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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183 views

The Mean Value Property and Affine Functions

I need some hints to solve the following: A function $f(t)$ on an interval $I = (a,b)$ has the mean value property if $f(\frac{s+t}{2}) = \frac{f(s)+f(t)}{2}$ where $s,t\in{I}$. Show that any ...
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574 views

Affine plane of order 4?

I cannot seem to construct an affine plane of order 4. I have the construction for order 3- but cannot seem to come up with or find the construction for 4 anywhere. Could someone show me a picture of ...
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53 views

finding two polynomials that their roots are a given line.

Given a field $F$ and $A = F^3$. we define $L$ to be the line that goes through the points: $(8,1,-1)$, $(5,0,-1)$. My object is to find two polynomials $q(X_1,X_2,X_3)$, $p(X_1,X_2,X_3)$ in ...
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515 views

Equiangular polygon inscribed in rectangle

In a drawing application I am writing, I would like to offer the opportunity for a user to draw an equiangular n-sided polygon inscribed in rectangular bounds drawn by their finger (this application ...
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10 views

Decomposition of 4x4 or larger affine transformation matrix to individual variables per degree of freedom.

There are a couple of problems and solutions where affine matrices are decomposed into their seperate tranformations. However they are all for the 2D case and I`m finding it difficult to generalise it ...
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17 views

Two discrete lines always intersect at a point

In my lecture notes we have the following: $K$ field Extension of the affine space. Relation between points and lines: Two discrete points define an unique line and two discrete lines always ...
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27 views

Question on Logic in translation

let $P,P'$ two affine subspace of $R^{3}$ have we equality between this two statement $$\exists\ u_{0}\in R^{3}\ \mbox{such that } t_{u_0}(P)=P'$$ $$\exists B,A\in PP' \mbox{such that } ...
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15 views

help me to check the following argument on a linear transformation

all, Just want to confirm the following argument: Assumption: we know that $S:=\{(x_1,x_2,x_3) : \|(x_1,x_2)\|_2 \leq (1+k)x_3\}$, for a given value $k$. By linear transformation: ...
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61 views

Dimension of the intersection of affine subspaces

Let $\alpha,\beta,a,b,c \in \mathbb{R}.$ Consider three affine planes in the affine space $\mathbb{R}^{3}$: $P_{1}$ of equation $x+2y+\beta z=a$, $P_2$ of equation $2x+y=b$, $P_{3}$ of equation ...
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28 views

$P+Q:=\varphi_{O}^{-1}\left(\varphi_{O}(P)+\varphi_{O}(Q)\right)$

let X is affine space and $\overrightarrow{X}$ is vector space associted to X $$\begin{array}{ccccc} & \varphi_{O} : & X & \longrightarrow & \overrightarrow{X}\\ & & ...
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24 views

Determine the expression for a continuous affine transformation

In this problem I'm doing, I'm being asked To determine the affine transformation matrix which maps triangle V to triangle W. I'm also being asked to determine this matrix's continuous ...
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47 views

Let $T(x)=Ax+v$, where $A\in GL (n,\mathbb{R})$ and $v \in \mathbb{R}^n$. Show that T sends affine subspaces of $\mathbb{A}^n$ to affine subspaces.

I'm not too familiar with Affine geometry so I feel as though I'm missing something from my solution: Let $E$ be an affine subspace defined by $E=\{P+ \textbf{v}\, |\, \textbf{v} \in U\}$ where $U$ ...
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15 views

Two affine subspaces parallel

Theorem: Two affine subspaces $V,V'$ of $(X,\overrightarrow{X})$ are said to be parallel $(V\parallel V')$ if there is a translation such that $t_{\vec{u}}(V)=V'$ $V\parallel V' ...
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26 views

What are the prerequisites to understand Affine Invariant Fourier Descriptors?

