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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
6 views

How can I prove $M+t$ is a hyperplane if $M$ is a maximal subspace

Let $M$ be a non-empty proper subset of a vector space $X$ over $\mathbb R$ and $t$ belongs to $X$, then $M$ is a maximal subspace if and only if $t+M$ is a hyperplane and $t$ belongs to $t+M$.
John MATH's user avatar
0 votes
0 answers
22 views

Basis/frame of reference for the coordinates of points in affine space $\mathbb{A}^n(k)$

I am currently reading a chapter on Affine Space from the Book titled "Algebra, Topology, Differential Calculus, and Optimization Theory for Computer Science and Machine Learning" (link) by ...
vxek's user avatar
  • 173
0 votes
0 answers
27 views

discriminant to distinguish parallel line and double line degenerate conic sections

A real affine conic section is the zero locus in $\mathbb{R}^2$ of the quadratic form $$q(x,y)=ax^2+2bxy+cy^2+2dx+2ey+f=0.$$ We may understand this as the $Z=1$ affine patch of the locus in the ...
ziggurism's user avatar
  • 16.3k
5 votes
3 answers
71 views

Generalized notion of perpendicularity, (not orthogonal)

In 3 dimensions, we might call 2 planes perpendicular iff their normals are orthogonal. But this does not coincide with the definition of orthogonal subspaces - the dot product of any pair of vectors ...
Shuri2060's user avatar
  • 4,313
1 vote
2 answers
43 views

Which affine transformations preserve rectangles?

I need a way to decide if a given affine transformation preserves arbitrary rectangles in $\mathbb{R}^2$, meaning after applying it to any rectangle, it is still a rectangle afterwards. Thought ...
VaNa's user avatar
  • 163
0 votes
0 answers
53 views

Euclidean first fundamental form as vector bundle valued, first order differential invariant

I'm self studying the book Cartan for beginners and i'm trying to solve the exercise 2.5.2. The setup is the following: Let $\operatorname{ASO}(n+s)$ the set of rototranslation in the $n+s$ ...
Marco's user avatar
  • 566
1 vote
1 answer
75 views

How to prove characterizations of affine basis without the notion of affine combinations?

Motivation: I would like to prove characterizations of the affine basis property without using the notion of affine combinations. I am working with the following definitions. Definition: For any ...
Max Flow's user avatar
  • 232
0 votes
0 answers
21 views

How do homomorphisms of affine spaces relate to homomorphisms of difference spaces?

I am looking at two (common) definitions: Definition 1: An affine space is a triple $(A, V, +)$ with $A \neq \emptyset$, with $(V, +_V)$ a vector space, and with $+ \colon A \times V \to A$ such that ...
Max Flow's user avatar
  • 232
0 votes
0 answers
47 views

proof that affine spaces are isomorphic to vector spaces?

Can someone please explain in detail the following proof that affine spaces are isomorphic to vector spaces? To clarify the notations the book has previously defined $R_{v}$ and $v_{x,y}$ as follows ...
Faber Bosch's user avatar
0 votes
0 answers
36 views

What's the fundamental theorem of affine geometry in case field has characteristic 2?

In the book of Marcel Berger, Geometry 1, the Fundamental theorem of affine geometry is stated as: Let $X,X'$ be affine spaces of same dimension $d\geq2$. Let $f:X \rightarrow X'$ be a bijection ...
fsepp's user avatar
  • 51
1 vote
0 answers
47 views

A sort of "minimal presentation " for a local ring essentially of finite type over a field

Let $k$ be a field of characteristic $0$. Let $(R,\mathfrak m)$ be a local ring essentially of finite type over $k$ (https://stacks.math.columbia.edu/tag/07DR). Then, $R$ is the homomorphic image of ...
strat's user avatar
  • 169
1 vote
0 answers
56 views

hyperplanes as intersection with cones

An affine hyperplane $H$ in $\mathbb{R}^{3}$ is an affine subspace in $\mathbb{R}^{3}$ of dimension 2, i.e. there exists a $h \in \mathbb{R}^{3}$ and a two-dimensional $\mathbb{R}$ vector subspace $U$ ...
Marius Lutter's user avatar
2 votes
2 answers
58 views

Check that the sets of points in $\mathbb R^3$ span the same affine subspace

Check that the sets of points $\{(1, 1, 0) , (2, 1, 0) , (3, 1, 1)\}$ and $\{(2, 1, 0) , (3, 1 , 0) , (3, 1, 1)\}$ span in $\mathbb R^3$ the same affine subspace. My try: Consider the difference ...
qerty149's user avatar
  • 116
3 votes
1 answer
77 views

