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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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3answers
116 views

Affine transformation from Points from 3d to 2d

I have a plane E in R^3. The points $$p0 = (1,1,1), p1 = (1,0,1), p2 = (0,1,1) $$ are on E. I cannot figure out an affine transformation α : E → R2 that produce the following outcome: $$f(p0) = (0,0), ...
2
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2answers
119 views

Find any affine transformation that swaps affine lines

The task is to find any affine transformation that will swap the following two lines: $$L_1:(1,1,1) + span((1,0,2))$$ $$L_2:(1,0,1) + span((1,0,-1))$$ From what I understand there is a number of ...
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2answers
36 views

Simple Affine Stretch of Triangle

I am having a little trouble with the following problem: Use explicit affine transformation, i.e., written down in matrix form, to stretch triangle having vertices $(\frac{1}{2},0), (\frac{1}{2},\...
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1answer
42 views

a circle and a parabola have 3 intersection points

Is it possible that a circle and a parabola on a euclidean plane have 3 intersection points and the center of the circle does not lie on the axis of parabola?
2
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1answer
32 views

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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1answer
29 views

Geometric interpretation of ranks of matrices gathering coefficients of 3 affine and associated vectorial planes

Given three planes $\pi_{1}$, $\pi_{2}$ and $\pi_{3}$ $\in \mathbb{R}^{3}$ with their respect cartesian equation of the form $A_{i}x + B_{i}y + C_{i}z + D_{i} = 0$, we can determine their relative ...
1
vote
1answer
123 views

Vanishing points from three collinear points

I would like to find the 2D vanishing point from a three collinear points as is shown in "Multiple View Geometry in Computer Vision" Example 2.19 (see here). What I did so far: 1 - I've extracted ...
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0answers
35 views

What is the name of a linear combination that has coefficients that sum up to zero?

Is there a special name for a linear combination where the coefficient add up to zero?
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2answers
197 views

Given four points, determine a condition on a fifth point such that the conic containing all of them is an ellipse

The image of the question if you don't see all the symbols The given points $p_1,p_2,p_3,p_4$ are located at the vertices of a convex quadrilateral on the real affine plane. I am looking for an ...
2
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3answers
86 views

How to find an affine map for given points

I am trying to understand affine maps. In the book I am using there is the following example, which I don't understand. Given are the following points: \begin{array}{lll} p_0 = \begin{pmatrix} 1 \\ ...
4
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3answers
5k views

How to define an affine transformation using 2 triangles?

I have $2$ triangles ($6$ dots) on a $2D$ plane. The points of the triangles are: a, b, c and x, y, z I would like to find a ...
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1answer
23 views

Explicit affine transformation between simplex and subsimplex

Take a simplex $\mathcal P_{n}$ with corner points $[00\dots00]$, $[10\dots00]$,$\dots$,$[11\dots11]$ in $\mathbb R^{n+1}$. Slicing by a hyperplane $\sum_{i=1}^{n+1}x_i=t$ where $0\leq t\leq n$ gives ...
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0answers
23 views

Understanding the connection between $PGL(3, K)$ and $AGL(2, K)$

Sorry if this is a trivial question, but I'm having trouble wrapping my head around the connection between these two groups. It seems intuitively clear to me that the group of invertible affine ...
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0answers
23 views

Uniform Bound on fibres of a finite morphism between affine varieties.

Let $X,Y$ be two affine varieties (irreducible zero loci of polynomials over algebraically closed field). Let $\phi : X\rightarrow Y$ be a finite morphism (i.e. K[X] is finitely generated $\phi ^*(K[Y]...
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0answers
36 views

What is an affine $0$ space?

I have encountered $\mathbb{A}^0$, where $\mathbb{A}^n$ is the affine $n$ space over some algebraically closed field. What exactly is $\mathbb{A}^0$? Does this only consist of $1$ element perhaps?
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2answers
13 views

How to convert half plane from parametric to general form?

I have: $ x=2+5t $ $y=5-t, t\geq 0 $ and I am supposed to convert it to general form. I know how to do it when I have plane, but that is half plane and I have no idea what will change. Can anyone ...
1
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1answer
32 views

smallest affine plane not generated by a field

What is the smallest affine plane not generated by a field? By "smallest" I mean has the least number of points. By "generated by a field", I mean planes of this form: $X = \mathbb{F}^2$ (the set ...
4
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1answer
100 views

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\...
2
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1answer
51 views

The complement of a point in $\Bbb{A}^n$ is compact in the Zariski topology.

