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.
Questions tagged [affine-geometry]
1202 questions
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Skew planes in $\mathbb{A}^4$
Can there be two skew planes in $\mathbb{A}^4$?
By this I mean two disjoint planes $\pi_1,\pi_2\subset\mathbb{A}^4$ such that their underlying direction vector spaces only intersect at zero.
Heitor Fontana
- 3,676
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Point Addition in Affine Space
The only operations defined on points in an affine space are
point-vector addition. this yields a new point.
point-point subtraction. this yields a vector.
This can be extended to an affine sum,
$\sum_i \alpha_i P_i$, where $\sum_i\alpha_i =…
wsaleem
- 141
2
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3 answers
The Sphere is not an Affine Space
In Eric Gourgoulhon's "Special Relativity in General Frames", it is claimed that the two dimensional sphere is not an affine space. Where an affine space of dimension n on $\mathbb R$ is defined to be a non-empty set E such that there exists a…
Araq
- 133
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Proof of ratio preserving on collineations
What I want to ask is possibly a rather classical result. I remember I read a book which says that if $f$ is a bijection (a.k.a. collineation of an affine space) of $\mathbb{R}^2$ sending collinear points to collinear points, then $f$ will preserve…
Easy
- 4,485
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Prove that $\mathbb{R}^n$ contains at most $n+1$ affinely independent points
I'm learning affine geometry and came across the following statement while reading the introductory paragraph on affine independence :
$\mathbb{R}^n$ contains at most $n+1$ affinely independent points.
Coming for linear algebra, the above…
user347616
2
votes
2 answers
Affinity which maps one plane to another
I'm learning affine geometry, specifically affine transformations, and need help with the following exercise :
Let $P_1 : -x + z = 3$ and $P_2 : x - 4y + 3z = 9$ be two planes of $\mathbb{R^3}$.
$(1)$ Give affine frames of the form $\mathcal{R} =…
user347616
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1 answer
Affine geometry and simply transitive action
I'm learning affine geometry, specifically the notion of action, and need help to understand the following example :
Let us take $\mathbb{A} = \{(x, y) \in \mathbb{R^2} : y > 0\}$ and, as the $\mathbb{R}$-vector space, $E = \mathbb{R^2}$. As the…
user347616
2
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1 answer
What does it mean to fix a point in an affine space?
In their book Metric Affine Geometry, Snapper and Troyer state on page 59:
It cannot be stressed enough that the affine space $X$ is not a vector space. Its points cannot be added and there is no way to multiply by scalars. No point in $X$ is…
Stan Shunpike
- 4,921
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Prove that this affinity is the identity mapping
Let $\phi : K^n \to K^n$ be an affinity, such that all lines are parallel to their image under $\phi$. Prove that if $\phi$ has two fixed points, then $\phi$ is the identity mapping.
My attempt:
I denote the two fixed points of $\phi$ by $p$ and…
rehband
- 1,921
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A confusion regarding Affine spaces
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 point in the Affine space! Hence, it seems to me…
freebird
- 11
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Get affine transformation matrix from two positions of the object
I have an object in 3d space which is represented by the set of vertices. Then I scale this object, rotate it and translate. After these operations I get the second set of vertices with new coordinates. I know the correspondence between the old set…
Mr.Jimm
- 13
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affine translation in direction of a vector
Suppose I have a line segment in 3D-space, having end-points $(a,b)$.
I want to translate this segment by $w$ units in the direction specified by 3 angles $\alpha,\beta,\gamma$ with respect to $x,y,z$ axis.
So, I compute vector $\vec{u}=b-a$ and…
miso
- 11
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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 map $p∩q$?
Attempt: I know that the intersection of…
GENERAL123
- 13
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On the solvability of the Affine Group
Let $n\in\mathbb{N}$. For each $A\in \mathcal{M}_{n\times n}(\mathbb{R})$ and $b\in\mathbb{R}^{n}$, define $T_{A,b}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ the affine transformation
$$T_{A,b}(x)=Ax+b$$
Set $G=\{T_{A,b}: det(A)\neq 0\}$. It is easy…
José Luis Camarillo Nava
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Affine manifolds References
Does anyone know of any online pdf lectures on affine manifolds? I'm interested in compact affine manifolds. Specifically, I'm looking for specific examples of compact affine manifolds where the affine charts are specified.
Gilbert Easton
- 575