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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2
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0answers
55 views

A question for epigraph and affine function

I'm working with a problem the epigraph of a real-valued function $f$ is a halfspace $\iff$ $f$ is a real-valued affine fuction. First, I quickly recall some definitions: A (closed) halfspace is a ...
-1
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1answer
62 views

Finding the number of solutions to two equations

I have a question: For the following system of linear equations, using Gaussian elimination, decide whether it has at least one solution. If it does, represent the general solution as an affine map ...
2
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1answer
41 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 ...
0
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1answer
627 views

Find 2D affine transform matrix given a pair of points

I have the coordinates of two points in an initial 2d coordinate system and the corresponding coordinates in a target system. Is is possible to determine the affine transform matrix from these values? ...
1
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1answer
1k views

Affine Transformation matrix for mapping a triangle to another

I'm trying to write a function in Matlab that will give me a matrix T that can be used to multiply points in homogeneous coordinates. This is achieved my mapping a ...
1
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1answer
33 views

How do we define affine combinations in affine spaces?

Suppose we're given a field $A$ and a vector space $X$ over it. Then an affine space over that is a set $P$ (of "points") equipped with an action $$+ : X \times P \rightarrow P$$ such that $0_X+p ...
0
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1answer
80 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 ...
3
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1answer
85 views

Example: Irreducible component - affine varieties

Again, I know how to prove the statement. But, I cannot find any example. Please help me for finding an example. Thank you:)
2
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1answer
86 views

Affine variety example.

I know how to show the statement. But I cannot find an example (the part I underlined by a yellow pen) please help me for finding an example. Note: For example, can I consider the following ...
0
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1answer
38 views

Relations between an affine space and a topological space

What is the relation between an affine space and a topological space? Is one a specialization of the other? Moreover, what do we call a point in geometry: an element of a topological space or an ...
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3answers
528 views

Origin in vector space?

In the wikipedia article about vector space I do not understand this sentence Roughly, affine spaces are vector spaces whose origin is not specified. A vector space does not need an origin. When ...
1
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1answer
371 views

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 ...
1
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1answer
43 views

Do all the solutions have to be in an affine variety?

An affine variety $V(X)$ is the zero-locus of a set of polynomials. So if the variety is generated by the polynomial $y-x=0$ in $\mathbb{R}^2$, then do all the solutions (i.e., every point satisfying ...
2
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0answers
58 views

How to describe the variety for lines through the origin

Let $I(x)$ be an ideal generated by the affine variety $X$. $I(X)=\langle xy,yz,zx\rangle$ gives the three coordinate axes through the origin (because it is the intersection of the $3$ planes $xy=0$, ...
3
votes
1answer
212 views

Tensor product of affine space and algebra

When reading about Quantum Mechanics, I always feel a bit disappointed when physicists consider that (for example) the 3-dimensional position of a particle must be decomposed into 3 coordinates $x, y, ...
0
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1answer
33 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 ...
2
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0answers
41 views

Semisimple part of a nilpotent connected affine algebraic group

These notes on affine algebraic groups mention the following theorem. Let $G$ be a connected nilpotent affine algebraic group (over an algebraically closed field $k$), and denote $G_s$ and $G_u$ ...
0
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0answers
116 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 ...
1
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1answer
124 views

Why a 2D Affine Transformation matrix is 3 by 3

The matrix which I get for Scaling , Shearing and Rotation are follows: Scale: Shear Rotation Why do we need Homogenous Co-ordinate to get the transformation matrix as listed below? (need a ...
3
votes
2answers
40 views

Varieties strict inclusion infinitely long

Let $F$ be a field, and consider the affine space $F^n$. Could there be affine varieties $V_1,V_2,\ldots\subseteq F^n$ such that $V_1\supset V_2\supset\cdots$, and no two are equal?
0
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1answer
92 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 ...
1
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1answer
129 views

Desargues Theorem and help with its significance

I am still trying to get a hang of drawing the picture. The only idea I get from the theorem is that if two triangles are in perspective from a point, then we the theorem , we also get that the ...
0
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1answer
36 views

Fixpoints of affine transformations

I want to find out all the possibilities what fixpoints of an affine transformation can be in 2-dim vector space. If the transformation is identity, then it is trivial - fixpoints describe the ...
0
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2answers
64 views

Identify a quadric

Could you tell me how to identify a given quadric? Given a conic section, I should find an orthonormal affine frame in $\mathbb{R}^2$ (with standard dot product) in which the equation has a canonical ...
1
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0answers
59 views

Linear algebra, affine space, and floor function

My question is mostly: is there a name for this kind of things. I am mostly interested by finding book or articles about what follows, but without even a word or a name, it is quite hard to search for ...
0
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1answer
57 views

Direction of traslation of affine movement

I have a doubt about this. We have an affine isometry of an affine space $X$ of dimension 3. Now, we know it's the composition of some movement (reflection, rotation, etc), with a traslation, and we ...
0
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1answer
58 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 ...
2
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0answers
38 views

Monotononically Increasing Water Filling Solution?

