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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.

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How to show that the only affine automorphism of $\mathbb{R}$ is the identity, using fundamental thorem of affine geometry?

Let the set of real numbers $\mathbb{R}$ be endowed with the natural affine structure (that means the group action is the usual addition). The question is how to prove that the only affine automorphism in affine space $\mathbb{R}$ is the identity,…
user5280911
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Affine transformation from a parallelogram to a square

I need to transform an area of an image so that a parallelogram defined as $((x0,y0), (x1,y1), (x2,y2), (x0+x2-x1, y0+y2-y1))$ is mapped on a square $((0,0),(0,128),(128,128),(128,0))$. Apparently, Python OpenCV can read and save images as well as…
Stepan
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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.
Anna Smith
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Barycenter with a negative weight

My question is about the barycenter. Given an affine space $(A,V)$, being $A$ a set and $V$ a $\mathbb{R}$-vector space, for $\{p_1,\dots,p_k\}$ and weights $\alpha_1,\dots,\alpha_k \in \mathbb{R}$, you define the barycenter as the element $g \in A$…
user183002
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What does mean here $\text{the vector space of its translations}$?

The dimension of an affine space is defined as the dimension of the vector space of its translations. What does mean here $\text{the vector space of its translations}$? We know that an affine space do not have fixed origin. In other word, in an…
MAS
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Reference for computations with affine subspaces

I'm writing a paper in logic that involves some affine geometry. Specifically, I need the fact that given (some finite representation of) affine subspaces $A,B$ of $\mathbb{Q}^n$ and an affine transformation $f\colon \mathbb{ℚ}^n \rightarrow…
Sebastian Zivota
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Find the equation of a non-degenerate parabola that contains $(0,0,1)$ and for which the axis is parallel to the $y$-axis

My attempt: General equation $$ ax^2+a'y^2+a''z^2+2byz+2b'xz+2b''xy=0.$$ (conic section is the intersection of this conic with the plane $z=1$). Expressing that $(0,0,1$) lies on this conic: $a''=0$. So we have…
MyWorld
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Basis of an affine subspace

Consider an affine subspace $D$ of an affine space or affine plane $\mathcal{A}$. Every set of points that are not elements of a proper affine subspace of $D$ is called a generating set of $D$. If every point $x$ of a set (of points) $S \subseteq…
MyWorld
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Prove a set $V$ is not algebraic

I need help showing that the set $V = \{(a,b) \in \mathbb{A}^2(\mathbb{C})\ \vert \ \vert a\vert^2 + \vert b\vert^2 = 1\}$ is not an algebraic subset of complex affine 2-space. I believe that I can just choose two complex numbers $a$ and $b$ with…
mathgeen
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smallest affine plane not generated by a field

What is the smallest affine plane not generated by a field? By "smallest" I mean has the least number of points. By "generated by a field", I mean planes of this form: $X = \mathbb{F}^2$ (the set of points) $$ L = \{\{(x,ax+b) \mid x \in…
Mathew
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How do you explain this proof?

I have this theorem with proof that i need to understand, however the proof in the book i am into is not complete. Thm: If $(p_1, p_2)$, $(q_1, q_2)$ and $(r_1, r_2)$ are vertices of a triangle, then the area of that triangle is the absolute value…
Ryan Liares Dagasuhan
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set of 4 points undergo affine transformation then why only one invariant is possible

consider 4 points in 2D in clockwise order as A,B,C,D forming a quadrilateral. If the quadrilateral undergoes affine transformation then affine invariant will be preserved which is given by area of triangle ACD/ABC as per …
user1371666
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Affine subspaces and parallelism

I'm learning affine geometry, specifically affine subspaces, and need help with the following exercise : We are given an affine space $\mathbb{A}^3$ with basis $(a_0, a_1, a_2, a_3)$ and two affine subspaces $P_1, P_2$ defined by the affine spans…
user347616
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Are two arbitrary vectors affine independent by definition?

I was wondering if I only have two vectors, then are they affine independent by definition? The affine independence definition is the following: $M=\{v_1,v_2,...,v_m\}$ vectors are affine independent if $\{v_j-v_1\}_{j\neq 1}$ are linearly…
KevinKim
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Is the "Uniqueness Property" of Affine Spaces an Axiom?

According to the wikipedia on Affine Spaces, an Affine Space $A$ is defined as an underlying set $A$ alongside a vector space $\vec{A}$ with right group action (free, transitive) $+$ of $\vec{A}$ on $A$. After laying out this definition, wikipedia…
user1770201
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