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
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
8
votes
1 answer
What is an affine space?
I am having trouble understanding what an affine space is.
I am reading Metric Affine Geometry by Snapper and Troyer. On page 5, they say:
"The upshot is that, even in the affine plane, one can compare lengths of parallel lines segments. If $A…
Stan Shunpike
- 4,921
5
votes
2 answers
Affine map $f : P_1 \to P_2$ between two planes
I'm learning affine geometry, specifically affine maps, and need help with the following problem :
We give the affine planes
$$P_1 = \{(x, y, z) \in \mathbb R^3 : 3x + 2y + z = 6\} \quad \text{and} \quad P_2 = \{(x, y, z) \in \mathbb R^3 : -x - 2y…
user347616
5
votes
1 answer
Affine space $-$ Understanding basic example
I'm learning affine geometry and I'm having a hard time understanding a basic example related to the definition of an affine space and the notion of an action.
Before giving the example which causes me problems I'm just going to restate the…
user347616
4
votes
1 answer
How many $k$-dimensional affine subspaces?
I'm trying to determine the number of $k$-dimensional affine subspaces of a $n$-dimensional affine space ($k \leq n$) $X$ over $\mathbb{F}_q$, where $q$ is the power of a prime number.
Let $V$ be the underlying $n$-dimensional vector space over…
user391447
4
votes
2 answers
Affine transformation with translation
Basically, the issue I am trying to solve is that of "zooming round a point in a graphic, so that this point stays put on the display".
First, here is how I display the curve within a rectangle on screen :
The curve has xmin, xmax, ymin and ymax…
Adeline
- 143
3
votes
1 answer
Reduction of an affine hyperplane
For a $0 \neq v \in \mathbb{R}^{n}$ and a number $b \in \mathbb{R}$, the set $W^{n} \left( v,b \right) = \left\{ x \in \mathbb{R}^{n} | \langle x| v\rangle =b\right\}$ is called an affine hyperplane in $\mathbb{R}^{n}$.
For any affine hyperplane $W…
Tom White
- 55
3
votes
3 answers
Definition of an affine space
Sorry if this is a beginner question but I have been trying to find a good definition of an affine space and can't seem to find one that makes intuitive sense. Hoping that one could help explain what an affine space is after defining it…
Lance
- 3,698
3
votes
1 answer
Affine Transformation ― correct direction of scale
due to the fact that I am not mathematician I hope the question wont be ejected cause of triviality. But here we go:
what is given:
In an svg graphic, I have an element on which several are transformations applied (the working example can be seen…
philipp
- 203
3
votes
0 answers
affine transformations, strategy for finding invariant straight lines
At first lets introduce some notation.
$\mathcal{A}^n$ is a $n-$dimensional affine space and $V$ is its associated vector space. For any affine subspace of $\mathcal{M}$, its associated vector space it would be denoted as $V_{\mathcal{M}}$.
For the…
karhas
- 675
3
votes
3 answers
Show that halfspace is not affine.
Let us define half-space as
$$ C = \{x\mid a^Tx\leq b\} $$
Intuitively (or geometrically), I understand why halfspace is not affine. But while I prove that half-space is convex, it seems to hold for affine case.
Let us choose any $x_1,x_2\in C$ and…
jakeoung
- 1,261
2
votes
1 answer
affine hull, how to understand the statements below?
I am new to affine space, I looked through the wikipedia page, and have problem understanding the statements below.
The affine hull of a set of three points not on one line is the plane
going through them.
The affine hull of a set of four points…
user2262504
- 954
2
votes
0 answers
Dual Translation plane
An affine plane $\mathcal{A}$ is called a translation plane if the translation group of $\mathcal{A}$ operates transitively on the point set of $\mathcal{A}$.
So how do we define the dual translation plane?
Thanks.
Ryan
- 179
2
votes
1 answer
A question for epigraph and affine function
I'm working with a problem the epigraph of a real-valued function $f$ is a halfspace $\iff$ $f$ is a real-valued affine fuction.
First, I quickly recall some definitions:
A (closed) halfspace is a set of the form $\{x:a^T\textbf{x} \le b \}$ for…
user
- 1,391
2
votes
0 answers
Affine set and linear equation
Prove or disprove the following statement.
For any affine set C in R^n, there exists a solution set of linear equation that express C.
Chris kim
- 850
- 7
- 19