset V(y-xz)$ and $M\subset\mathbb A_k^2$ such that $$ V(y-xz)\setminus L \simeq \mathbb A_k^2\setminus M\ . $$ Hint suggests that I use the projection $(x,y,z)\mapsto(x,y)$. ...
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1answer
17 views

Are quasiaffine subsets of $\mathbb{A}_F^n$ always necessarily open or closed?

Something I was wondering about lately, suppose $\mathbb{A}_F^n$ is affine space over a field $F$ which is algebraically closed. Are the quasiaffine subsets $Z\subseteq\mathbb{A}_F^n$ always either ...
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0answers
16 views

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 ...
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1answer
18 views

A set $S\subseteq\mathbb{A}^n$ is quasi-affine iff $S=Z\setminus V$ for closed $Z$ and $U$?

I'm confused by a remark in note I'm reading. It essentially says, Let $S\subseteq\mathbb{A}^n$ be a subset of affine $n$-space over an algebraically closed field. It's clear that $S$ is ...
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1answer
46 views

Distinction between point and vector outside of US ( particularly Germany and Eastern Europe )

There was a long discussion in a forum I visit in where a calculus teacher was being critical of Stewarts Calculous for making a distinction between points and vectors. He argued that no such ...
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1answer
28 views

Making a matrix full rank through affine transformations

If I have (finite) $k$ vectors, $u_1,...,u_k\in\mathbb{R}^N$ that are in general linearly dependent is it possible to take positive affine transformations of the form: $$u'_i=\alpha_i u_i ...
2
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1answer
23 views

Surface area of transformed sphere

So if I have a sphere with center C and radius R and then apply one or more affine transformations (so any combination of rotating, scaling and translating), how would I go about finding the surface ...
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1answer
19 views

Application of Desargues' theorem for constructions

I found this interesting document (german) on the internet. On page 8 it says: "Draw a line segment between two given points only using compass and ruler, while the distance between the two points is ...
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1answer
31 views

Prove that the set of all fixed points is a hyperplane

Let $A$ be a $n$-dimensional affine space ($ 2 \leq n$) and let $\Phi:A\to A$ be a bijective affine mapping, which isn't the identity mapping, with the following property: For all points $p$ and $q$ ...
2
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1answer
37 views

Decomposition of shear matrix into rotation & scaling

How can I decompose the affine transformation: $$ \begin{bmatrix}1&\text{shear}_x\\\text{shear}_y&1\end{bmatrix}$$ into rotation and scaling primitives? $$ ...
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1answer
31 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: ...
2
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1answer
22 views

How to orthogonalize a set of 2x2 matrices?

I have set of 2D affine transformations of images and I need to modify the transformations such way that they become as close to rotations as possible to minimize distortions of images. Let the ...
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1answer
21 views

The set if affine?

The Set $\{ Ax + b | Fx = g \}$, is it affine? How can I prove it? My answer is yes, the intuition is that $\{ x | Fx = g \}$ is a solution space of equation $Fx = g$, thus it is a linear subspace. ...
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0answers
18 views

When the sum of coefficients of two linear combinations are equal.

I recently was looking a set of polynomials (the Legendre polynomials up to degree $n$) that form a basis for the space of polynomials $\{a_{0} + a_{1}x + \dots + a_{n}x^{n}: a_{i} \in \mathbb{R}\}$ ...
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1answer
40 views

Affine connection

The affine connection is not in general defined uniquely by the smooth structure and the Riemannian metric. Can you give some demonstration with some examples?
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0answers
11 views

Is the below formula equivalent?

$K$ is a simplicial complex: Is $\{\sigma \in K | \sigma \cap conv(\{a, b\}) = \emptyset\}$ equivalent to $\{ \sigma \in K | \sigma \cap \{ a \} = \emptyset \} \cap \{ \sigma \in K | \sigma \cap \{ ...
2
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1answer
36 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 ...
2
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0answers
25 views

Linearizable reductive group action.

Let $k$ be a 0 characteristic field, and $G$ a reductive group in $GA_2(k)$. How is it possible to deduce that $G$ is conjugated to a subgroup of $GL_2(k)$ ?
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2answers
30 views

affine hull, how to understand the statements below?

I am new to affine space, I looked through the wikipedia page, and have problem understanding the statements below. The affine hull of a set of three points not on one line is the plane going ...
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0answers
9 views

Affine transformation step by step

So I have the before and after 3d coordinates of an object that has been translated and rotated. I need to calculate the matrix to return the object to it's original position. I've been reading ...
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0answers
14 views

V is an affine subspace iff for any two distinct points V contains the line dtermined by these points

Since it can be shown that the barycenter of n weighted points can be obtained by repeated computations of barycenters of two weighted points, a nonempty subset V of E is an affine subspace iff for ...
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0answers
8 views

the set $V$ of barycenters ${\Sigma}_{i\in I}{\lambda}_ia_i$ is the smallest affine subspace containing $(a_i)_{i\in I}$

Given an affine space $(E,E^{\to})$, for any family ${(a_i)}_{i\in I}$ of points in E, the set V of barycenters ${\Sigma}_{i\in I}{\lambda}_ia_i$ is the smallest affine subspace containing ...
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1answer
15 views

Let $a,b$ be affine combinations of points from a set $S$. Then is the affine combination of $a,b$ also an affine combination of points from $S$?

Let A be an affine space, $a,b$ affine combinations of points from a finite subset $S$ of A. Then is the affine combination of $a,b$ also an affine combination of points from $S$? I found it ...
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2answers
70 views

Is there any difference between a flat manifold and an affine space?

