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.

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Kernel of the homomorphism of an affine group onto the general linear group is the translation space

Let $E$ be an affine space attached to a $K$-vector space $T$. Let $G$ be the group of all bijective affine mappings of $E$ onto itself. Write $\phi$ for the group homomorphism associating the $K$-...
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1answer
16 views

Condition for affine independence

Let $E$ be an affine space attached to a $K$-vector space $T$. For a family $(x_i)_{i\in I}$ of elements of $E$ and any $a\in E$, the set $$\left\{\sum_{i\in I}\lambda_i(x_i-a)+a\ |\ \lambda\in K^{(I)...
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Model a rectangular sheet's movements in the 3D space

I have sheet of paper with width $w$ and length $l$ . Assume the paper is rigid and can not be folded. If we restrict to $2D$, the possible movements and rotations can be modeled using a affine ...
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The direction of parallel projection of an affine camera?

In this book on page 173, it states Given an affine camera of form: $$P_A = \begin{bmatrix} m_{11}&m_{12}&m_{13}&t_1\\ m_{21}&m_{22}&m_{23}&t_2\\ 0&0&0&1\end{...
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1answer
20 views

Condition for a subset of an affine space to be affine

Let $E$ be an affine space attached to a $K$-vector space $T$ and $a \in V\subset E$. Consider the bijection $f:E\rightarrow T$ send $x$ to $x-a$, where $x=(x-a)+a$. Clearly $f(E)$ is a $K$-subspace ...
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1answer
17 views

Faithful or free action in the definition of affine space

For a vector space $T$ over a field $K$, Bourbaki defines affine space as a homogeneous $T$-set $E$ on which $T$ acts faithfully. The definitions found both on Wikipedia and nlab say that the action ...
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23 views

Bijection $\psi$ between $\varepsilon \mathbb{Z}^2$ and $\varphi(\varepsilon \mathbb{Z}^2)$ with bounded distances $d(p,\psi(p))$?

Let $\varphi: \mathbb{R}^2\to\mathbb{R}^2$ be a rigid transformation (translation + rotation) and let $\varepsilon \mathbb{Z}^2 = \{ (\varepsilon \cdot k_1, \varepsilon \cdot k_2 ) \in \mathbb{R}^2 | ...
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Arrange a point relative to another within 3D space

According to this accepted answer on Stack Overflow, you ... need to set the coordinates m41, m42 and m43 of an SCNMatrix4 B (representing the last column and the first three rows of a 4x4 matrix) to ...
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Contact theory affine differential geometry

I am studying affine differential geometry to plan curves, that is $[\gamma_s(s), \gamma_{ss}(s)]=1$, and I need to show the following result: "Two curves having the same affine tangent also have ...
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Geometric construction of vanishing points from a length ratio.

Can someone please explain why the vanishing point can be constructed in such a way? The textbook I am studying shows this example without much explanation. From my studies, I know using the cross ...
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Recovery of affine properties from images

In the textbook I am studying Multiple View Geometry in Computer Vision, Second Edition. Richard Hartley, Andrew Zisserman. pg 49, it is stated that one can recover affine properties from images by ...
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Decomposing the affine transformation matrix with the SVD

In the textbook I am studying Multiple View Geometry in Computer Vision, Second Edition. Richard Hartley, Andrew Zisserman. pg 40, it is stated that The affine matrix $A$ can always be decomposed as $...
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Can we define Affine space by Affine combination (without the vector space )?

The Affine space is defined with the help of vector space. Is it possible to make Affine combination the basic operation of Affine space, to define the Affine space? so that we dont need a vector ...
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Incommensurable segments on an affine line

I would really appreciate some help with my following question. Thanks in advance. Massi. Area: foundations of classical geometry Domain: incommensurable segments Background: in Euclid's geometry of ...
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1answer
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Linear and Affine functions

Wanted to be clear on my understanding on affine functions, and if we have a matrix $A \in \mathbb{R}^{n \times m}$ and a vector $c \in \mathbb{R}^{n}$, I know an affine map is given by: $$f: \mathbb{...
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What does the “origin” mean when introduce the affine space?