$\mathbf{I}$ is the $K\times K$ identity matrix. $\mathbf{h}_i\in\mathbb{C}^{M\times1}\quad\forall1\leq i\leq K$ are column vectors. Consider the solution of the convex optimisation problem over ...
0
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1answer
60 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 ...
3
votes
1answer
283 views

Prove, in this figure, that $EFGH$ is a parallelogram

In the following figure, $ABCD$ is a parallelogram, and $O$ is any point. Parallelograms $OAEB, OBFC, OCGD, ODHA$ are completed. Prove that $EFGH$ is a parallelogram. We can obtain a fairly trivial ...
3
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0answers
340 views

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 ...
3
votes
2answers
284 views

How to determine the period of a binary matrix?

As a part of my research project, I encounter a difficulty to compute the period of a binary cat matrix $C$, which is a square matrix of dimension $n$ with the following properties $|\rm{det}(C)|=1$ ...
1
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1answer
335 views

Affine transformation matrix coefficients

In an affine transformation $x \mapsto Ax+b$, $b$ represents the translation; but what does the matrix $A$ represent exactly? In a 2D example, $A$ is a $2\times 2$ matrix, but what does each term ...
0
votes
1answer
123 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 ...
0
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1answer
35 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 $\mathbf c$”?
0
votes
2answers
492 views

Get affine transformation matrix from two positions of the object

I have an object in 3d space which is represented by the set of vertices. Then I scale this object, rotate it and translate. After these operations I get the second set of vertices with new ...
0
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0answers
239 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 ...
2
votes
1answer
664 views

Finding an affine combination of a point on a triangle

I have a problem involving affine combinations that I can't figure out how to solve. Given the above picture, write q as an affine combination of u and w. Now, I understand how to write the ...
0
votes
2answers
32 views

How do you rotate a vector by $90^{\circ}$?

Consider a vector $\vec{OA}$. How will I rotate this vector by $90^{\circ}$ and represent in algebraically?
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3answers
60 views

Prove, using affine geometry, that in this figure $\Delta DEF$ is always equilateral

Consider the following figure: $\Delta DAC, \Delta CEB, \Delta AFB$ are isosceles. $\angle ADC = \angle CEB = \angle AFB = 120^{\circ}$. Prove that $\Delta DEF$ is equilateral. Now, there is a ...
0
votes
1answer
60 views

Prove for any four points: $|AB|^2 + |CD|^2 -|BC|^2 - |AD|^2 = 2\cdot \vec{AB}\cdot \vec{DB}$

Let $A, B, C, D$ be four points in space. Prove $$|AB|^2 + |CD|^2 -|BC|^2 - |AD|^2 = 2\cdot \vec{AC}\cdot \vec{DB}$$ Clearly, $$AB = B-A$$ $$CD = D-C$$ $$AD = D-A$$ If I directly substitute the ...
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2answers
69 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 ...
0
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1answer
106 views

alternative definition of Affine map

Let $f:X\longrightarrow Y$ be a function on real vector spaces (note that $X,Y$ have arbitrary dimensions). If $T(x)=f(x)-f(0)$ is linear, $f$ is called an affine map. Prove that $f$ is affine if ...
3
votes
1answer
340 views

Affine group, semi-direct product and linear transformations

According to wikipedia the Affine group is the semi-direct product of a vector space $V$ and the general linear group $GL(V)$. Here is the definition of the semi-direct product in terms of matrices ...
0
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1answer
152 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 ...
2
votes
2answers
110 views

Prove that $\text{(BE)}\|\text{(JF)}$ using vectors.

Problem Let $\text{ABC}$ be a triangle and let $\text{I}$ , $\text{J}$ and $\text{K}$ be points such that: $\vec{\text{BI}}=\frac{1}{2}\vec{\text{IC}}$, $\vec{\text{AJ}}=2\vec{\text{JB}}$ and ...
2
votes
0answers
173 views

categorical description of the Minkowski sum of polytopes

Consider the category $\textbf{Poly}$ of polytopes, where the objects are convex hulls of finite subsets of $\mathbb{R}^d$ for arbitrary $d \in \mathbb{N}$ and where the morphisms are affine maps ...
2
votes
1answer
113 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!
4
votes
1answer
100 views

Bijection $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ preserves collinearity $\iff \ \ f(x)=Ax+b$

I don't know how to prove the following: Bijection $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ preserves collinearity $\iff \ \ f(x)=Ax+b$, where $A \in GL_2(\mathbb{R})$, $b$ is a fixed vector in ...
0
votes
1answer
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: ...