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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21 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
60 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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2answers
120 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
84 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 ...
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0answers
112 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
33 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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73 views

What if segments are not infinitely divisible?

I almost got myself mixed up I a philosophical discussion again. Somebody was talking about the Planck time and length which are, according to him, the minimal possible time and distance, and how ...
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0answers
45 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 ...
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1answer
59 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
39 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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1answer
381 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? ...
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1answer
953 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 ...
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1answer
29 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 ...
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1answer
59 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
81 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:)
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1answer
84 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 ...
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1answer
35 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
306 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 ...
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1answer
345 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 ...
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1answer
42 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 ...
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0answers
55 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
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0answers
180 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, ...
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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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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 $G_u$ ...
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0answers
106 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
86 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 ...
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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?
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1answer
85 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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1answer
105 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 ...
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1answer
33 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 ...
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2answers
57 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 ...
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0answers
57 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 ...
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1answer
52 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 ...
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1answer
53 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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0answers
35 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 ...
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1answer
50 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
258 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 ...
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292 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 ...
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2answers
229 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 $|\rm{det}(C)|=1$ ...
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1answer
255 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 ...
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1answer
101 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
33 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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2answers
284 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 ...
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0answers
217 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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1answer
634 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 ...
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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 ...
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1answer
56 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
55 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
94 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 ...