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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Does this map define a rational map?
$\phi(x,y)=\frac{y-x^2}{x^2}$ for $\phi:X\to\mathbb{A}^1(\mathbb{C})$
$X$ being a variety $X=V(\langle x^5-x^4+2x^2y-y^2\rangle) \subset \mathbb{A}^2(\mathbb{C})$
B DIll
- 33
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Affine subspaces proposition
I try to prove that a subset of an affine space is an affine space iff it contains the line through every two distinct points of it.
guest
- 645
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There are exactly 8 isometries $F$ with $F(l_{1})=l_{2}$, $F(l_{2})=l_{3}$, $F(l_{3})=l_{1}$ and $l_{1} \cap l_{2} \cap l_{3} $ is fixed point.
$l_{1}, l_{2}, l_{3} $ are 3 pairwise orthogonal lines in $\mathbb{E_3}$
Prove that there are exactly 8 isometries $F$ with $F(l_{1})=l_{2}$, $F(l_{2})=l_{3}$, $F(l_{3})=l_{1}$ and $l_{1} \cap l_{2} \cap l_{3} $ is a fixed point.
I have no idea how…
bob
- 1,256
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Find a plane equation from a line and a point
Consider the point $B(1,0,1)$ and the line $R=(1,1,1) + \alpha[1,1,-1]$.
Find the equation of a plane that passes through $B$ and contains $R$.
What I tried doing was simply setting two different values for $\alpha$, since if the points are part of…
markoff
- 35
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1 answer
Relative postion of a plane and a hyperplane in $\mathbb{R^4}$
I know what happens in $\mathbb{R^2}$ and $\mathbb R^3$.
In $\mathbb{R^2}$ , two lines either intersect in a point or they are parallel.
In $\mathbb{R^3}$, two lines (or a line and a plane) can intersect in a point, be parallel or be askew. Two…
markoff
- 35
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Find a plane perpendicular to the intersection of two affine subspaces
Consider the two following affine subspaces of $\mathbb{R^3} $:
$$S=\{x,y,z)\mid 2x+y+z=1\}$$ and $$T=\{(x,y,z)\mid x-y+2z=0\} $$
Find the plane $H$ perpendicular to the intersección of $S$ and $T$, that passes through the point $p(-1,0,1)$.
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Find the dimension of an affine subspace
Find the dimension of the affine subspace of $\mathbb{R^5}$ generated by the points
$$p=(-1,2,-1,0,4)$$
$$q=(0,-1,3,5,1)$$
$$r=(4,-2,0,0,3)$$
$$s=(3,-1,2,5,2)$$
Is it as trivial as simply finding $\vec{pq}, \vec{qr}, \vec{rs}, \vec{sp}$ and finding…
markoff
- 35
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Rotation of a hyperbola in affine geometry
Given the hyperbola $x^2 - 3xy + y^2 + 4x - 5y + 2 =0$
I have translated this by $x+\frac{7}{5}$ and $y-\frac{2}{5}$ and got
$x^2 - 3xy + y^2 = \frac{9}{5}$
Now, the bit where I'm stuck;
I have rotated by $\frac{\pi}{4}$ and got
$5x^2 - y^2 = 18/5$…
Alice
- 185
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Two meanings to affine independence? (help me clear up my misunderstanding)
I must be misunderstanding something. Let's look at the following two definitions for a set of points $S=\{v_1,v_2,...,v_k\}$ to be affinely independent:
1) S is affinely independent if the set $\{v_2-v_1, v_3-v_1, ..., v_k-v_1 \}$ is linearly…
JQX
- 159
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1 answer
Affine hull of two points in R4
I try to describe an affine hull of two points (1,3,2,4) and (1,4,2,3) so i try to make the linear equation which describe it .
KostasC
- 101