The affine-geometry tag has no wiki summary.
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Why in the affine space can not we use Grassmann formula?
For example, in space three-dimensional affine space generated by two skew lines is all the space three-dimensional, since they are not coplanar.
For this reason it is not worth the Grassmann ...
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0answers
18 views
Distance between two affine lines using determinant of Gramian matrix.
I've a task to find the distance in $E^4$ between:
$L = [1,2,-1,4] + \text{lin}((1,2,-1,0))$
and
$M = [2,3,1,5] + \text{lin}((2,1,0,2))$
My efforts to find the correct solution:
Let
...
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3answers
67 views
Closed subset of an affine variety… is it affine?
Preliminaries
So, first of all let me give you the definitions I'm dealing with. Let $k$ be an algebraically closed field, and $\mathbb{A}^n = k^n$.
An affine variety is a closed and irreducible ...
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0answers
41 views
$f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$
Suppose, that $f: E^3 \rightarrow E^3$ is an isometry, and $\det f = 1$ and $f'\neq id$
Please help me prove, that $f$ is a composition of rotation about an axis and moving along this axis.
I don't ...
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3answers
120 views
Product of two algebraic varieties is affine… are the two varieties affine?
Let $X_1$ and $X_2$ two algebraic varieties such that their product $X_1\times X_2$ is affine. Are $X_1$ and $X_2$ affine then?
If this is not true, could you give a counterexample?
-1
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0answers
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A question about hyperplanes in affine geometries [closed]
List all hyperplanes in
$\operatorname{AG}_3(2)$
$\operatorname{AG}_4(2)$
What is the main idea while listing?
Can you explain please?
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0answers
6 views
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.
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1answer
22 views
On affine spaces, distances, angles, and coordinates
Let's define an affine space as a pair $(A, V)$, where $A$ is a set and $V$ is a vector space, together with a map $V\times A \rightarrow A, \;\; (v, a) \mapsto v + a,$ such that
$\forall \, a \in ...
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1answer
25 views
affine variety of infinitely many polynomials can be represented as an affine variety of its finite subset
Let $f_1,f_2,\cdots$ be an infinite sequence of polynomials in $k[x_1,\cdots,x_n]$ and let $V(f_1,f_2,\cdots)=\{(a_1,\cdots,a_n)\in k^n:f_i(a_1,\cdots,a_n)=0$ for $i=0,1,\cdots\}$. Show that there is ...
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0answers
15 views
Affine subspaces in characteristic 2
Let $K$ be a field and $E$ an affine space over $K$. A subset $V$ of $E$ is an affine subspace (i.e. affine combinations of points of $V$ are in $V$) iff for any pair of distinct points of $V$ the ...
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1answer
24 views
Invariants of point sets in an affine space
A distance between a pair of points in an affine space is invariant under translation, rotation and reflection.
An angle in a triangle whose corners are tree points is also invariant under scaling.
...
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1answer
22 views
How to find equation system describing affine space, having base of linear space and a vector
How to find equation system describing affine space, having base of linear 'overspace' and a vector?
Suppose that I've vectors $\alpha$ and $\beta$, so that $W=\text{lin}(\alpha, \beta)$, and a ...
0
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1answer
32 views
Intersection of affine subspaces of finite codimension in Hilbert space
I'm wondering whether the following assertion is true:
Any two affine subspaces of the same finite codimension in a ($\infty$-dimensional) Hilbert space either are parallel or have nonempty ...
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0answers
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Existence of a special set of q+2 points in the finite affine plane over $\mathbb F_q$
I am working in the finite affine plane over $\mathbb F_q$ with $q=2^n$.
Such a plane has $q^2$ points, $q^2+q$ lines, each line has $q$ points, and by a point is passing $q+1$ lines.
There are $q+1$ ...
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1answer
119 views
Two affine varieties are not isomorphic
Given the affine variety $A:=Z(y^{2}-P(x)) \subset \mathbb{C} ^{2} $, where $P(x)$ is a polynomial with $\deg P \geq 2$, I need to show that $A$ is not isomorphic to $ \mathbb{C}$.
I know it ...
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0answers
32 views
Prove something is affine?
For any subspace $K$ and any point $u$, prove $K+u$ is affine. Or if you have an affine set $V$ and point $u$, then prove $V-u$ is a subspace.
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1answer
27 views
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.
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0answers
34 views
The explicit set-description for the intersection of a convex and affine hull
When given two finite sets of points in $\mathbb{R}^n$, how does one get an explicit description of their convex (resp. affine) hull?
