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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69 views

who found that translation in N space is the same as shearing in N+1 space?

According to the wikipedia, Although a translation is a non-linear transformation in a 2-D or 3-D Euclidean space described by Cartesian coordinates, it becomes, in a 3-D or 4-D projective ...
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2answers
336 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.) ...
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183 views

Affine group, semi-direct product and linear transformations

According to wikipedia the Affine group is the semi-direct product of a vector space $V$ and the general linear group $GL(V)$. Here is the definition of the semi-direct product in terms of matrices ...
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4answers
95 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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4answers
59 views

A function is convex and concave, show that it has the form $f(x)=ax+b$

A function is convex and concave, it is called affine function. That is the function: $$f(tx+(1-t)y)=tf(x)+(1-t)f(y),\, \, t\in (0,1) $$ Force $y=0$(suppose $0$ is in the domain of $f(x)$), we ...
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46 views

The affine line with two points removed

To which affine variety $V$ is $\mathbb{A}^1 \setminus \{0, 1\}$ isomorphic to? What would be the isomorphism in this case? Any help would be appreciated.
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2answers
79 views

Is there any difference between a flat manifold and an affine space?

What is the difference, if any, between a flat manifold (in which the Riemann tensor vanished identically) and an affine space?
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894 views

Is perspective transform affine? If it is, why it's impossible to perspective a square by an affine transform, given by matrix and shift vector?

I'm a bit confused. I want to program a perspective transformation and thought that it is an affine one, but seemingly it is not. As an example, I want to perspective a square into a quadrilateral (as ...
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2answers
31 views

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 ...
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1answer
38 views

If $n\geq 2$, why is $k[\mathbb{A}^n\setminus\{p\}]=k[\mathbb{A}^n]$?

Assuming $n\geq 2$, why is the coordinate ring of affine $n$-space over an algebraically closed field $k$ unchanged if we delete a point? That is, if $p\in \mathbb{A}^n$, why does ...
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21 views

Prove that this affinity is the identity mapping

Let $\phi : K^n \to K^n$ be an affinity, such that all lines are parallel to their image under $\phi$. Prove that if $\phi$ has two fixed points, then $\phi$ is the identity mapping. My attempt: I ...
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81 views

Find intersection multiplicities

Let curves $A$ and $B$ be defined by $x^2-3x+y^2=0$ and $x^2-6x+10y^2=0$. Find the intersection multiplicities of all points of intersection of $A$ and $B$. If we let $f=x^2-3x+y^2$ and ...
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3answers
187 views

Origin in vector space?

In the wikipedia article about vector space I do not understand this sentence Roughly, affine spaces are vector spaces whose origin is not specified. A vector space does not need an origin. When ...
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1answer
67 views

Geometric Deformations

There are geometric transformations such as translation, rotation and uniform scaling (Affine transformations). I am interested in knowing whether there is a separate class of transformations that ...
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2answers
164 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
134 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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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 ...
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1answer
119 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 ...
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5answers
182 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, ...
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1answer
408 views

What is the relation between complex numbers and transformation matrices?

I read addition and multiplication with complex numbers can be represented as translation and rotation in a 2D plane. I am using this to move around objects on the screen. I have an offset number, ...
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1answer
218 views

This is the most difficult question I could get without using mass point geometry

In triangle ABC, points D and E are on sides BC and CA respectively, and points F and G are on side AB with G between F and B. BE intersects CF at point O_1 and BE intersects DG at point O_2. If FG ...
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22 views

A set $S\subseteq\mathbb{A}^n$ is quasi-affine iff $S=Z\setminus V$ for closed $Z$ and $U$?

I'm confused by a remark in note I'm reading. It essentially says, Let $S\subseteq\mathbb{A}^n$ be a subset of affine $n$-space over an algebraically closed field. It's clear that $S$ is ...
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51 views

Distinction between point and vector outside of US ( particularly Germany and Eastern Europe )

There was a long discussion in a forum I visit in where a calculus teacher was being critical of Stewarts Calculous for making a distinction between points and vectors. He argued that no such ...
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1answer
32 views

Application of Desargues' theorem for constructions

I found this interesting document (german) on the internet. On page 8 it says: "Draw a line segment between two given points only using compass and ruler, while the distance between the two points is ...
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20 views

Alternative characterization of a finite dimensional affine set

As the definition in the S. Boyd's textbook: My question is the following representation: What is the relationship between this representation and the definition above it? EX: sum of elements ...
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1answer
680 views

Affine Transformation matrix for mapping a triangle to another

I'm trying to write a function in Matlab that will give me a matrix T that can be used to multiply points in homogeneous coordinates. This is achieved my mapping a ...
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1answer
25 views

How do we define affine combinations in affine spaces?

