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.
Questions tagged [affine-geometry]
1202 questions
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Affine plane, necessity of line uniqueness
Let $\mathcal{A} = (\mathcal{P},\mathcal{L})$ where $\mathcal{P}$ is a set of points and $\mathcal{L}$ consists of subsets of $\mathcal{P}$ be a affine plane if
each two points of $\mathcal{P}$ lie in a unique line
Let $L\in \mathcal{L}$ be a line…
dietervdf
- 4,524
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1 answer
Showing space closed under affine combinations is translation of vector space
I'm struggling to reconcile two different definitions of an affine space. The definition in my course notes is:
An affine space in $\mathbb{R}^n$ is a non-empty subset closed under affine combinations; that is, $X$ is an affine space if, whenever…
user53515
- 91
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Express a point as an affine combination of another two points(3D collinearity)
So, given the points A(1,2,2), B(2,4,2) and C(3,6,2) I have to show that they are collinear.
If they are collinear then I must express one point as an affine combination of the other two points.
I have searched everywhere and I can't find an…
southpaw93
- 229
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The set if affine?
The Set $\{ Ax + b | Fx = g \}$, is it affine?
How can I prove it?
My answer is yes, the intuition is that $\{ x | Fx = g \}$ is a solution space of equation $Fx = g$, thus it is a linear subspace.
The $Ax + b$ is a linear transformation plus a…
user2262504
- 954
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1 answer
Let $a,b$ be affine combinations of points from a set $S$. Then is the affine combination of $a,b$ also an affine combination of points from $S$?
Let A be an affine space, $a,b$ affine combinations of points from a finite subset $S$ of A. Then is the affine combination of $a,b$ also an affine combination of points from $S$?
I found it hard to show because it involves two affine…
pxc3110
- 1,410
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Dumb question regarding affine transformations
So, we can write an affinity $\phi$ as $$\phi(x) = Ax + b$$
for some linear transformation $A$ and vector $b$.
What exactly does it mean for an affinity to be a "scaling"? Is this a mapping of the form $\phi(x) = \lambda x + b$ for some scalar…
rehband
- 1,921
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About affine map on three dimensional euclidean space
An affine map $(t,M)$ with $t\in R^3$ and matrix $M$ maps $x\in R^3$ into $t+Mx$. It has property $P$ if for any $x$ with $|x|\leq 1$ then $|t+Mx|\leq 1$.
Our goal is to characterize the set of such maps with property $P$, that is characterized the…
gondolf
- 101
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Prove, using vectors, that this quadrilateral is a rhombus
Consider the following quadrilateral $ABCD$, with $E, F, G, H$ as the midpoint of $AD, DC, CB, BA$ respectively such that $\Delta ECH$ and $\Delta AGF$ are equilateral. Prove that $ABCD$ is a rhombus. Determine its angles.
I wanted to approach…
Gerard
- 4,264
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3 answers
Prove, using affine geometry, that in this figure $\Delta DEF$ is always equilateral
Consider the following figure:
$\Delta DAC, \Delta CEB, \Delta AFB$ are isosceles. $\angle ADC = \angle CEB = \angle AFB = 120^{\circ}$.
Prove that $\Delta DEF$ is equilateral.
Now, there is a proof for this using coordinate geometry. And its an…
Gerard
- 4,264
0
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1 answer
Prove for any four points: $|AB|^2 + |CD|^2 -|BC|^2 - |AD|^2 = 2\cdot \vec{AB}\cdot \vec{DB}$
Let $A, B, C, D$ be four points in space. Prove
$$|AB|^2 + |CD|^2 -|BC|^2 - |AD|^2 = 2\cdot \vec{AC}\cdot \vec{DB}$$
Clearly,
$$AB = B-A$$
$$CD = D-C$$
$$AD = D-A$$
If I directly substitute the values, ignoring the distance operator, I get:
$$|AB|^2…
Gerard
- 4,264
0
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1 answer
Affine coordinates of a line
Can you help me figuring out how to solve the next problem?
If the points M and N have affine coordinates $(m_1,m_2,m_3)$ and $(n_1,n_2,n_3)$ with respect to some points A,B,C, then the points X of the line MN have the affine coordinates…
Jane Doe
- 177
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0 answers
Affine subspace parallel proof (wanting a second opinion)
Let $X$ an affine space and $S,T$ two affine subspace of $X$. I want your opinion on my proof about the statement
If $S \cap T = \emptyset$ its okay. If $S \cap T \neq \emptyset$, we have $P \in S \cap T$. Let $P_S \in S$ and $P_T \in T$, then by…
Tohiea
- 405
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How do homomorphisms of affine spaces relate to homomorphisms of difference spaces?
I am looking at two (common) definitions:
Definition 1: An affine space is a triple $(A, V, +)$ with $A \neq \emptyset$, with $(V, +_V)$ a vector space, and with $+ \colon A \times V \to A$ such that the following conditions…
Max Flow
- 232
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1 answer
Equivalent statement for affine subspaces
We take $A$ to be a nonempty subspace of standard Euclidean space, then we have the following statements equivalent:
A is an affine subspace
$$\forall x,y\in A, t\in\mathbb{R}, (1-t)x+ty\in A$$
It looks like just following the definition but here…
79999
- 157
- 1
- 8
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Image of an extreme point by an affine map
Let $C$ be a convex subset of an affine space $E$.
Is it true that the image of an extreme point of $C$ by an affine map $f: E \longrightarrow F $ is an extreme point of $f(C)$ ?
Kieran McShane
- 398