E. g. Suppose
$
A = \{e_1, e_2, e_3, e_4\},\;
B = ...
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2answers
48 views
Find the line passing thought the point $p=(1,2,0)$, paralel to the plane…
Find the line passing thought the point $p=(1,2,0)$, paralel to the plane $P=\{x,y,z \mid x+2y-z=-4\}$ and crossing the line $L=\{(x,y,z):x+2y=2, y+z=4\}$
So I've tried to put the equation of plane ...
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4answers
43 views
Find the equation of plane containing line described by
Please help me in this really easy task
Find the equation of plane containing line described by
$x+3y-2z=1$, $2x-y+2z=3$, containing point $(1,1,3)$
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0answers
43 views
Separation of Euclidean Space
Consider a finite collection $\mathcal{H}$ of hyperplanes of $\mathbb{R}^n$ that have a common line. Given some $A \subseteq \mathbb{R}^n$ that is homeomorphic to a subset of $\bigcup\mathcal{H}$, ...
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2answers
59 views
Equation of plane — point/vector pedagogy
Suppose we have a point $\mathbf P$ and a vector $\mathbf n$ in plain ordinary 3D space. Here I am deliberately using upper-case letters for points, and lower-case points for vectors, since they are ...
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1answer
57 views
Variety Affine Space
I have the following question which i'm not sure how to work out...
$For\ f=6x^2y-xy^2-2y^3+1\ and\ \ h=3x-2y\ \in \mathbb{C}[x,y]$
Show that V(f,h) is empty.
What can you say about the ideal ...
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3answers
134 views
Showing that if $fg=gf$ and $fh=hf$, then $gh=hg$, where $f$, $g$, and $h$ are affine functions
Given real numbers $a$ and $b$ ($a \ne 0$), let $f_{a,b}$ be the function $\mathbb{R} \to \mathbb{R}$ defined by $x \mapsto ax+b$. The set of such functions is a permutation group on $\mathbb{R}$, ...
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4answers
85 views
Sufficient condition for a function to be affine
If for a function $f:\mathbb{R}\to\mathbb{R}$, I can prove for any real $x,y$, that $f(\frac{x+y}{2})=\frac{f(x)}{2}+\frac{f(y)}{2}$, can I say for sure that it is affine, as in of the form ...
3
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1answer
120 views
Affine Independence $\iff$ Linearly Independent
I guess I'm having some trouble getting my head around the notion of affine independence. As I've been taught, a set of vectors $\{\vec{x_1},\ldots,\vec{x_n}\}\subset \mathbb{R}^d$ is affinely ...
1
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1answer
63 views
Intersection of affine subspaces is affine
So if I have two affine subspaces, each is a translate ( or coset) of some linear subspace. I want to show that the intersection of such affine subspaces is again affine, particularly in ...
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3answers
157 views
The vanishing ideal $I_{K[x,y]}(A\!\times\!B)$ is generated by $I_{K[x]}(A) \cup I_{K[y]}(B)$?
Let $K$ be a field, $x=(x_1,\ldots,x_m)$, $y=(y_1,\ldots,y_n)$, $A\!\subseteq\!\mathbb{A}^m_K$, $B\!\subseteq\!\mathbb{A}^n_K$. Does there hold $$I_{K[x,y]}(A\!\times\!B)=\langle\langle I_{K[x]}(A) ...
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2answers
78 views
Equation of a line in homogenous coordinates given 2 points in affine coordinates
So if I have 2 points $A$ and $B$ such that $F(A) = (1; a, a^3)$, and $F(B) = (1; b, b^3)$. how do I find the equation of this line in homogeneous coordinates?
So I know how to get a line the ...
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2answers
44 views
Affine geometry about parallelogram
If ABCD is a parallelogram and M,N,P,Q are points on it sides then MNPQ is a paralellogram iff the diagonals intersect at a common point (i.e the diagonals of MNPQ and ABCD intersect at the same ...
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0answers
64 views
Convexity of affine function.
Can someone help me with a proof that affine function preserves convexity?
Given that $f$ is convex, $A$ is in $\mathbb{R}^{M\times N}$ and $b$ is in $\mathbb{R}^m$ then show that $g(x) = f(Ax+b)$ is ...
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1answer
73 views
affine set convex set
How to show the following:
Let $C$ be a convex subset of $\mathbb R^d$. Then $\operatorname{int} C \neq \emptyset$ if and only if
$\operatorname{aff} C = \mathbb R^d$ where $\operatorname{aff} C$ is ...
