# 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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1 vote
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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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24 views

### 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 ...
• 1,865
11 views

### 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 ...
1 vote
32 views

### (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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• 8,904
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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 ...
• 1
1 vote
43 views

### 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: ...
• 3,078
1 vote
60 views

### 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 ...
• 3,078
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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 ...
• 1,248
1 vote
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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 ...
• 1,248
1 vote
42 views

### 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 ...
• 3,725
24 views

### 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 ...
• 465
1 vote
58 views

### 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 ...
• 1,865
1 vote
67 views

### 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 ...
• 77
1 vote