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]
1273 questions
267
votes
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
- 4,412
131
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
- 1,415
55
votes
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
- 4,047
54
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
- 876
36
votes
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
- 3,649
27
votes
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.
25
votes
4 answers
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
- 2,202
23
votes
2 answers
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
- 437
23
votes
6 answers
Why is the affine hull of the unit circle $\mathbb R^2$?
In Boyd & Vandenberghe's Convex Optimization, the affine hull of a subset $C \subset \mathbb R^n$ is defined 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,…
Palace Chan
- 1,277
22
votes
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
- 263
21
votes
3 answers
Since the Curvature tensor depends on a connection (not metric), is it the relevant quantity to characterize the curvature of Riemannian manifolds?
The definition of the Riemann curvature tensor does not include a metric. So, if we have a smooth manifold(not a Riemannian manifold), we can define the Riemannian curvature tensor for it by just giving it a connection (not the Levi-Civita…
TheQuantumMan
- 2,678
19
votes
3 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
- 293
- 1
- 2
- 6
16
votes
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
- 1,351
16
votes
1 answer
"An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps."
I was reading the Wikipedia article for complex affine spaces, which says the following:
Affine geometry, broadly speaking, is the study of the geometrical properties of lines, planes, and their higher dimensional analogs, in which a notion of…
The Pointer
- 4,686
16
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
- 1,865