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.

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What is the precise definition of the Darboux tangent to a surface? [closed]

What is the definition of Darboux tangents of a surfaces? The book "Affine Differential Geometry" mentions it, but it does not give a precise definition.
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The intuition behind this explicit form of an affine hull?

I have come across an explicit formula of an affine hull from here. I'm trying to prove that formula in 2. below. However, my approach is quite muscular and not natural, i.e. I have to know the ...
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Set of complementary subspaces as an affine space

Let $V$ be a vector space and $W$ a fixed vector subspace of $V$. Denote by $\mathcal{E}$ the set of vector subspaces $U$ of $V$ such that $V=U\oplus W$. It is stated in the Examples section here that ...
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What does it mean for a set of points to be affinely independent in the context of range spaces?

For some background, there are many range spaces with finite VC-dimension that arise naturally in discrete and computational geometry. One example is the set of all points in d-dimensional Euclidean ...
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(Non)Linear transformation [closed]

I was thinking of how prints on clothes get stretched while wearing them and if it is a linear transformation, so one can easily design a piece of cloth (e.g. a sock)such that the image will be ...
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Order of Affine non-soluble groups

2-Transitive groups has been classified. The complete table has been mentioned in this Textbook ( see e.g., Table 7.3 and Table 7.4, page no. 194-197). Table 7.3 contains Affine 2-transitive groups ...
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Order of group $A\Gamma L_d(F)$ of Affine semi-linear transformation.

Question What is the order of a group $A\Gamma L_d(q)$ of Affine semilinear transformation ? (here $q$ is a prime power of some prime $p$).
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Discussion of Affine space in William L. Burke (page-14)

The parameterization along the line passing through two points in an affine space is not unique. Transformations of the parameter $u$ $ u \to k u +b$ Change one uniform parameterization into another. ...
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Undestandig of definition of an affine subspace

Consider the affine space $\mathbb{R}^{3}$. Then for example the set $ \{(x,y,z):3x-3y+z=0\}$ is an affine subspace. But $ \{(x,y,z):3x-3y+z=2\} $ and $ \{(x,y,z):x^2\} $ are not affine subspaces. We ...
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Maps between Euclidian spaces that preserve lines are affine?

Let $f:\mathbb{R}^n\longrightarrow\mathbb{R}^n$ be a continuous function and $n\geq 2$ such that the image of every line of the form $$x+tv $$ where $x\in \mathbb{R}^n$, $t\in \mathbb{R}$, $v\in B(0,1)...
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$A \subset \mathbb{R}^n$ is an affine subspace $\Leftrightarrow \sum \mu _ia_i \in A$ whenever $a_i \in A$, $\sum \mu_i =1$

Lemma: $A \subset \mathbb{R}^n$ is an affine subspace $\Leftrightarrow$ If $\sum \mu_i =1$, and if $a_i \in A$ for any $i =1,...,k$, then we have $\sum \mu _ia_i \in A$. Definition: A subspace $A$ ...
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Showing that the set difference of two affine varieties need not be an affine variety [duplicate]

I was wondering what would be an example of two affine varieties $A, B$ over the field of complex numbers $\mathbb{C}$ such that $A\setminus B$ is not an affine variety? I was initially thinking about ...
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Need help finding G. Pick's 1917 article "Über affine Geometrie iv: Differentialinvarianten der Flächen gegenüber affinen Transformationen"

I'm studying affine geometry and I need help finding this article "Über affine Geometrie iv: Differentialinvarianten der Flächen gegenüber affinen Transformationen" written by G. Pick in ...
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Bound on dimension of tangent space of an affine variety [duplicate]

I've been reading through my notes and the following fact is stated without any proof or justification: For an affine variety $X\subset\mathbb{A}^n$ and a point $p\in X$ we have $$dim_k T_pX\geq dim_p ...
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Prove: affine transformation maps "line at infinity" to "line at infinity"

I'm studying Computer Vision and my lecturer stated that: The affine transformation maps "line at infinity" to "line at infinity". I'm trying to prove it as part of my ...
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Is any manifold that can be parametrized with only one chart an affine space?

From Wikipedia: An affine space is a set A together with a vector space $\overrightarrow{A}$, and a transitive and free action of the additive group of $\overrightarrow{A}$ on the set A. Now let's say ...
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Is an affine transformation of a multivariate Student-t distribution also a Student-t distribution?

