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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26 views

Express a point as an affine combination of another two points(3D collinearity)

So, given the points A(1,2,2), B(2,4,2) and C(3,6,2) I have to show that they are collinear. ...
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80 views

What are sliding vectors mathematically?

What is the mathematical definition of sliding vectors and their operations, as used in mechanics? What kind of mathematical structure do they form? Does the operation of constructing the "space" of ...
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1answer
26 views

The set if affine?

The Set $\{ Ax + b | Fx = g \}$, is it affine? How can I prove it? My answer is yes, the intuition is that $\{ x | Fx = g \}$ is a solution space of equation $Fx = g$, thus it is a linear subspace. ...
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22 views

When the sum of coefficients of two linear combinations are equal.

I recently was looking a set of polynomials (the Legendre polynomials up to degree $n$) that form a basis for the space of polynomials $\{a_{0} + a_{1}x + \dots + a_{n}x^{n}: a_{i} \in \mathbb{R}\}$ ...
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11 views

Is the below formula equivalent?

$K$ is a simplicial complex: Is $\{\sigma \in K | \sigma \cap conv(\{a, b\}) = \emptyset\}$ equivalent to $\{ \sigma \in K | \sigma \cap \{ a \} = \emptyset \} \cap \{ \sigma \in K | \sigma \cap \{ ...
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12 views

Affine transformation step by step

So I have the before and after 3d coordinates of an object that has been translated and rotated. I need to calculate the matrix to return the object to it's original position. I've been reading ...
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24 views

V is an affine subspace iff for any two distinct points V contains the line dtermined by these points

Since it can be shown that the barycenter of n weighted points can be obtained by repeated computations of barycenters of two weighted points, a nonempty subset V of E is an affine subspace iff for ...
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8 views

the set $V$ of barycenters ${\Sigma}_{i\in I}{\lambda}_ia_i$ is the smallest affine subspace containing $(a_i)_{i\in I}$

Given an affine space $(E,E^{\to})$, for any family ${(a_i)}_{i\in I}$ of points in E, the set V of barycenters ${\Sigma}_{i\in I}{\lambda}_ia_i$ is the smallest affine subspace containing ...
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1answer
15 views

Let $a,b$ be affine combinations of points from a set $S$. Then is the affine combination of $a,b$ also an affine combination of points from $S$?

Let A be an affine space, $a,b$ affine combinations of points from a finite subset $S$ of A. Then is the affine combination of $a,b$ also an affine combination of points from $S$? I found it ...
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31 views

An easy definition of an $n$-dimensional affine cube

In a few weeks I'm giving a presentation on the History of Ramsey Theory and I want to start off with Hilbert's cube lemma. The only problem is that the pre-requisites for this course is only ...
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1answer
21 views

Transform gradient to reference element

Minimal example of the problem How can you transform the gradient to the reference element?
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30 views

Dumb question regarding affine transformations

So, we can write an affinity $\phi$ as $$\phi(x) = Ax + b$$ for some linear transformation $A$ and vector $b$. What exactly does it mean for an affinity to be a "scaling"? Is this a mapping of the ...
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17 views

Interesting observation WRT 1,2,3-dimensional convex polytopes and higher dimensional ones as counterpart

When I experimenting with qhull utility and Quickhull Algorithm, I found that in $\mathbb{R}^d, d \in \{1,2,3\}$ space the number $F$ of convex hull's $(d - ...
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19 views

Adjacency of convex hull facets

Let $C$ be a $d$-dimensional convex polytope and $p$ is a point outside of it. $C=\{f_c\}$ defined by set of facets $f_c=\{p_c,A_c\}$ where $p_c$ is a tuple of vertices and $A_c$ is a set of ...
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69 views

Why are affine subspaces also sometimes called linear manifolds?

According to Wikipedia, an affine subspace is a subset of a vector space closed under affine linear combinations. That is, linear combinations whose scalar coefficients sum to 1. It's not clear to ...
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1answer
60 views

Affine Bundles vs Affine Spaces

I went through the wiki article on affine spaces and had a quick look on the affine bundle wiki article but I don't understand what the affine map is in the case of affine bundles over vector bundles. ...
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34 views

equiaffine arc length, moving frame, and affine curvature

I am trying to learn affine geometry, and I'm having some trouble getting started with the following problem. Compute (a) the equiaffine arc length, (b) the moving frame, and (c) the affine ...
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1answer
17 views

Newbie with barycentric coordinates: why one is zero when on a vertex?

I'm trying to calculate if a 2D point lies inside a triangle and I solved the following system: ...
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17 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
52 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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78 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
25 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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17 views

Preservation of aligned points

If a mapping transforms aligned points into aligned points is necessarily an affine application? In other words, are there any mapping with that property without preserve the ratios between aligned ...
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1answer
44 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
30 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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101 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
72 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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1answer
73 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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14 views

How is affine space analogue for lattices called?

Lattices are so like vector spaces that it seems natural to have an affine space construction for them. Unfortunately I could not find how such a construction is called. Could you please help me? ...
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1answer
45 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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1answer
46 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
73 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
25 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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186 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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2answers
52 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
111 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 ...
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1answer
73 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: ...
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1answer
37 views

Affine coordinates of a line

Can you help me figuring out how to solve the next problem? If the points M and N have affine coordinates $(m_1,m_2,m_3)$ and $(n_1,n_2,n_3)$ with respect to some points A,B,C, then the points X of ...
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0answers
97 views

Multi-affine function

Suppose i have a three-variable function f(x1,x2,x3), f:R^3 -> R. If it is linear for x1,x2 and x3 we can say it has the form f(x1,x2,x3) = c1x1 + c2x2 + c3x3 where c1,c2,c3 in R. We can ...
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1answer
97 views

Why in the affine space can not we use Grassmann formula?

For example, in space three-dimensional affine space generated by two skew lines is all the space three-dimensional, since they are not coplanar. For this reason it is not worth the Grassmann ...
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1answer
55 views

Invariants of point sets in an affine space

A distance between a pair of points in an affine space is invariant under translation, rotation and reflection. An angle in a triangle whose corners are tree points is also invariant under scaling. ...
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1answer
86 views

Intersection of affine subspaces of finite codimension in Hilbert space

I'm wondering whether the following assertion is true: Any two affine subspaces of the same finite codimension in a ($\infty$-dimensional) Hilbert space either are parallel or have nonempty ...
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2answers
69 views

Affine geometry about parallelogram

If ABCD is a parallelogram and M,N,P,Q are points on it sides then MNPQ is a paralellogram iff the diagonals intersect at a common point (i.e the diagonals of MNPQ and ABCD intersect at the same ...
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1answer
177 views

Distance between affine space and point

Let $A,A'$ two affine subspaces of a finite Euclidean Vectorspace $V$. Let $p,p'$ two points, such that $d(A,p)=d(A',p')$. $\dim(A)=\dim(A')$ I would like to show that there exists a movement ...
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103 views

distance of an affine subspace to a polytope

I wonder how to prove the following statement. Let $V$ be a $d$-dimensional normed space with $d \geq 3$, let $P \subset V$ be a $(d-2)$-dimensional polytope. Then there is an $\epsilon > 0$ such ...
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1answer
1k views

How to prove we could use mass point geometry to solve all the triangle problem involving ratio between line segment and transversal in a triangle?

what is an easy way to prove that use mass point geometry to solve a problem in the link i provide that is involving cevians in a triangle is same as using the other way in euclidean geometry or ...
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1answer
50 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 ...