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.
1,160
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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 $t+M$.
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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 ...
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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 ...
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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 ...
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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 ...
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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 $n+s$ ...
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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 ...
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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 ...
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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
...
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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 ...
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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 ...
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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 $U$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
...
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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 ...
- 3,537
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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. ...
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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, ...
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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 $\mathcal{D} / k$ ...
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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 ...
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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, ...
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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 $ \...
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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 $\...
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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 ...
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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 ...
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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$G=2$
$\...
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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 $$\{x_1,\cdots,x_n\},\{y_1,\cdots,y_n\},\{u_1,\cdots,u_n\},\{v_1,\cdots,...
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Finite subsets of vector space under affine group action
For a field $K$ and natural numbers $n, m$, define $A(n,m,K) \in \mathbb{N} \cup \{\infty\}$ as follows: For an $n$-dimensional vector space $V$ over $K$, the group of affine automorphisms of $V$ acts ...
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Geometric and linear independence in affine and linear spaces (Proof Validation)
I am looking to prove the following statement:
Let $\{a_i\}^n_{i=0} \subseteq L$ be a set of points, then we have
$\{a_i\}^n_{i=0}$ is geometrically independent $\Longleftrightarrow \{a_i - a_0\}^n_{...
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Point of defining affine spaces.
So I’m doing my first bachelor mathematics. And for our linear algebra and geometry course. We defined abstract affine spaces/surfaces axiomatically. To then prove that with papus’s axioma (I think) ...
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Isomorphisms of causal structure of space with Lorentz form
Let $E$ be a real vector space, and $q$ be a quadratic form on $E$ of signature $(-1,1,\cdots,1)$. Let us call time vectors the vectors $v$ such that $q(v) < 0$. The set of time vectors has two ...
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Factoring a linear transformation into translation, rotation and back again.
An affine transformation from a vector $(x,y)$ represented by $n_1=[x,y,1]^T$ to $n_2$ can be written with a matrix
$$n_2 = M n_1$$so that:
$$M = \begin{bmatrix}a&b&c\\d&e&f\\0&0&...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 ...
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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),\...
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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+.......
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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 ...
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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{...
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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/...
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