Suppose we're given a field $A$ and a vector space $X$ over it. Then an affine space over that is a set $P$ (of "points") equipped with an action $$+ : X \times P \rightarrow P$$ such that $0_X+p ...
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289 views

Rigorous definition and relations between point/vector/affine space/vector space/basis/frame/coordinate system

I am trying to understand the exact relation between all these things: point vector affine space vector space basis frame coordinate system Can you explain me rigorously (in the mathematical ...
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1answer
40 views

Do all the solutions have to be in an affine variety?

An affine variety $V(X)$ is the zero-locus of a set of polynomials. So if the variety is generated by the polynomial $y-x=0$ in $\mathbb{R}^2$, then do all the solutions (i.e., every point satisfying ...
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1answer
76 views

Desargues Theorem and help with its significance

I am still trying to get a hang of drawing the picture. The only idea I get from the theorem is that if two triangles are in perspective from a point, then we the theorem , we also get that the ...
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1answer
515 views

Finding an affine combination of a point on a triangle

I have a problem involving affine combinations that I can't figure out how to solve. Given the above picture, write q as an affine combination of u and w. Now, I understand how to write the ...
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1answer
202 views

Line-preserving transformations

Is there a name for the class of transformations on the Euclidean plane (or projective plane) that preserves lines? They are not all affine transformations; consider a perspective projection $p$ in ...
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32 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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1answer
338 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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1answer
172 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 ...
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1answer
105 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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1answer
39 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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121 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 $A_i = R_i \times B_i$, $R_i$ ...
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1answer
116 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 ...
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1answer
226 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 ...
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1answer
51 views

Confusion regarding convex and affine set

I am a bit confused regarding convex and affine set. When they mention set, does it mean the set consisting of all the points belonging to the line or shape respectively?
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1answer
300 views

Minimising a matrix equation to find 'best fit' affine matrix

Here is my problem: I have an image divided into segments. Each segment consists of pixels with coordinates (x,y) called vector $v$, each pixel has a length 3 vector RGB called $I(v)$. I want to ...
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23 views

A confusion regarding Affine spaces

Take an Affine space $\Bbb{A}$ over the field $K$. How would you determine the points satisfying any polynomial $f(x)$? If there is no fixed origin, points can be given names with reference to ANY ...
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1answer
34 views

linearization of $\log(|x|)$

I am trying to convexify $\log(|x|)$. I think its concave. So I am trying to get an affine upper bound through linearization. But the problem is there are two concave functions because of the absolute ...
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1answer
31 views

Angle of two planes in $\mathbb E_4$

I have two planes (given in parametric form) in $\mathbb E_4$: $\alpha$: (7,3,5,1) + t(0,0,1,0) + s(3,3,0,1) and $\beta$: (1,5,4,1) + r(0,0,0,-1) + p(2,0,0,1), and I have to find angle between them. I ...
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145 views

Tensor product of affine space and algebra

When reading about Quantum Mechanics, I always feel a bit disappointed when physicists consider that (for example) the 3-dimensional position of a particle must be decomposed into 3 coordinates $x, y, ...
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52 views

Linear algebra, affine space, and floor function

My question is mostly: is there a name for this kind of things. I am mostly interested by finding book or articles about what follows, but without even a word or a name, it is quite hard to search for ...
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0answers
42 views

affine translation in direction of a vector

Suppose I have a line segment in 3D-space, having end-points $(a,b)$. I want to translate this segment by $w$ units in the direction specified by 3 angles $\alpha,\beta,\gamma$ with respect to ...
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1answer
74 views

The graph of a regular function is an algebraic set, and intersection of hypersurfaces is finite?

i have some problems with these exercises, can you give me a hint? Let $f:\mathbb A^n_k\rightarrow\mathbb A^m_k$ be a regular function. If $X\subset\mathbb A^n_k$ is an algebraic set, show that the ...
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121 views

Number of vector and affine subspaces of dimension $ k$ of $E$ over $\mathbb{F_q}$

Problem (comments after): Let $\mathbb{F_q}$ be a finite field of cardinal $q$ and $\mathcal{E}$ an affine espace of dimension $n$ directed by the vector space $E$. Show that: ...