I need to implement Affine Invariant Fourier Descriptors on matlab, the objective is to compare two objects one reference and other transformed by affine transformation for recognition, my problem is ...
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22 views

Computing the dimension of the affine hull of a set of vectors

I have a few hundred vectors that live in $R^d$ with $d$ larger than 100. I'm interested in finding the dimension of the affine hull of these points. Computing the dimension of the linear subspace ...
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111 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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30 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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31 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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12 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 \{ ...
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14 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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29 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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12 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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16 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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34 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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23 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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36 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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18 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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20 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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72 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 ...
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68 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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39 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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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: ...
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18 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
53 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? ...
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86 views

5 Parameter Affine Transformation

I am working on computing affine transformation using Gradient Ascent Method, so the Inverse compositional algorithm. However, I am stuck in one probably simple step but I fail to understand them. ...
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1answer
25 views

Non constant function of two points invariant under Affine transformation proof

Here is the question; Prove that there does not exist any nonconstant function of pairs of distinct points $P,Q\in\mathbb{R}^2$ or of triples of distinct non collinear points $P,Q,R\in\mathbb{R}^2$ ...
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18 views

Preservation of aligned points

If a mapping transforms aligned points into aligned points is necessarily an affine application? In other words, are there any mapping with that property without preserve the ratios between aligned ...
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1answer
50 views

Constructing a similarity matrix between points

I have two images with two sets of corresponding points. In order to align the images I'm trying to compute the similarity matrix that describes the relationship between the corresponding points. I ...
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1answer
31 views

About affine map on three dimensional euclidean space

An affine map $(t,M)$ with $t\in R^3$ and matrix $M$ maps $x\in R^3$ into $t+Mx$. It has property $P$ if for any $x$ with $|x|\leq 1$ then $|t+Mx|\leq 1$. Our goal is to characterize the set of such ...
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104 views

$3$D transformation matrix to $2$D matrix

I have a $3$D affine transformation $4\times 4$ matrix. I need to convert it (project) to a $2$D affine transformation $3\times 3$ matrix, which looks like this: $3$D rotations are irrelevant and ...
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1answer
76 views

Show existence of linear transformation from subset to subspace embedded in $\mathbb{F}_2^n$

Assume I have a subset $X$ (not necessarily a subspace) of $\mathbb{F}_2^n$, of size $\leq 2^{n-1}$. It seems likely to me that there should exist a bijective linear transformation taking $X$ to a ...
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16 views

How is affine space analogue for lattices called?

Lattices are so like vector spaces that it seems natural to have an affine space construction for them. Unfortunately I could not find how such a construction is called. Could you please help me? ...
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1answer
49 views

Why all norms define the same relative interior?(Convex Optimization, Stephan Boyd)

When I was reading 'Convex Optimization', Stephan Boyd, I was stopped by Example 2.2. Before Example 2.2 is started, following definition is coming. If the affine dimension of a set ...
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47 views

Finding affine transformation

Find affine transformation which takes the ellipse $x^2+4y^2+2x-8y+3=0$ to the form of the ellipse ${x^2 \over 9}+{y^2 \over 16}=1$. So I took the quadric and reached to a standard form: ${(x+1)^2 ...
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1answer
79 views

Affine stratification of Grassmannian $\mathbb{G}(1,\mathbb{P}^3)$

Let $G=\mathbb{G}(1,\mathbb{P})$ be the Grassmannian variety of lines in $\mathbb{P}^3$. I have to do an affine stratification of $G$. In order to do this we consider the flag $\mathcal{F}$ of the ...
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1answer
27 views

Find $\vec{QB}$ in terms of $\mathbf c$

I've managed to work out “$\vec{AM}$ in terms of $\mathbf a$ and $\mathbf b$” to be $3\mathbf a+\mathbf b$. But how can I work out “$\vec{QB}$ in terms of $\mathbf c$”?
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190 views

Prove, using vectors, that this quadrilateral is a rhombus

Consider the following quadrilateral $ABCD$, with $E, F, G, H$ as the midpoint of $AD, DC, CB, BA$ respectively such that $\Delta ECH$ and $\Delta AGF$ are equilateral. Prove that $ABCD$ is a ...
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2answers
54 views

Determining the rotation shape

Consider a large number of points distributed on the circumference of a circle with radius r. If I rotate each point with a randomly chosen Euler angle around a randomly chosen coordinate inside this ...
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1answer
114 views

Proving a vector equality in a triangle without using Thales' theorem.

Problem Let $\text{ABC}$ be a triangle, and $\text{M}$ and $\text{N}$ are points where: $\vec{\text{AM}}=\frac{1}{3}\vec{\text{AB}}$ and $\vec{\text{AN}}=\frac{1}{2}\vec{\text{AB}}$ and ...
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74 views

Problem with vector calculation.

Problem Let $\text{ABC}$ be a triangle and let $\text{A'}$ , $\text{B'}$ and $\text{C'}$ be respectively the center of $\text{[BC]}$ , $\text{[AC]}$ and $\text{[AB]}$. Prove that: ...