The book "Invitation to Algebraic Geometry" says the following: The complement of a point in $\Bbb{A}^n$ is compact in the Zariski topology. Why is this this the case? This is thing that is asked ...
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1answer
30 views

Understanding the symbol <A,B> in affine spaces

I'm trying to solve this exercise: A subset F of an affine space is an affine subspace if and only if for all points A and B of F, the inclusion <A, B> ⊂ ...
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1answer
32 views

Dimension of an affine plane

In geometry, an affine plane is defined as a system of points which fullfill: 1) Any two distinct points lie on a unique line. 2) Given any line and any point not on that line there is a unique line ...
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0answers
44 views

Lines in projective and affine space

I'm trying to understand lines in affine and projective space in order to solve problems 2.15 and 4.13 in Algebraic Curves by William Fulton: https://www.google.com/url?sa=t&source=web&rct=j&...
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0answers
30 views

Why is $\mathbb{R}^2$ endowed with the taxicab metric isomorphic to the infinity distance model of the cartesian real plane? (Hartshorne exercise 8.9) [duplicate]

My question deals with the following exercise from Hartshorne's Euclid: Geometry and Beyond: Following our general principles, we say that two models $M,M'$ of our geometry are isomorphic if there ...
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1answer
841 views

Relation between SVD and affine transformations (2D)

I am trying to decompose affine transformations in elementary operations (rotation, scaling, shear), leaving translation aside for the moment, and in particular, to understand the relation of affine ...
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0answers
11 views

Why is the stabilizer of a line segment a cylinder?

There is a passage in a paper I'm reading discussing the stabilizer of an edge. For an edge (passing through the origin), its stabilizer (in $\operatorname{GL}_2(\mathbb{R})$) must fix the ...
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0answers
31 views

Find a set of polynomials whose common zero set is $\{(1, 2), (0, 5)\}$.

Find a set of polynomials $\{P_1, \dots, P_n\}$, all of whose coefficients are real numbers, whose common zero set is the given set. I know what a zero set is, but I think my confusion comes from ...
3
votes
2answers
274 views

Identifying the Plane at Infinity in the World Necessitates Determining the Affine Geometry of the World?

Page 18 of my computer vision textbook, Multiple View Geometry in Computer Vision (Second Edition), by Hartley and Zisserman, states the following: 1.8 Auto Calibration $\vdots$ Generally ...
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0answers
21 views

$W=\mathbb{P}(S)$ for some (k+1)-dimensional subspace $S$ of $\mathbb{k}^{n+1}$

Let $W$ be a subset of $\mathbb{P}_n(\mathbb{k})$. It is known that for every affine map $A$ (with which $W$ intersects) $A \cap W$ is a k-dimensional affine space. Is it true that $W=\mathbb{P}(S)$ ...
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0answers
22 views

3D Affine Rotation Matrix from Orthogonal Vectors

How does one define an affine rotation matrix in order to rotate a 3D volume to align with a new coordinate system? The current coordinate system is $\mathbf{x}, \mathbf{y}, \mathbf{z}$ and I want to ...
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2answers
76 views

How is this a coherent definition of “hyperplane”?

My textbook says the following: A hyperplane is a set of the form $$\{ x \mid a^T x = b \},$$ where $a \in \mathbb{R}^n$, $a \not= 0$, and $b \in \mathbb{R}$. Analytically it is the ...
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0answers
17 views

Proving that $\mathbb{K}$ is a skew field (see post for detailed description of $\mathbb{K}$).

Consider a Desarguesian affine space $\mathcal{A}$. Choose a fixed point $o$ and a fixed line $\mathbb{K}$ through $o$. Select another, arbitrary point $e$ on $\mathbb{K}$. For each point $x \in \...
2
votes
2answers
268 views

Affine hull of the intersection of two convex sets

Is it true that the affine hull of the intersection of two convex sets is the intersection of the affine hulls of these sets? Where the intersection of the two convex sets is non empty? Many thanks!
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1answer
32 views

There cannot be a concept of parallelism in a homogeneous projective space?

Page 4 of my computer vision textbook, Multiple View Geometry in Computer Vision (Second Edition), by Hartley and Zisserman, states the following: Affine geometry. We will take the point of view ...
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0answers
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Homotheties in an affine Desarguesian space: conjugation and factors

Definition: a class of conjugated homotheties (conjugated by a translation) is called a factor. I really don't understand this definition of a factor. I believe it has something to do with the ...
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1answer
257 views

Is the Centroid and Circumcenter of a triangle affine invariant?