I note from Wikipedia that an affine transformation of a Multivariate Normal Distribution is also a Multivariate Normal Distribution, as shown here https://en.wikipedia.org/wiki/...
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Writing an integer $m$ as the sum of $2m$ integers contained in $[-2,2] \cap \mathbb Z$

Suppose that $m$ is a positive integer. I want all the possible ways to write $$ m=k_1+\ldots+k_{2m} $$ where $k_i \in [-2,2] \cap \mathbb Z$. For example, if $m=1$, I can write $$ 1=1+0 \quad \text{...
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examples of linearlized polynomial and affine polynomials

I am reading the definition of linearlized polynomials and affine polynomials, and found it a bit difficult to understand. Could anyone give a very simple and intuitive examples of them? Thanks in ...
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How came the coefficients of vectors in the Solution?

Let (A,a1,a2) and (B,b1,b2) be two affine coordinate systems (illustrated below).Represent the point P in both systems. As Solution is in the Image . I want to Know that how are these values ...
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Baby rudin example 10.32

This is the definition which we need for this example : . There is the example: I don't understand why is $\partial\Sigma$ equal to $\Sigma(\partial D)$ and then why is $\Sigma$($\partial D$) equal ...
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How to change coordinate system from one plane to another?

I have two planes plane1 and plane2 which equations are known. Given a point c with coordinates (x,y,z) of the plane plane1, is there any way to get the correspondent point c' on plane2? I've tried ...
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Calculate matrix in affine transformation

Trying to solve the following question: Consider an image of size $M\times N$ that undergoes a transfor-mation consisting of only rotation and translation (No scaling or shear were applied). We were ...
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affine transformation preserves the property of parallelism among lines

I'm a bit confused about when vectors are parallel. I was trying to prove the following theorem: Let $f(\vec{x})=A\vec{x}+\vec{b}$ be an affine transformation. Then $f$ preserves the property of ...
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Need some help with understanding symmetry group of some $X \subset \mathbb{R}^n$ and its normal subgroup's quotient group.

I'm a student who is currently studying Finite Groups of Isometries in Euclidean Geometry. Symmetry Group is the first topic of this section, but I am having trouble understanding the concept clearly. ...
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Equality of vectors in an affine space?

A real affine space is a triple $(\mathbb A, \varphi, \mathbb V)$ where: $\mathbb A$ is a set; $\mathbb V$ is a $\mathbb R$-vector space; $\varphi: \mathbb A\times \mathbb A\rightarrow \mathbb V$, $...
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Finding the formula of an affine function using the parallelogram identity

Here is the question It asks me to find u(x,y) and I solved it using u(x,y) = Ax + By + C but I doubt this is the way question wants me to find the solution. I want to find the solution using the ...
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Axioms for vector space and affine space under the same system and notation

Background: Here are common axiomatic definition of a vector space and an affine space: A vector space is a set together with two operations satisfying the following eight axioms. For addition: ...
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Minimal assumptions that must made in order to properly defining segment.

Let $v$,$w$ be two points. the segment $[v,w]$ is usually defined as a connected set when $v,w\in\mathbb R^n$: $[v,w]=\{x|| x=a v+(1-a)w\ \ \text{for some} \ a\in[0,1] \}$. The question is: if we ...
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Show that there is an isometry $f:\mathbb{R}^n\to \mathbb{R}^n$ such that $f|_X$ is $g$ where $g :X \to X$ is isometric on subset $X$.

Here's the problem. If $X \subset \mathbb{R}^n$ is any subset and $g:X\to X$ is an isometric map, show that there is a unique isometry $f:\mathbb{R}^n\to \mathbb{R}^n$ such that $f|_X$ is $g$. I ...
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If $a,b \in A$ and $A$ is an affine subspace, show that $A-a = A-b$

I would like to get some help with my answer to this problem. My answer: First, consider if $a=b$. This automatically gives us that the statement is true. Secondly, we consider when $a \neq b$. Pick ...
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Meaning of "homology" in the context of an affine geometry excercise.

Reading through my notes of Affine Geometry I stumbled upon an exercise which cites the concept of a homology and I wish to understand how -in the context of the exercise- a homology is defined. The ...
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Shear mapping defined through triangle ABC with A, B fixed points

I am stuck on the following problem: Let $\mathcal{A}^2$ be an affine plane with the associated vector space $V$ and the points $A, \: B=\tau_u A, \: C=\tau_v A \in \mathcal{A}^2$ be the corners ...
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Embedding of affine space in vector space

Let $E$ be an affine space over a field $k$ and let $V$ its vector space of translations. Denote by $X=\operatorname{Aff}(E,k)$ the vector space of all affine-linear transformations $f:E\to k$, that ...
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Is a finitely generate $k$-algebra that has no nilpotent element ($k$ is a field) an integral domain?

