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]
1075 questions
218
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5 answers
What is the difference between linear and affine function
I am a bit confused. What is the difference between a linear and affine function? Any suggestions will be appreciated
user34790
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102
votes
8 answers
What are differences between affine space and vector space?
I know smilar questions have been asked and I have looked at them but none of them seems to have satisfactory answer. I am reading the book a course in mathematics for student of physics vol. 1 by Paul Bamberg and Shlomo Sternberg. In Chapter 1…
user41451
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47
votes
3 answers
Rotation Matrix of rotation around a point other than the origin
In homogeneous coordinates, a rotation matrix around the origin can be described as
$R = \begin{bmatrix}\cos(\theta) & -\sin(\theta) & 0\\\sin(\theta) & \cos(\theta) & 0 \\ 0&0&1\end{bmatrix}$
with the angle $\theta$ and the rotation being…
Dschoni
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43
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4 answers
$\mathbb{A}^{2}$ not isomorphic to affine space minus the origin
Why is the affine space $\mathbb{A}^{2}$ not isomorphic to $\mathbb{A}^{2}$ minus the origin?
user6495
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29
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2 answers
What *is* affine space?
In my recent reading of various books and notes on algebraic geometry and scheme theory, I have come across three definitions of affine $n$-space over a field $k$:
$\mathbb{A}_k^n$ is $k^n$ 'without an origin';
$\mathbb{A}_k^n$ is simply $k^n$ with…
Tim
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27
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5 answers
What are affine spaces for?
I'm studying affine spaces but I can't understand what they are for.
Could you explain them to me? Why are they important, and when are they used? Thanks a lot.
Surfer on the fall
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22
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When is a vector "glued" to the origin?
Let $V$ be a real finite-dimensional vector space (I guess this forces $V$ to be $\mathbb{R}^n$). My intuition is that a vector $v\in V$ must be "glued" to the origin, since the origin is the only canonical thing that $V$ has (not even the basis is…
étale-cohomology
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20
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3 answers
What is the difference between projective geometry and affine geometry?
I recently started reading the book Multiple View Geometry by Hartley and Zisserman. In the first chapter, I came across the following concepts.
Projective geometry is an extension of Euclidean geometry with two lines always meeting at a point.
In…
rotating_image
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20
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Why is the affine hull of the unit circle $\mathbb R^2$?
In Boyd's "Convex Optimization" it defines the affine hull of a subset $C$ of $\mathbb R^n$ as
$$\text{aff} C = \left\{\theta_1 x_1 + \ldots +\theta_k x_k \mid x_1, \ldots x_k \in C, \theta_1 + \ldots \theta_k = 1 \right\}.$$
Then, it claims…
Palace Chan
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17
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What does it mean to be "affinely independent", and why is it important to learn?
I was studying linear optimization and i saw the term Affine independence. I came across this http://www.cis.upenn.edu/~cis610/geombchap2.pdf while trying to get a better understanding of the topic.
What does it mean to be Affinely independent ? Why…
RuiQi
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15
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2 answers
What is the difference between affine and projective transformations?
I'm trying to grasp the difference between the affine and projective transformations.
I got the point of the "line at infinity", but their matrix representation is not yet clear enough.
Here's the affine transformation $A$
$$
A =…
Maystro
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15
votes
2 answers
Proof $\mathbb{A}^n$ is irreducible, without Nullstellensatz
As the title suggests, could anyone either provide me with or direct me to a proof that affine n-space $\mathbb{A}^n$ is irreducible, without using the Nullstellensatz?
This is an exercise in a second course on representation theory, so if there is…
Sandy
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14
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1 answer
Is every convex-linear map an affine map?
Let's say that a map $f: V \rightarrow W$ between finite-dimensional real vector spaces is convex-linear if $f(\lambda x + (1-\lambda)y) = \lambda f(x) + (1-\lambda)f(y)$ for all $\lambda \in [0,1]$.
Let's say that a map $f: V \rightarrow W$ between…
Tom Jonathan
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14
votes
3 answers
Hole in the axioms of Hartshorne's "Foundations of Projective Geometry"?
I'm currently working my way through Foundations of Projective Geometry by Hartshorne, and he states the axioms characterizing an affine plane as:
An affine plane is a set $\mathbb{X}$ together with a collection…
Alec Rhea
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14
votes
1 answer
fitting points into partitions of a square
A friend of mine came up with the following problem:
Let $\{X_1, X_2, ..., X_n\}$ be an arbitrarily finite partition of the unit square $[0, 1]^2$. Let $\{P_1, P_2, ..., P_m\}$ be a finite set of points in $[0, 1]^2$.
Can all the points $P_i$ be…
Carlos Esparza
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