Explanation of Barycentric coordinates

Let $ABC$ be a triangle in the plane $\mathbb{R}^2$ (in other words, the points $A,B,C$ are affinely independent). Let $AA_1, BB_1, CC_1$ be the medians of this triangle meeting at the point $M$. Find ...
End points's user avatar
0 votes
1 answer
41 views

affine combinations of Bernstein basis polynomials that are nonnegative and sum to 1

The $i^\text{th}$ degree-$n$ Bernstein basis polynomial is defined as $$ \begin{equation} b_i^n(x) = \binom{n}{i} (1-x)^{n-i} x^i. \end{equation} $$ The Bernstein basis polynomials have many ...
brentertainer's user avatar
1 vote
0 answers
27 views

Find all affines mappings $A$ with the property $A(p) = q$ and $A(q) = p$

The lines $p : y = 1$ and $q : x = −2$ are given in the affine plane $\mathbb{R}^2$. Find all affines mappings $A$ with the property $A(p) = q$ and $A(q) = p$. Hint: Where is such an affine mapping ...
GENERAL123's user avatar
1 vote
1 answer
86 views

Is the tetrahedron is the unique polytope having the property that all its vertices are adjacent?

$\textbf{Definition 1:}$ A polytope $P \subseteq \mathbb{R^n}$ is the convex hull of a finite points $x_1,...,x_n \in \mathbb{R}^n$, i.e, $P=\text{conv}(\{x_1,...,x_k\})$. The dimension of P is the ...
Alejandro David's user avatar
0 votes
2 answers
59 views

Show that the intersection of a hyperplane and a plane is a line given the following conditions.

I have been trying to prove that the abovementioned intersection is a line but I do not get to the conclusion. The problem gives the following restrictions. The dimension of the affine space A must be ...
Lucía's user avatar
  • 21
2 votes
1 answer
94 views

When do affine transformations preserve ratios of n-dimensional hypervolume?

Affine transformations of the plane preserve ratios of areas; that is, if $Area(F) = k \cdot Area(G)$, then $Area(T(F)) = k \cdot Area(T(G))$. They do not, however, preserve ratios of length, unless ...
SRobertJames's user avatar
  • 2,722
0 votes
0 answers
38 views

Prove every parallelogram preserving transformation can be constructed from an isometry and two stretches

Prove that any transformation of the Euclidean plane which preserves parallelograms can be constructed from the composition of a rigid transformation (an isometry) and two stretches, using synthetic ...
SRobertJames's user avatar
  • 2,722
0 votes
1 answer
16 views

Defining a mapping which rotates a plane

Let $n = (n_1, n_2, n_3) \in \mathbb{R}^3 \setminus \left\{0\right\}$. I want to define a linear map $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$, $f(x)=Ax$, which maps the xy-plane $z = 0$ to the plane ...
mathslover's user avatar
  • 1,352
0 votes
1 answer
50 views

Proving $\mathbb{P}^2-\{[0:0:1]\}$ is not affine

I'm assuming the field to be algebrically closed I know that there are other questions on this site that address the problem in the title, but the answers are just hints like: "you have to prove ...
Kandinskij's user avatar
  • 3,537
0 votes
1 answer
52 views

Prove every point on the plane is a unique affine combination of the vertices of any triangle

Given three non-colinear points on the plane, prove that any point on the plane can be uniquely represented as an affine combination of them (this is barycentric coordinates). My proof is below. ...
SRobertJames's user avatar
  • 2,722
0 votes
0 answers
44 views

Is the Euclidean distance monotonic under affine transformation?

Let us say we have an Affine transformation: $\vec{\gamma}=\eta \vec{\tau} +\vec{\kappa}$, where $\eta$ is a diagonal matrix. Specifically, $\vec{\tau}{=}[\tau_x \ \tau_y\ \tau_z]^T$ forms a sphere, ...
Abir's user avatar
  • 1
0 votes
0 answers
37 views

Some problem about affine space and algebraic variety

Actually, I am reading something about control theory, but it contains lots of mathematical descriptions that I can't understand. The contents are the following: Consider the system $\mathcal{D} / k$ ...
Xiaolong Huang's user avatar
0 votes
1 answer
72 views

