# 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.

266 questions
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### Transformations that map points inside the sphere to points inside the sphere

I am trying to figure out what is the most general linear transformation that maps points inside the unit sphere to points inside the unit sphere. I am slightly abusing the word linear here by ...
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### Affine plane curves classification

Define an affine plane conic as $\operatorname{Spec}A$ where $A=k[x,y]/(f)$ and $f$ a quadratic polynomial with no multiple factors. Define an equivalence relation on the set of alline plane conics ...
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### Relationship between hyperalgebra (algebra of distributions) of an affine group scheme to its cohomology

Let $G$ be an affine group scheme, and $\mathrm{Dist}(G)$ its hyperalgebra. I am wondering what is the relationship between $\mathrm{Dist}$(G) and $G$ interms of Cohomology? Is there a cohomology ...
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### Is the complement of a complex affine algebraic set in an irreducible complex affine algebraic set (path) connected in the euclidean topology?

Let $n \ge 2$, and $V$ be an affine algebraic set in $\mathbb C^n$ and $W$ be an irreducible affine algebraic set in $\mathbb C^n$, with $V \subsetneq W$ ; then is it true that $W \setminus V$ is ...
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### The Universal Property of Projective Space

Since the theories of affine and projective geometry are specified by certain axioms, we can consider the category $\mathcal{A}$ and $\mathcal{P}$ of affine and projective geometries, whose morphisms ...
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### Most general space on which we can do calculus

I have two somewhat related questions: Question 1: What is the most general space (set of objects) on which we can do calculus? Is it a normed space, or can we relax the conditions a bit further? ...
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### Differences and similarities between Euclidean and Minkowski geometry

I am trying to get my head around the differences and similarities between Euclidean and Minkowski plane geometry. AS far as I understand it they are both affine geometries meaning the parallel ...
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### Calculate the singular points of affine curve

I want to calculate the singular points of the affine curve $$f(X,Y)=(1+X^2)^2-XY^2 \in \mathbb{C}[X,Y]$$ The point $P=(x,y)$ is singular $\Leftrightarrow$ If $x=0$ we find $y=0$ and then from the ...
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### 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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### Cardinality of quasiaffine variety

The excercise 1.4.8(a) of Hartshorne's Algebraic Geometry says Show that any variety of positive dimension over $k$ has the same cardinality as $k$. Using Hartshorne's notation, we define a quasi-...
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### Measuring Similarity of Affine Transformations

I am currently working on a problem where a calibration Algorithm provides me with an Affine Transformation that transforms a 2D Image to it's assumed Position in a 3D Volume. To evaluate the accuracy ...
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### Meaning of affine transformation

From Wikipedia, I learned that an affine transformation between two vector spaces is a linear mapping followed by a translation. But in a book Multiple view geometry in computer vision by Hartley ...
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