This is almost the same question as Origin in vector space?, and I am also confused by the statements in the wikipedia article of Affine space: St1: Roughly, affine spaces are vector spaces whose ...
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Connection between euclidean spaces (affine space + inner product on translation space) and euclidean geometry

I guess it's not a coincidence that euclidean spaces (affine space + inner product on the translation space) and euclidean geometry share the word euclidean. Do the axioms of euclidean spaces allow a ...
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Stuck in understanding a custom affine transformation “RoIRotate”

I'm currently reading text spotting (text detection + recognition) research paper and one of it's component is named "RoIRotate" which basically rotates the oriented text to axis aligned ...
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about irreducible quadratic polynomial in $k[x, y]$ [duplicate]

(a) Let $Y$ be the plane curve $y=x^{2}$ (i.e., $Y$ is the zero set of the polynomial $f=$ $\left.y-x^{2}\right) .$ Show that $A(Y)$ is isomorphic to a polynomial ring in one variable over $k$ (b) Let ...
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is this theorem true for case quasi-projective?

Definition. If $X, Y$ are two varieties, a morphism $\varphi: X \rightarrow Y$ is a continuous map such that for every open set $V \subseteq Y,$ and for every regular function $f: V \rightarrow k$, ...
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Automorphisms of $\mathbb{A}^{n}$

Let $\varphi: \mathbf{A}^{n} \rightarrow \mathbf{A}^{n}$ be a morphism of $\mathbf{A}^{n}$ to $\mathbf{A}^{n}$ given by $n$ polynomials $f_{1}, \ldots, f_{n}$ of $n$ variables $x_{1}, \ldots, x_{n} .$ ...
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Is this lemma true? [duplicate]

Lemma : Let $K$ be a field that is not algebraic closed then for every Algabraic $\operatorname{set} X \subseteq A_{K}^{n}$ there exist exist a polynomial $f \in K\left[x_{1}, \ldots, x_{n}\right ]$ ...
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image of rational function

let $X=Z(z^{2}-x y) \subset \mathbb{A^{3}}$ and $ \varphi: X \rightarrow \mathbb{A^{1}} $ such that $\varphi(x, y, z) \mapsto \frac{x}{z} .$ find domaim and range of rational function $\varphi$ . ...
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Transformation in 2 dimension problem $T:=\begin{pmatrix}1 & 0 & -2\\0 & 1 & -2 \\ 0 & 0 & 1\end{pmatrix}$

The Problem My Questions I have got this problem here and solution. I am not understanding how we got the matrix T in eqn 16 and also the final result. Again, I am not understanding what is done in $...
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Finding the best-fit affine transformation Ti,j from i to j

Im trying to code up a function for finding the best fit affine transform between two curves i and j for checking the similarity between two curves. My main problem at the moment is getting the two ...
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1answer
18 views

Characterising the affine image of a polar convex body

Let $S\subseteq \mathbb R^n$. Its polar is defined by $$S^\circ = \{z\in\mathbb R^n:z^Tx\le 1,\forall x\in S\} $$ Now, let's say we are interested in some affine image of $S$ (and its polar). Let $T =...
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61 views

If a line and its points are removed from a projective geometry, is the resulting is affine geometry?

I saw that it's written in a couple of places, but couldn't find proper proof for this. I saw it for example in: here and here I would like to find a reference to a proof of this (as I'm sure some ...
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17 views

Is a projective reference in Pn needed for an affine subspace to be set (by means of a group action)?

Assuming a group action $A \times V \rightarrow{A}$ to set an affine space $A$, and assuming that the space $V$ is a translation-space, that is a vector space; then, is a projective reference needed (...
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29 views

Affine-space $A_{n}$ as group-action object vs as projective space subspace

Are the affine-space $A_{n}$ as a group-action object vs affine-space $A_{n}$ as subspace (open set complement of a projective hyperplane: coordinate projective hyperplane, etc.) of a projective space ...
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How to show that the following subsets of V are vector or affine subspaces.

Consider the vector space V of polynomials with real coefficients of degree less than or equal to 2. Say, justifying the answer, if the following subsets of V are vector or affine subspaces. $(i)$ The ...
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1answer
22 views

Operation precedence in coordinate transformation

Consider two orthogonal, right-handed frames whose orientations and positions are as depicted in the below picture. In order to bring the axes of frame {A} onto those of frame {B}, we can first ...
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1answer
42 views

Determine Jacobi Matrix of a unit simplex

Let $\Phi$ be a affine linear mapping with $\Phi(K)=\hat K$, where $\hat K$ is the unit simplex and $\Phi$ is of the form $$\Phi(\hat x)=A\hat x+b$$ with $A\in\mathbb R^{d\times d}$ and $b\in\mathbb R^...
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87 views

coordinate form of an affine transformation

I am having trouble with this seemingly simple question: Write the standard coordinate form of an affine transformation in $\mathbb{A}^{2}(\mathbb{R})$ that maps the point (1, −2) to the point (0, 10),...
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Matrixes of affine transformations of $K^{2}$

Take $\begin{bmatrix}a&0&c\\k_{1}&1&k_{2}\\b&0&d\end{bmatrix}$ and $\begin{bmatrix}1&k_{1}&k_{2}\\0&a&c\\0&b&d\end{bmatrix}$ for vectors $\begin{...
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1answer
55 views

How can I see a visual representation of the difference between Euclidean and Affine geometry?