Birational morphism of the affine line

Let $k$ be an algebraically closed field. Suppose $K$ has characteristic nonzero, can we characterize the birational morphism of $A^1$ the way we characterize isomorphisms(all linear maps)? I know it ...
user avatar
0 votes
0 answers
45 views

What is official name for converting orthogonal coordinates to affine coordinates, but then treating the affine axes as perpendicular

Suppose we had the equation y=x^3 , and then represented our data points as vectors. Let the unit vectors of our axes be u=1+0i and v=0+1i, then our data point vectors have the form d[n]=x[n]u+y[n]v, ...
Edward Solomon's user avatar
1 vote
1 answer
42 views

Lines crossing 3-degree polynomial curve

I am trying to solve the following exercise, in the context of basic affine geometry (I saw it in a exercise sheet from a first course on affine and lineal geometry): Consider the affine plane $ \...
M159's user avatar
  • 642
1 vote
0 answers
48 views

Properties of $3\times 3$ matrices

Given two sets of corresponding points $\mathbf{P}_1, \mathbf{P}_2 \in \mathbb{R}^{d \times n}$, where $\mathbf{P}_1, \mathbf{P}_2$ are pointclouds expressed as matrices, we can derive a matrix $\...
Audrey's user avatar
  • 81
0 votes
2 answers
94 views

Zariski or algebraic criterion for generic point of a projective k-variety or are both equivalent? (near elementary alg.geom.)

To begin with, I am currently on what one could call elementary / classical algebraic geometry. OK in algebraic geometry almost everything is above elementary level ... but I don't imply simple. The ...
Ulysse Keller's user avatar
0 votes
0 answers
101 views

Extending a map $\mathbb{A}^1-\{0\}\to \mathbb{P}^1$

I know this question has been asked a couple of times on this site, but both of the times $\mathbb{P}^1$ was regarded as a space of homogeneous coordinates and not as $\mathbb{A}^1\cup \infty$ and I ...
Kandinskij's user avatar
  • 3,537
2 votes
2 answers
96 views

Intersection multiplicity bezout's theorem

I was reading algebraic curves by fulton, where I have doubt in applying bezouts theorem. Let $\quad F:=x^2+y^2-z^2$ $$ G:=x^2+y^2-2 z^2 $$ be Two curves in $ \mathbb{P}^2$, deg of $F=$ deg$G=2$ $\...
A  Narode's user avatar
0 votes
0 answers
26 views

How can I find a smoothest complex analytic function with a (finite) set of prescribed function values?

How can I find the smoothest complex analytic function $$x+yi \to u+vi$$ with a finite set of points having prescribed values $$\{x_1,\cdots,x_n\},\{y_1,\cdots,y_n\},\{u_1,\cdots,u_n\},\{v_1,\cdots,...
mathreadler's user avatar
  • 25.4k
0 votes
0 answers
61 views

Finite subsets of vector space under affine group action

For a field $K$ and natural numbers $n, m$, define $A(n,m,K) \in \mathbb{N} \cup \{\infty\}$ as follows: For an $n$-dimensional vector space $V$ over $K$, the group of affine automorphisms of $V$ acts ...
Vvabi's user avatar
  • 17
0 votes
0 answers
43 views

Geometric and linear independence in affine and linear spaces (Proof Validation)

I am looking to prove the following statement: Let $\{a_i\}^n_{i=0} \subseteq L$ be a set of points, then we have $\{a_i\}^n_{i=0}$ is geometrically independent $\Longleftrightarrow \{a_i - a_0\}^n_{...
Thomas Christopher Davies's user avatar
1 vote
0 answers
42 views

Point of defining affine spaces.

So I’m doing my first bachelor mathematics. And for our linear algebra and geometry course. We defined abstract affine spaces/surfaces axiomatically. To then prove that with papus’s axioma (I think) ...
jyly's user avatar
  • 11
1 vote
1 answer
33 views

Isomorphisms of causal structure of space with Lorentz form

Let $E$ be a real vector space, and $q$ be a quadratic form on $E$ of signature $(-1,1,\cdots,1)$. Let us call time vectors the vectors $v$ such that $q(v) < 0$. The set of time vectors has two ...
Plop's user avatar
  • 2,194
0 votes
0 answers
36 views

Factoring a linear transformation into translation, rotation and back again.

