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.
1,214
questions
0
votes
0
answers
9
views
Prove continuity of the affine extension mapping between geometric simplicial complexes
Let $\Delta_1$ and $\Delta_2$ be geometric simplicial complexes. Let $K_1$ and $K_2$ be their associated abstract simplicial complexes. Let $f: V(K_1) \to V(K_2)$ be a simplicial mapping.
We define ...
- 1
3
votes
0
answers
89
views
Is there a function whose maximizers remain the same after any affine transformations?
Let $f: \mathbb{R_+}^n\to \mathbb{R_+}$ be a function that is strictly increasing in each of its arguments. Let $M_f$ be the set of its maximizers on some fixed compact subset $D\subseteq \mathbb{R_+}^...
- 10.8k
1
vote
0
answers
39
views
an isometry preserving an equilateral triangle will preserve its vertices
Prove that an isometry preserving an equilateral triangle will
preserve its vertices, i.e if $\Delta ABC$ is an equilateral triangle
and $f$ is an isometry s.t $f(\Delta ABC)=\Delta ABC$ then
$\{A,B,C\...
- 158
0
votes
0
answers
29
views
What affine transformation does the projective transformation correspond to?
In the projective space $P^3(\mathbb{K})$ with the frame
$(S_0,S_1,S_2,S_3;E)$, consider a projective transformation as
follows: $$\begin{cases} tx_0'=x_0\\ tx_1'=-x_1\\ tx_2'=-x_2\\
tx_3'=x_3.\end{...
- 158
-2
votes
1
answer
65
views
A diophantine equation with no solution in positive integers $x,y$ i.e $(y(y+1)+1)^2+1\neq 100x$
Hi I ask separately a question regarding the question where I sktech a special case of the Brocard-Ramanujan problem :
Problem :
Let $x,y$ be positive integers shows that :
$$(y(y+1)+1)^2+1=100x\...
2
votes
2
answers
51
views
Affine Map as a Morphism of Affine Vector Spaces
I've recently took interest in morphism and category theory and I'm amazed how it offers a very general notion. However, I'm struggling to apply this for the affine vector spaces.
I've seen that a ...
- 31
-1
votes
2
answers
78
views
Jensen's inequality with affine combination
From page 217 of 'Convex functions' by Arthur Wayne Roberts, Dale Varberg (exercise F) i want proof that: $f:\mathbb{R}\longrightarrow\mathbb{R}$ is affine iff
$$ f\biggl(\displaystyle\sum\limits_{i=1}...
- 387
0
votes
2
answers
61
views
Projective varieties contained in dense open subsets
Let $X$ be a smooth irreducible projective variety over the complex numbers. Let $U$ be a nontrivial dense open subset of $X$.
Does there exist a projective curve $C$ inside $U$?
My attempt:
Let's ...
- 103
0
votes
0
answers
46
views
One place at infinity
I am currently reading a book titled "Numerical Semigroups," which can be found here. I have a question regarding a definition provided by the author. To begin, let's define $F=y^n+a_1(x)y^{...
- 123
0
votes
0
answers
25
views
Weight of a monomial
I have a question for the mathematicians in affine algebraic geometry: Given an algebraically closed field $k$, we define the projective $n$-space as the quotient space $\mathbb{P}^n = (k^{n+1} - \{0_{...
- 123
1
vote
1
answer
49
views
Midpoint polygon for odd $n$
Given $n$ distinct points $M_1, M_2, M_3, \dots, M_n$ in an affine space $U$; in case $n$ is odd, it's possible to find $A_1,\dots,A_n \in U$ such that $M_1$ is the midpoint of $\overline{A_1A_2}$, $...
- 893
0
votes
1
answer
33
views
An ellipse $\mathcal{E}$ touches a fixed ellipse $\mathcal{C}$ at $A$, prove the length of semi-major axis of $\mathcal{E}$ is constant
$\mathcal{C}$ is an ellipse with center $O$ and semi-major axis length $=a$, semi-minor axis length $|OB|=b$.
$A$ is a point moving on $\mathcal{C}$.
$E$ is a point on $\mathcal{C}$ such that $OE$ is ...
- 3,047
0
votes
1
answer
59
views
How to recognize affinely dependent or independent?
I have been trying to understand example of affinely dependent (AD) or affinely independent(AI).
But from the above images how are leftmost two figures AI and is rightmost AD? I am not able to ...
user1290851
0
votes
3
answers
81
views
Why affinely independent points in $\mathbb{R^d}$ is $d+1$?
Below text quoted from Jiff Matousek book:
Affine dependence of $a_1 ,\dots, a_n$ is equivalent to linear dependence of the $n-1$ vectors $a_1 - a_n, a_2 - a_n,
\dots, a_{n-1}-a_n$ .
Therefore, ...
user1290851
0
votes
1
answer
47
views
Find all points M such that MA<MB.
In the Euclidean affine plane $\mathbb{E}$, let $A$ and $B$ be two distinct points. Prove that the set {$M \in \mathbb{E} \mid MA < MB$} is a half-plane determined by the perpendicular bisector of ...
- 11