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.

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2
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1answer
58 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.
2
votes
2answers
89 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 ...
1
vote
4answers
94 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 $(1,1,3)$
1
vote
0answers
64 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 $\bigcup\mathcal{H}$, ...
1
vote
2answers
162 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 ...
0
votes
1answer
69 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 ...
5
votes
3answers
513 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 $\mathbb{R}$, ...
3
votes
4answers
165 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
votes
1answer
551 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
vote
1answer
309 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 ...
5
votes
3answers
403 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) ...
0
votes
3answers
184 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 ...
0
votes
2answers
67 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 ...
1
vote
0answers
355 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 ...
1
vote
1answer
133 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 ...
3
votes
1answer
256 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
votes
1answer
126 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 ...
0
votes
1answer
118 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 ...
1
vote
1answer
152 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
votes
1answer
38 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 ...
8
votes
2answers
11k 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
votes
1answer
105 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 ...
1
vote
1answer
102 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 ...
0
votes
2answers
190 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
votes
1answer
180 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 ...
4
votes
1answer
2k 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 ...
0
votes
1answer
170 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 ...
1
vote
1answer
67 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 ...
4
votes
1answer
155 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
votes
1answer
151 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 ...
1
vote
1answer
35 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 ...
1
vote
1answer
322 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. ...
1
vote
1answer
119 views

How to get around non-commutativity of matrix multiplication?

I have a problem with a matrix equation/transformation problem which I need solving. I have two transformations $A_1$ and $A_2$, both of which can be expressed as $A_i = R_i \times B_i$, $R_i$ ...
0
votes
1answer
179 views

The Mean Value Property and Affine Functions

I need some hints to solve the following: A function $f(t)$ on an interval $I = (a,b)$ has the mean value property if $f(\frac{s+t}{2}) = \frac{f(s)+f(t)}{2}$ where $s,t\in{I}$. Show that any ...
1
vote
1answer
112 views

Detecting the type of singularity with the Jacobian

Say we have a plane curve $\mathcal{C} = V(f(x,y)) \subset \mathbb{A}^2_{\mathbb{C}}$. The partial derivatives tell us about the singularities: if they all vanish at a point $p \in\mathcal{C}$ then ...
3
votes
1answer
65 views

Set of all affine maps between two polytopes

Let $V$ be a finite-dimensional real vector space, let $P, Q \subseteq V$ be polytopes with $P \subseteq Q$. (Let a polytope be defined as the convex hull of finitely many points.) I'm interested in ...
3
votes
1answer
209 views

Fitting Shape in Circle for Shape Classification

I need to classify arbitrary 2D shapes. The classification should be invariant to at least affine transform. To achieve this invariance, I decided to "normalize" each shape by fitting it to a unit ...
2
votes
1answer
430 views

Applications of the fundamental theorems of affine and projective geometry.

The fundamental theorem of affine/projective geometry says that a bijection between two finite dimensional spaces that preserves the relation of collinearity is a (semi-) affine/projective ...
2
votes
1answer
90 views

non-affine functions

it is obvious that if $f$ is an affine function, then $f$ has this property: there exist two function $g$ and $h$ such that $f(t+s)=g(t)+h(s)$ for all $t,s \in\mathbb{R}$. My question is: is there ...
2
votes
1answer
107 views

Finding a coset

I'm given $V$ a vector space over a field $\mathbb{F}$. Letting $v_1$ and $v_2$ be distinct elements of $V$, define the set $L\subseteq V$: $L=\{rv_1+sv_2 | r,s\in \mathbb{F}, r+s=1\}$. (This is the ...
1
vote
1answer
112 views

Affine Subspace Confusion

I'm having some trouble deciphering the wording of a problem. I'm given $V$ a vector space over a field $\mathbb{F}$. Letting $v_1$ and $v_2$ be distinct elements of $V$, define the set $L\subseteq ...
3
votes
1answer
319 views

Affine Spaces and Affine transformations

Can anyone please recommend a book that describes Affine Spaces and Affine Transformations? Many books i saw described it very briefly. Can anyone please suggest a book that deals with it in detail? ...
7
votes
4answers
789 views

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 + ...
3
votes
0answers
123 views

Measuring Similarity of Affine Transformations

I am currently working on a problem where a calibration Algorithm provides me with an Affine Transformation that transforms a 2D Image to it's assumed Position in a 3D Volume. To evaluate the accuracy ...
1
vote
5answers
181 views

How to precisely distinguish vectors and points? [duplicate]

Possible Duplicate: Distinction between vectors and points I have a doubt about the distinction between points and vectors. I know there's already a topic about that here in the web site, ...
10
votes
4answers
2k views

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.
1
vote
1answer
213 views

use homography to rotate around x/y axes

I need to construct a homography out of a 3x3 rotation matrix. I am fundamentally misunderstanding some part of how homographies are constructed. I have been assuming that a homography is ...
2
votes
1answer
335 views

Turning affine planes into projective planes

How can we show that an affine plane of order $n$ can always be turned into a projective plane of order $n$? Say I start with an affine plane, and split it into $n+1$ parallel classes, add a point ...
0
votes
1answer
514 views

Affine plane of order 4?

I cannot seem to construct an affine plane of order 4. I have the construction for order 3- but cannot seem to come up with or find the construction for 4 anywhere. Could someone show me a picture of ...
1
vote
2answers
322 views

n-dimensional affine space that is not isomorphic with the n-dimensional real space

I'm looking for an example of $n$-dimensional affine space that is isomorphic with $\mathbb{R}^n$ as affine space but not with respect to other properties (for example it has different ordering etc.) ...