I have just started studying different types of geometry with the first to I have covered being Euclidean geometry and Affine geometry. I am aware that Euclidean geometry is what we have used the most ...
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Projective reference in projective space $P_{n}$

Given a projective space $P_{n}$, a reference on it would need $n+1$ linearly independent points plus a $(n+2)th$ point, which is a combination of the former, and performs as to give a fixed position ...
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1answer
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$y = \beta_0 + \beta_1 x_1 + \dots + \beta_p x_p$, unknowns $(x_1, \dots, x_p, y)$: Describes a hyperplane of affine space $\mathbb{R}^{p + 1}$?

Let's say we have an equation $y = \beta_0 + \beta_1 x_1 + \dots + \beta_p x_p$, where the unknowns are $(x_1, \dots, x_p,y)$. It is said that this describes a hyperplane $H$ of the affine space $\...
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Why is this affine transformation legal?

If $K$ is a centrally symmetric body, and $\varepsilon$ is its maximal ellipsoid, then $K \subset \sqrt{n}\varepsilon$. To check this, we may assume $\varepsilon = B^n_2$, i.e. the unit ball of ...
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1answer
27 views

Affine mapping task

I'm currently refreshing my Mathematics knowledge to pass an exam, so I found a list of tasks and got stuck on this one. Could anyone please help me to understand, what am I supposed to even start ...
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How to find $k$-dimensional affine subspaces?

Let $V$ be an affine space of dimension n over the finite field $\Bbb{F}_q$ of $q$ elements. How many $k$-dimensional affine subspaces are there in $V$? So this is a question I came across while ...
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149 views

Generalized Helly's theorem

Let $X_1,\ldots,X_n$ be convex sets in $\mathbb{R}^d$. Suppose the intersections of every $d + 1 − k$ of them contain an affine $k$-dimensional subspace. Prove that there exists an affine $k$-...
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1answer
47 views

Show the set is affine subspace.

About 2 days ago I was learning stuff about affine geometry and yesterday I got stuck with the following problem Suppose that $S$ is a subset of affine space A. Show the set: $$ \langle S \rangle \...
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1answer
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Line passing through the origin and incident to two other lines

I came across this silly exercise but it's giving me a headache, because I cannot figure whether or not I'm not doing it right. I have two lines of cartesian equations given $$ r : \left\{ \begin{...
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1answer
71 views

Show that convex hull can be represented as an intersection of half-spaces

Yesterday I've stuck with the following problem: Suppose that $a_0, a_1, ..., a_n$ are affine independent points of $n$-dimensional affine space. $H_i$ is the hyperplane that passes through all these ...
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Prove that any ratio-preserving map of collinear points is affine

This is problem 2, chapter 2 from CAGD 5th edition by Gerald Farin. In order to prove that any ratio-preserving map $Φ$ is affine I tried to prove that it leaves barycentric combinations invariant. ...
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How to prove for parallel translation or homothety?

Let an affine transformation $G: A^2 → A^2$ maps each line to a line parallel to it or to the same line. Prove that G is either a parallel translation or a homothety. I know that under a homothety, ...
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1answer
33 views

How to write the equation of a line in affine space

Write the equation (in coordinates $x_1, x_2)$ of a line in $A^2$, where A denotes Affine space: passing through the point (2, −3) and parallel to the vector (5, 2). Can we use the equation y=mx +c to ...
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66 views

Mathematical model for classical mechanics

Even knowing some basics of maths and physics I get puzzled when I try to systematise some concepts for better understanding. One is basically on how all the mathematical concepts comprise the model ...
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2answers
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All affine sets are affine subspaces (and vice-versa)

Show that a set $∅ \neq A ⊂ \mathbb{R}^n$ is affine if and only if it is an affine subspace. Definitions I know: Affine set: If a set $A \subset\mathbb{R}^n$ is affine, then for all $x,y\in A$, we ...
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1answer
30 views

Affine transformation matrix for conversion between line and rectangle

Is there a way using an affine transformation matrix to convert between a rectangle of zero height (i.e. effectively having 2 different end/corner points) and a rectangle of > zero height (i.e. ...

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