In Hartshone section 1 exercise 1.5, he wants me to prove that If $B$ if a finitely generated $k$-algebra with no nilpotent elements, then $B$ is isomoprhic to the affine coordinate ring of some ...
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Are the following sets of coordinates affinely equivalent?

Given the coordinates $\{(0,t^{k}),(t^{k},0)\}$ for $k\in\{1,...,n\}$ and for $t>1$, and $\{(\alpha s^{k-1},0),(0,\alpha s^{k-1})\}$ for $k\in \{1,...,n\}$ and for $s>1$ and $\alpha>0$, are ...
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Rewriting affine quadric equation by a coordinate change

I have the projective quadric given by the equation $$Q(X,Y,Z)=YZ-Z^2$$ in $\mathbb{P}(V)$, where $V$ is a $3$-dimensional vector space, and would like to find the equation of the affine conic ...
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The plane described by the equation $ax+by+cz+d=0$ is invariant under the affine map $f$ if and only if $(a,b,c,d)$ is an eigenvector of $M$.

My lecture notes include the following result with no proof: Theorem: Let $f:\mathbb{R}^3\rightarrow\mathbb{R}^3$ be an affine map with corresponding linear function $\phi:\mathbb{R}^3\rightarrow\...
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Affine map vs. affine transformation

I am reading the Wikipedia page about affine space here. To fix notation, we let $A$ be underlying set and $\vec{A}$ be the vector space that acts on the set (free and transitively). The page mentions ...
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Affine map without fixed points is injective

My question is: given an affine endomorph map $f$ in such way that there no exists fixed points (that is, $f(x)\neq x$ for all $x$). Then, can we assert that $f$ is injective? Thanks in advance!
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Fixed points of affine and linear transformations

Let $\mathbb {K}$ be a field. Let $f: \mathbb {K}^2 \rightarrow \mathbb {K}^2; x \mapsto Ax+b$ be an affine transformation. Suppose $f$ has a fixed point line (i.e. a line such that every point on ...
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Reflect a point across a line using affine transform

I'm trying to reflect a point (6,12) across the line y=7 using an affine transformation. Logically, the line ...
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In affine geometry, will "promoting" a linearly independent set of n vectors and a linearly independent set of n linear forms determine a metric?

This is something which always baffled me about discussions of affine geometry. I was told "you have no metric available". But it seems obvious that there is always a metric available. In ...
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2 answers
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Show that there exists a regular function that is not a quotient of polynomials

Let $k$ be an algebraically closed field. Consider the Zariski-closed subset $Y=Z(xw-yz)\subset k^n$. On $U=Y\setminus Z(y,w)$ we consider the function $$ f\colon U\to k, P=(a,b,c,d)\mapsto \begin{...
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Proving a generalization of Desargues’ theorem in dimension $\geq 2$

I need help with this generalization (I think) of Desargues’ theorem in higher dimensions: Let $E$ be a vector space of dimension $n \geq 2$ and let $D$, $D’$ and $D’’$ be three different projective ...
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How to find the number of intersection points between affine curves with multiplicity 2?

How many intersection points with multiplicity 2 are there in the intersection of the affine curves $y^{3}-2 y x+1=0$ and $3 x+2 y+3=0$ (defined over $\mathbb{C}) ?$ I cant even find an intersection ...
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Why vectors can't be translated in the context of affine space

While I'm studying the Affine Space lecture materials for the Computer Graphic Course, it stated that point can be translated to different places since it represents a position, however, the vector ...
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Loomis and Sternberg Problem 1.15

Any hints are appreciated. Thanks in advance. My Attempt: Either $\alpha = \beta$ or $\alpha \neq \beta$. The fist case is logically equivalent to $A=B$. If I can prove that the second case is ...
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Affine Scaling unboundedness interpretation

I'm learning Affine Scaling algorithm and it is written for checking unboundedness the algorithm does this: If $-X^{2}_{k}r^{k} > 0$ then it's unbounded, where $k$ is the k-th iteration. I want ...
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What is view confusion in perspective projection?

I read the definition of view confusion that if any object exist behind the COP (center of projection), then it can be projected onto the view-plane seems like upside down and backward. My question ...
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