# Tagged Questions

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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### A set $S\subseteq\mathbb{A}^n$ is quasi-affine iff $S=Z\setminus V$ for closed $Z$ and $U$?

I'm confused by a remark in note I'm reading. It essentially says, Let $S\subseteq\mathbb{A}^n$ be a subset of affine $n$-space over an algebraically closed field. It's clear that $S$ is quasi-...
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### Distinction between point and vector outside of US ( particularly Germany and Eastern Europe )

There was a long discussion in a forum I visit in where a calculus teacher was being critical of Stewarts Calculous for making a distinction between points and vectors. He argued that no such ...
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### What's the intuition behind the definition of the tangent space of $\Bbb R^2$?

I'm reading a book on differential forms and on page one it defines the tangent space to $\Bbb R^n$. In what follows I've translated the statements into two dimensions for simplicity. Let $p$ be a ...
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### Show that the singular locus $\Sigma$ of an affine variety $V$ contains no irreducible component of $V$.

This is a problem from Ideals, Varieties, and Algorithms by Cox et. al. Let $V=V_1\cup \cdots \cup V_r$ be a decomposition of variety into its irreducible components. Let $\Sigma$ be the ...
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### What is the affine space and what is it for?

These two topics already exist: (preface: got in contact with affine space through computer graphics subject in university) What are affine spaces for? What are differences between affine space and ...
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### Relationship between affine functions and affine sets?

A function $f: \mathbf{R}^n \to \mathbf{R}^m$ is affine if it is a sum of a linear function and a constant ($f(x) = Ax + b$). A set $S \subseteq \mathbf{R}^n$ is affine if for any $x_1,x_2 \in S$ and ...
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### Why affine variety not vector space variety?

I am new to algebraic geometry. A basic question baffles me: why is the setting the affine space not the vector space?
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### The hexagon in Pappus' theorem and its relation to the more usual form

I am reading about geometry. Pappus' affine great (compared to a weaker small) theorem is introduced as follows: If the angles of a hexagon lie alternatingly on two intersecting straight lines and ...
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### Explanation of $\ker (\bar{m}-\bar{id})^2 \cap \{x_0=1\}$

Let $m$ be an isometry on $\mathbb{R}^2$ which is a composition of a reflection and a translation. The way to find the axis of the isomtry is by solving: $$\ker (\bar{m}-\bar{id})^2 \cap \{x_0=1\}$$ ...
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### why should add one column using Moore-Penrose pseudoinverse

I have a code from someone that I dont understand: This code is written in matlab and the function is to estimate linear geometric transformation [1] of a matrix using pinv. The size of first matrix ...
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I'm just looking for a reference or the proof that every affine map $f:V\rightarrow W$ between two possible different linear spaces $V$ and $W$: $$f[\lambda x+ (1-\lambda) y]=\lambda f(x)+(1-\lambda)... 0answers 25 views ### Non-affinely parametrized geodesics Consider a non-affinely parameterised geodesic, i.e., a geodesic whose tangent vector field obeys \nabla_X X = fX for some function f. Prove that one may reparameterise the geodesic so the tangent ... 0answers 38 views ### Vanishing points from three collinear points I would like to find the 2D vanishing point from a three collinear points as is shown in "Multiple View Geometry in Computer Vision" Example 2.19 (see here). What I did so far: 1 - I've extracted ... 0answers 34 views ### Do affine spaces have coordinate transformations? I asked a question on Physics SE and there seemed to be some confusion as to whether affine spaces could have coordinate transformations. Specifically, the particular space I was working with was \... 0answers 26 views ### Determining if a set is projective or not In \mathbb{P}^3 define the following sets:$$X=\{w_0w_1^2=w_2^2w_3-w_3^3\}\\Y_1=\{y_3=0\}\\Y_2=\{\sum_{i=0}^3 w_i=0\}\\Y_3=\{w_0+w_1+w_2+2w_3=0\} Does the set $Z=X\cap Y_3\setminus((X\cap Y_2)\cup(... 1answer 114 views ### Affine transformation invariants and lie groups Is it possible to generate geometric properties which are invariant under affine transformations? I'm trying to learn about lie groups and lie algebras with the example of the lie group of affine ... 0answers 99 views ### Intersections of convex hulls Given a set of$n$points$\{A_1, \ldots , A_n\}$of the plane and every possible triangle formed with$3$points$A$, I would like to describe the intersections fo theses triangles. By intersection, ... 0answers 51 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 ... 0answers 912 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 ... 0answers 41 views ### Linearizable reductive group action. Let$k$be a 0 characteristic field, and$G$a reductive group in$GA_2(k)$(the group of automorphisms of k[x,y] as k-algebra). How is it possible to deduce that$G$is conjugated to a subgroup of$...
Take an Affine space $\Bbb{A}$ over the field $K$. How would you determine the points satisfying any polynomial $f(x)$? If there is no fixed origin, points can be given names with reference to ANY ...
### linearization of $\log(|x|)$
I am trying to convexify $\log(|x|)$. I think its concave. So I am trying to get an affine upper bound through linearization. But the problem is there are two concave functions because of the absolute ...