An affine transformation from a vector $(x,y)$ represented by $n_1=[x,y,1]^T$ to $n_2$ can be written with a matrix $$n_2 = M n_1$$so that: $$M = \begin{bmatrix}a&b&c\\d&e&f\\0&0&...
mathreadler's user avatar
  • 25.4k
0 votes
0 answers
49 views

Smooth affine plane curve: Differentials

$\newcommand{\dd}{\partial}$Let $f(u,v)$ be a smooth bivariate polynomial over $\mathbb{C}$. Let $X=\{(u,v)\in\mathbb{C}^2\vert f(u,v)=0\}$ be the corresponding affine smooth plane curve. In this ...
Peter's user avatar
  • 757
1 vote
0 answers
37 views

The relative interior is the union of a nested sequence of nonempty compact convex subsets

I read the following statement and its proof on Wikipedia: If $A \subset \mathbb{R}^n$ is nonempty and convex, then its relative interior $\text{relint}(A)$ is union of nested sequence of nonempty ...
Kim Seong Hyeon's user avatar
0 votes
0 answers
82 views

On the dimension of affine space.

Let $\mathbb{A}$ be a non empty set and $V$ be a vector space on a field $\mathbb{K}$ Definition 1. An application $$f\colon\mathbb{A}\times\mathbb{A}\to V\quad (P,Q)\mapsto f(P,Q)$$ defines an ...
NatMath's user avatar
  • 794
0 votes
1 answer
40 views

Equivalent statement for affine subspaces

We take $A$ to be a nonempty subspace of standard Euclidean space, then we have the following statements equivalent: A is an affine subspace $$\forall x,y\in A, t\in\mathbb{R}, (1-t)x+ty\in A$$ It ...
79999's user avatar
  • 135
0 votes
2 answers
35 views

Collineation of affine plane extends uniquely to collineation of projective plane

I'm trying to prove the following result, which is exercise 4.1 in Projective Planes by Hughes & Piper: If $\alpha$ is a collineation of an affine plane $A = P^l,$ there is a unique collineation $...
Display name's user avatar
  • 4,924
0 votes
0 answers
54 views

Dimension of affine hull of a convex set.

I am trying to prove the following. $C$ is a nonempty convex subset of $\mathbb{R}^n$. Show that $\operatorname{dim}(\operatorname{aff}(C))$ is the maximum of the dimensions of the simplexes contained ...
Muhammet Ali CANŞI's user avatar
0 votes
0 answers
35 views

Differentiability of map between affine maps via cartesian coordinate systems.

In the exercise 2.13.4 of the book Meccanica Analitica: Meccanica Classica, Meccanica Lagrangiana e Hamiltoniana e Teoria della Stabilità by Valter Moretti, a concept of diffeomorphism is defined for ...
Генивалдо's user avatar
0 votes
1 answer
60 views

Every parameterizable affine variety is irreducible

I want to show that an affine variety $V \subset \mathbb{C}^n$ which is parameterizable in the following sense: There exist Polynomials $g_1, \ldots, g_n \in \mathbb{C}[x]$ such that: $$ V=\{(g_1(t),\...
Raoul Luqué's user avatar
0 votes
1 answer
62 views

Barycentric coordinates on an affine space

In the book 'Geometry of Quantum States' the author states that in an $n$-dimensional affine space, select $n+1$ points $x_i$ so that an arbitrary point $x$ can be written as: $x=\mu_0x_0+\mu_1x_1+.......
Lelouch's user avatar
-1 votes
1 answer
179 views

Affine change of coordinates.

I am reading Fulton's book on algebraic curves.In the second chapter they have defined what they call affine coordinate change map between two affine spaces.It is defined in the following manner: Let ...
Kishalay Sarkar's user avatar
0 votes
1 answer
55 views

Let $\Theta \subset \mathbb{R}^k$, $\theta_0 \in \Theta$. Show that $\operatorname{aff}(\Theta) = \operatorname{aff}(\Theta - \theta_0) + \theta_0$

Here is my working. So I am trying to show $\operatorname{aff}(\Theta) = \theta_0 + \operatorname{aff}(\Theta - \theta_0)$ for any $\theta_0 \in \Theta \subset \mathbb{R}^k$. We know $\operatorname{...
user601297's user avatar
  • 1,086
0 votes
0 answers
25 views

What are the direction vectors of 14 equiangular lines in $R^7$ and the direction vectors of 84 equiangular lines in $R^{21}$?

What are the direction vectors of 14 equiangular lines in $R^7$ and the direction vectors of 84 equiangular lines in $R^{21}$? Reference: https://www.sciencedirect.com/science/article/pii/...
Elizabeth Amanda's user avatar

1
2 3 4 5
24