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1answer
89 views
fixed point projective geometry
I am thinking about the following:
Let $\sigma:\mathbb C P^n\rightarrow\mathbb C P^n$ be a projectivity with $\sigma\circ\sigma=id_{\mathbb C P^n}$. I define the set of all fix points by ...
3
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1answer
59 views
Cross-ratio relations
The way I define the cross-ratio in projectve geometry:
Let $P_0,P_1,P_2,P_3$ being four points on a projective line G, such that $P_0,P_1,P_2$ are pairwise distinct. Let $\pi:\mathbb KP^1\rightarrow ...
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1answer
73 views
Cross-ratio projective geometry
I have 4 points $P_0=[1:2], P_1=[3:4], P_2=[5:6], P_3=[7,8]$ in $\mathbb KP^1$ and would like to evaluate the cross-ratio.
It is given by the following:
$\pi:\mathbb KP^1\rightarrow G$ is the unique ...
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1answer
52 views
Lines projective space
I have a question concerning the answer of Georges Elencwajg in Lines in projective space
There he states that the line $\overline {AB}=\mathbb P(\Lambda)\subset \mathbb P^n$ has its points of the ...
2
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1answer
32 views
Projectivities $\pi:\mathbb KP^1\rightarrow\mathbb KP^1$
I am a little bit confused conerning the following example of projectivities $\pi:\mathbb KP^1\rightarrow\mathbb KP^1$.
On the affine part $\mathbb K\subseteq \mathbb KP^1$ they are exactly the ...
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1answer
337 views
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
2
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1answer
63 views
Projective geometry well defined bijection
I consider the sphere $\mathbb S^n:=\{x\in\mathbb R^{n+1}: \|x\|=1 \}$ and the equivalence relation $x\sim y:\Leftrightarrow x=\pm y$.
How can it be shown that the inclusion $\mathbb ...
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1answer
53 views
projective geometry hyperplane
For $j=0,\ldots,n$ consider the affine hyperplane $A_j:=e_j+\langle e_0,\ldots,e_{j-1},e_{j+1},\ldots,e_n\rangle$ in $\mathbb K^{n+1}$ and the associated embedding $\tau_j:\mathbb ...
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2answers
64 views
Showing that a rigid motion preserves distances
For a linear map $\Phi : \mathbb{R}^n \to \mathbb{R}^m$ which has the form $[ x \mapsto Ax]$ for an $n \times m$ matrix $A$.
A rigid motion is an affine map where $A$ is orthogonal. How would you ...
2
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1answer
102 views
What does affinity means? (In categorical terms)
I know that traditionally, an affine space is "what is left from a vector space, after removing the origin". Given any set $X$ and a ring $K$, we can consider the set of all formal affine linear ...
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1answer
239 views
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 ...
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1answer
57 views
Distance between affine space and point
Let $A,A'$ two affine subspaces of a finite Euclidean Vectorspace $V$. Let $p,p'$ two points, such that $d(A,p)=d(A',p')$. $\dim(A)=\dim(A')$
I would like to show that there exists a movement ...
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0answers
55 views
Examples of (integral) affine manifold
An $n$-dimensional affine manifold $M$ is a topological manifold which admits a system of charts such that the coordinate changes are affine transformations i.e. in $GL(n,\mathbb{R}^n)\rtimes ...
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1answer
54 views
Motion in affine geometry
$V$ is a finite euclidean vectorspace and $\sigma:V->V$ is a motion, this means that $d(\sigma(a_i),\sigma(a_j))=d(a_i,a_j)$ for an affine coordinate system $a_0,...,a_n$
I know the following two ...
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1answer
130 views
Centre of a quadric
I found the following sentence in my linear linear algebra book (affine and projective geometry): $Q:V \to \mathbb{K}$ is a quadric (quadratic function) and $\alpha\in Aff(V)$. $Aff(V)$ is the set of ...
2
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1answer
74 views
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 ...
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1answer
23 views
$\alpha$ affine iff graph is affine subspace
I am just checking different analogous of $\alpha:V \longrightarrow W$ being affine. I have problems with this one:
$\alpha:V \longrightarrow W$ affine $\iff$ $G_\alpha=\{(v,\alpha(v): v\in V)\}$ is ...
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1answer
84 views
Affine subspace iff statement
I cannot find a proof for the following statement, though the statement itself seems common enough:
$X$ is an affine subspace of $\mathbb{R}^n \iff \forall a\in X: X - \{a\}$ is a linear subspace.
...