As the title says, are the centroid and circumcenter of a triangle affine invariant? And how would I go about proving it? Thanks.
1
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1answer
24 views

Understanding affine cypher / Euclidean math

$$e(x) = (ax + b)$$ $$d(y) = a^{-1}(y-b)$$ I need to prove that $$e(x)=d(y)$$ iff $$(a^2)=1$$ and $$b(a+1)=0,$$ so I tried working with $d(ex)$ to show they can be equivalent: $$ax + b = a^{-1}(ax + ...
2
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1answer
737 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 separate transformations. However, they are all for the 2D case and I`m finding it difficult to generalise ...
20
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3answers
20k views

Rotation Matrix of rotation around a point other than the origin

In homogeneous coordinates, a rotation matrix around the origin can be described as $R = \begin{bmatrix}\cos(\theta) & -\sin(\theta) & 0\\\sin(\theta) & \cos(\theta) & 0 \\ 0&0&...
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3answers
45 views

Name for ratio of three co-linear points in affine geometry

Given three points $a_1$, $a_2$, $a_3$ in an affine space such that $\dim \langle a_1, a_2, a_3\rangle = 1$ (i.e. they are co-linear) and $a_1\ne a_2$, my professor defined the "simple ratio" (making ...
2
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0answers
43 views

Definition of a Product

For notational convenience I want to introduce a short-hand notation: Let $A\in\mathbb R^{n\times(k+1)}$ and $x\in\mathbb R^k$. My "product" should represent the affine-transformation $A\bullet x = Sx ...
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0answers
14 views

Proof regarding affine spaces over fields

Consider an affine space $\mathcal{A} = \operatorname{AG}(n,\mathbb{K})$, with $\mathbb{K}$ a field. Show that $\mathcal{A}$ contains $\operatorname{AG}(2,2)$ if and only if $\operatorname{char}(\...
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0answers
34 views

Is this axiomatization of affine plane categorical?

First I'll give some definitions. Hilbert's plane axioms of incidence: We consider a set $P$ (plane) and a family $\mathcal{L}$ (a family of lines) with axioms: For any two distinct points $a,b$ ...
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2answers
23 views

An automorphism of an affine space preserves parallelism: question on proof

Definitions: An automorphism of an affine space is a permutation of the set $\mathcal{P}$ of points that preserves lines and planes (if the dimension it at least 3). This is the proof given in my ...
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0answers
7 views

Proving statements regarding bases of affine spaces

! Definitions: Basis of an affine subspace; Consider a $d$-dimensional affine subset $D$ of an affine space, and a set of points $S$ in $D$. If $S$ is a generating set for $D$, then $S$ ...
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1answer
38 views

Checking if a certain transform is affine and computing its inverse

$$T(x,y)=(2x+y+5,3x+2y+2)$$ Check if $T$ is affine and compute $T^{-1}$. I'm not sure if I'm using the right methods here or if I am missing something, but I went about proving that $T(A)+T(B)=T(A+B)$...
1
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1answer
45 views

Point Addition in Affine Space

The only operations defined on points in an affine space are point-vector addition. this yields a new point. point-point subtraction. this yields a vector. This can be extended to an affine sum, $\...
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0answers
17 views

Is there any incompatibility between affine spaces and Hilbert spaces? [closed]

I was wondering if there is such a thing like a Hilbertian affine space. I've seen the definition of an Euclidian affine space, which is: An affine space (A, V, φ) is an Euclidean affine space if ...
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2answers
40 views

Prove that General Affine Transformations preserve ratios of lengths

Let $A$ be a matrix with determinant 1. Then we call a general affine transformation, a transformation of the form \begin{align*} \begin{bmatrix}x'\\y'\end{bmatrix}=A\begin{bmatrix}x\\y\end{bmatrix}+\...
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votes
1answer
129 views

prove that hyperbolic cone is affine set (check my solution)

Hyperbolic cone $C_P$ with $P$ positive definite matrix is a set that satisfies the following $C_P = \{x: x^TPx\leq (c^Tx)^2\}$. I need to prove that this set is affine. I know that this set is convex,...
0
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1answer
17 views

Affine space and convex sets in the context of Euclidean space

I am a bit confused as to the relationship between the ideas of vector space, affine space, and convex sets in the context of Euclidean space $\mathbb{R}^d$. As of now, this is how I see it. $\mathbb{...