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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Distance between two affine lines using determinant of Gramian matrix.

I've a task to find the distance in $E^4$ between: $L = [1,2,-1,4] + \text{lin}((1,2,-1,0))$ and $M = [2,3,1,5] + \text{lin}((2,1,0,2))$ My efforts to find the correct solution: Let ...
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0answers
88 views

Prove something is affine?

For any subspace $K$ and any point $u$, prove $K+u$ is affine. Or if you have an affine set $V$ and point $u$, then prove $V-u$ is a subspace.
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65 views

Separation of Euclidean Space

Consider a finite collection $\mathcal{H}$ of hyperplanes of $\mathbb{R}^n$ that have a common line. Given some $A \subseteq \mathbb{R}^n$ that is homeomorphic to a subset of $\bigcup\mathcal{H}$, ...
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0answers
403 views

Convexity of affine function.

Can someone help me with a proof that affine function preserves convexity? Given that $f$ is convex, $A$ is in $\mathbb{R}^{M\times N}$ and $b$ is in $\mathbb{R}^m$ then show that $g(x) = f(Ax+b)$ is ...
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1answer
379 views

Affine subspace iff statement

I cannot find a proof for the following statement, though the statement itself seems common enough: $X$ is an affine subspace of $\mathbb{R}^n \iff \forall a\in X: X - \{a\}$ is a linear subspace. ...
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2answers
2k views

Definition of an affine subspace

I am reading this introduction to Mechanics and the definition it gives (just after Proposition 1.1.2) for an affine subspace puzzles me. I cite: A subset $B$ of a $\mathbb{R}$-affine space $A$ ...
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1answer
286 views

Research Paper and Affine Subspace

I was reading a research paper titled Purity and Reid's Theorem by A.Blass and J.Irwin and i have the problem with the explanation of the proof of the first theorem, that is theorem 1.1. In the proof ...
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1answer
33 views

Finding a line $L\subset V(y-xz)\subset\mathbb A^3_k$

I want to find lines $L\subset V(y-xz)$ and $M\subset\mathbb A_k^2$ such that $$ V(y-xz)\setminus L \simeq \mathbb A_k^2\setminus M\ . $$ Hint suggests that I use the projection $(x,y,z)\mapsto(x,y)$. ...
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1answer
29 views

Are quasiaffine subsets of $\mathbb{A}_F^n$ always necessarily open or closed?

Something I was wondering about lately, suppose $\mathbb{A}_F^n$ is affine space over a field $F$ which is algebraically closed. When I say a quasi-affine set, I mean a set that is locally closed, so ...
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2answers
27 views

fixed points of an affine transformation is unique iff $1 \notin SP(\vec{f} )$

Let $f$ be Affine transformation from $E$ to $E$ (always we assume it finite dimensional ) and $\overrightarrow{f}$ is the linear mapping associated to $f$. Then the map $f$ has a unique fixed ...
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2answers
21 views

Number of parameters to specify an affine transformation in n dimensions

In general, how many parameters does it take to specify an affine transformation in $n$ dimensions, and how does one go about proving this? For example, in 2 dimensions it takes 6 parameters, and in ...
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1answer
31 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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1answer
44 views

Affine transformation, if $L_1, L_2 - $ skew lines, $f(L_1), \ f(L_2) $ are parallel, then $f$ is not injective

Could you tell me how to prove that if $f$ is affine transformation, $L_1, L_2 $ are skew lines, $f(L_1), \ f(L_2) $ are parallel, then $f$ is not injective?
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1answer
69 views

Variety Affine Space

I have the following question which i'm not sure how to work out... $For\ f=6x^2y-xy^2-2y^3+1\ and\ \ h=3x-2y\ \in \mathbb{C}[x,y]$ Show that V(f,h) is empty. What can you say about the ideal ...
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1answer
361 views

Affine transformation matrixes

I could use some advise with the following problem: Lets say there is a cuboid that has two distinguished points - that is one of its vertexes ($A$) and the other one is somewhere on the surface ...
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1answer
25 views

how to obtain transformation matrix A in y = Ax + b notation?

I'm trying to obtain original transform matrix A and its translation vector b From y=Ax+b equation. I have original values of vectors before transform and translation (x) and vectors after transform ...
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1answer
35 views

Making a matrix full rank through affine transformations

If I have (finite) $k$ vectors, $u_1,...,u_k\in\mathbb{R}^N$ that are in general linearly dependent is it possible to take positive affine transformations of the form: $$u'_i=\alpha_i u_i ...
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1answer
33 views

Prove that the set of all fixed points is a hyperplane

Let $A$ be a $n$-dimensional affine space ($ 2 \leq n$) and let $\Phi:A\to A$ be a bijective affine mapping, which isn't the identity mapping, with the following property: For all points $p$ and $q$ ...
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1answer
43 views

dimension of quotient space

I am confused about the following: In Wiki: => dim(vector space) - dim(subspace) = dim(quotient space) In S. Boyd's textbook of cvx (p.22) => dim(subspace) = dim(affine set) Problem: ...
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1answer
48 views

Affine connection

The affine connection is not in general defined uniquely by the smooth structure and the Riemannian metric. Can you give some demonstration with some examples?
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2answers
34 views

affine hull, how to understand the statements below?

I am new to affine space, I looked through the wikipedia page, and have problem understanding the statements below. The affine hull of a set of three points not on one line is the plane going ...
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1answer
53 views

Formula of signed distance from hyperplane to point

Let $H$ be a hyperplane defined by the points $p_1, p_2, ..., p_n$ and single point $x$ generally out of the hyperplane. Is there any formula to calculate the signed distance between $x$ and $H$? I ...
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1answer
201 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
32 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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1answer
38 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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1answer
48 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
201 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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2answers
135 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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2answers
31 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
56 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
51 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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1answer
79 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 ...
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1answer
35 views

Effect of Moving within the Feasible Region

$f:\mathbb{R}^n\rightarrow\mathbb{R}$ is a concave function with local maximum at $\mathbf{x}^*$ in a convex, closed feasible set $\mathcal{F}\subset\mathbb{R}^n$. Now consider a suboptimal point ...
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3answers
205 views

Equation of a line in homogenous coordinates given 2 points in affine coordinates

So if I have 2 points $A$ and $B$ such that $F(A) = (1; a, a^3)$, and $F(B) = (1; b, b^3)$. how do I find the equation of this line in homogeneous coordinates? So I know how to get a line the ...
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1answer
119 views

Cross-ratio projective geometry

I have 4 points $P_0=[1:2], P_1=[3:4], P_2=[5:6], P_3=[7,8]$ in $\mathbb KP^1$ and would like to evaluate the cross-ratio. It is given by the following: $\pi:\mathbb KP^1\rightarrow G$ is the unique ...
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2answers
242 views

Showing that a rigid motion preserves distances

For a linear map $\Phi : \mathbb{R}^n \to \mathbb{R}^m$ which has the form $[ x \mapsto Ax]$ for an $n \times m$ matrix $A$. A rigid motion is an affine map where $A$ is orthogonal. How would you ...
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1answer
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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1answer
552 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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1answer
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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1answer
502 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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1answer
25 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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0answers
11 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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1answer
45 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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26 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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20 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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45 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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0answers
11 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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23 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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20 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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22 views

Discrete Geometry (Polytopes)

I have to try to prove the following: Let $V = {v_1,...,v_n} \subset R^d$ be a point configuration affinely spanning $R^d$ (i.e., $aff(V) = R^d)$. Let H be the collection of hyperplanes spanned by ...