What is the difference, if any, between a flat manifold (in which the Riemann tensor vanished identically) and an affine space?
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0answers
29 views

An easy definition of an $n$-dimensional affine cube

In a few weeks I'm giving a presentation on the History of Ramsey Theory and I want to start off with Hilbert's cube lemma. The only problem is that the pre-requisites for this course is only ...
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1answer
20 views

Alternative characterization of a finite dimensional affine set

As the definition in the S. Boyd's textbook: My question is the following representation: What is the relationship between this representation and the definition above it? EX: sum of elements ...
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5answers
894 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 ...
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1answer
19 views

Transform gradient to reference element

Minimal example of the problem How can you transform the gradient to the reference element?
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0answers
29 views

Dumb question regarding affine transformations

So, we can write an affinity $\phi$ as $$\phi(x) = Ax + b$$ for some linear transformation $A$ and vector $b$. What exactly does it mean for an affinity to be a "scaling"? Is this a mapping of the ...
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1answer
20 views

Prove that this affinity is the identity mapping

Let $\phi : K^n \to K^n$ be an affinity, such that all lines are parallel to their image under $\phi$. Prove that if $\phi$ has two fixed points, then $\phi$ is the identity mapping. My attempt: I ...
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0answers
15 views

Interesting observation WRT 1,2,3-dimensional convex polytopes and higher dimensional ones as counterpart

When I experimenting with qhull utility and Quickhull Algorithm, I found that in $\mathbb{R}^d, d \in \{1,2,3\}$ space the number $F$ of convex hull's $(d - ...
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0answers
18 views

Adjacency of convex hull facets

Let $C$ be a $d$-dimensional convex polytope and $p$ is a point outside of it. $C=\{f_c\}$ defined by set of facets $f_c=\{p_c,A_c\}$ where $p_c$ is a tuple of vertices and $A_c$ is a set of ...
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0answers
20 views

A confusion regarding Affine spaces

Take an Affine space $\Bbb{A}$ over the field $K$. How would you determine the points satisfying any polynomial $f(x)$? If there is no fixed origin, points can be given names with reference to ANY ...
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2answers
20 views

Number of parameters to specify an affine transformation in n dimensions

In general, how many parameters does it take to specify an affine transformation in $n$ dimensions, and how does one go about proving this? For example, in 2 dimensions it takes 6 parameters, and in ...
0
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1answer
49 views

Formula of signed distance from hyperplane to point

Let $H$ be a hyperplane defined by the points $p_1, p_2, ..., p_n$ and single point $x$ generally out of the hyperplane. Is there any formula to calculate the signed distance between $x$ and $H$? I ...
1
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1answer
32 views

linearization of $\log(|x|)$

I am trying to convexify $\log(|x|)$. I think its concave. So I am trying to get an affine upper bound through linearization. But the problem is there are two concave functions because of the absolute ...
2
votes
1answer
39 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: ...
2
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1answer
49 views

Computing the Zariski cotangent space

I'm an extreme beginner with algebraic geometry and am trying to get used to things. Say I have some (algebraically closed) field $k$, in $k^2$ I want to compute the Zariski cotangent space, let's say ...
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0answers
68 views

Why are affine subspaces also sometimes called linear manifolds?

According to Wikipedia, an affine subspace is a subset of a vector space closed under affine linear combinations. That is, linear combinations whose scalar coefficients sum to 1. It's not clear to ...
0
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1answer
51 views

Affine Bundles vs Affine Spaces

I went through the wiki article on affine spaces and had a quick look on the affine bundle wiki article but I don't understand what the affine map is in the case of affine bundles over vector bundles. ...
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0answers
26 views

equiaffine arc length, moving frame, and affine curvature

I am trying to learn affine geometry, and I'm having some trouble getting started with the following problem. Compute (a) the equiaffine arc length, (b) the moving frame, and (c) the affine ...
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1answer
30 views

Angle of two planes in $\mathbb E_4$

I have two planes (given in parametric form) in $\mathbb E_4$: $\alpha$: (7,3,5,1) + t(0,0,1,0) + s(3,3,0,1) and $\beta$: (1,5,4,1) + r(0,0,0,-1) + p(2,0,0,1), and I have to find angle between them. I ...
0
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1answer
17 views

Newbie with barycentric coordinates: why one is zero when on a vertex?

I'm trying to calculate if a 2D point lies inside a triangle and I solved the following system: ...
2
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1answer
52 views

intersection multiplicity from Shafarevich

In Basic Algebraic Geometry, Shafaravich proves the following theorem: Theorem. If $X$ is an irreducible affine curve, and $P \in X$ is a nonsingular point, then there is a function $t$, regular at ...
3
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1answer
60 views

local parameter on an irreducible affine algebraic curve

On page 14 of Shafarevich's Basic Algebraic Geometry 1, it is stated that for an irreducible affine algebraic curve $X: f(x,y) = 0$, and a nonsingular point $P \in X$, there is a regular function $t$ ...
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0answers
12 views

Dual representations of affine span of a set of affine transformers.

I am not well versed in the literature of affine transformers or Farkas' Lemma. I just know the basics of the two concepts. Are there any dual representation of affine span of a set of affine ...
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2answers
48 views

intersection multiplicity at non-zero point

Compute the intersection multiplicity of $f=x+y-2$ and $g=x^2+y^2-2$ at $(1,1)$. Is this the same as the intersection multiplicity of $f(x+1)$ and $g(x+1)$ at $(0,0)$ which I have computed to be 2? ...
3
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2answers
97 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 ...
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1answer
80 views

Find intersection multiplicities

Let curves $A$ and $B$ be defined by $x^2-3x+y^2=0$ and $x^2-6x+10y^2=0$. Find the intersection multiplicities of all points of intersection of $A$ and $B$. If we let $f=x^2-3x+y^2$ and ...