All Questions
Tagged with mg.metric-geometry ag.algebraic-geometry
87 questions
-1
votes
0
answers
65
views
Axes of symmetry and symmetry group of the tangent cone to an open, connected, convex subset of the Euclidean space
Given a closed convex set $K\subset \mathbb{R}^d$ and a point $x\in K$ the tangent cone to $K$ at $x$ is defined by
\begin{equation}
T_xK:=\overline{\{v\in \mathbb{R}^d: \exists \lambda \geq 0 \text{ ...
- 1,465
6
votes
0
answers
172
views
Does there exist a plane curve such that it has the heart curve as catacaustic?
Given a curve $C$ and a fixed point $L$ (the light source), the catacaustic of $C$ with respect to $L$ is the envelope of light rays coming from $L$ and reflected from the curve $C$.
The catacaustic ...
- 565
21
votes
1
answer
975
views
Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points
The following question resisted attacks at Math SE, so I thought I would try posting it here.
Is the following conjecture true or false:
Given any five coplanar points, we can always draw at least ...
- 3,527
1
vote
0
answers
84
views
Number of polyhedral covers of a triangulation of $S^2$
For a given triangulation (combinatorial Type I. or Type II.) of a $2$-sphere, what is the number of unique polygonal covers with $n$ polygons where ($n$ goes from $2$ to $N$)?
Under polygonal cover, ...
- 183
7
votes
1
answer
347
views
A corollary of the affine Desargues axiom
Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms:
Any distinct points $x,y\...
- 41.8k
2
votes
0
answers
121
views
Integral geometric meaning of diameter
Let $X\subset \mathbb CP^n, n>2$ be a complex smooth algebraic hypersurface.
Any hyperplane section $H\cap X$ is connected and has diameter $Diam(H\cap X)$
in the inner metric induced from the ...
- 2,934
2
votes
1
answer
321
views
de Rham's trisection method - English
I want to learn more about de Rham's trisection method in
De Rham, Georges, Un peu de mathématiques à propos d'une courbe plane, Elemente der Mathematik 2 (1947): 73-76. http://eudml.org/doc/140463.
...
11
votes
3
answers
1k
views
What is the minimum-curvature curve interpolating a given set of points in the plane?
We are given a set $X$ of $n\ge 3$ points in $\mathbb{R}^2$, belonging to the boundary of the convex hull of $X$ itself. Let $\Gamma(X)$ be the set of all convex, simple closed curves in $\mathbb{R}^2$...
- 1,677
4
votes
0
answers
119
views
Writing the $\ell^{p/(p-1)}$ unit sphere as a semi-algebraic set for $p\in\Bbb N$
The $\ell^p$ unit sphere $\{x\in\Bbb R^n\mid |x_1|^p+\cdots+|x_n|^p=1\}$ with $p\in\Bbb N$ is a semi-algebraic set, and its polar dual is
$$(*)\quad \{x\in\Bbb R^n\mid |x_1|^q+\cdots +|x_n|^q=1\},$$
...
- 13.6k
0
votes
0
answers
252
views
Computation of scalar curvature from a Riemannian metric
I want to compute the scalar curvature for points on an empirical manifold (sampled data).
I have already an algorithm that learns the Riemannian metric and computes geodesics, so from the metric I ...
0
votes
1
answer
247
views
Projecting a given point onto a random $2$-dimensional plane in more than $3$ dimensions
We are given $\mathbf{p}\in\mathbb{R}^d$, where $d\gg 1$. Let $\mathbf{v}$ be a point selected uniformly at random from the unit $(d-1)$-sphere $\mathcal{S}^{d-1}$ centered at the origin $\mathbf{0}\...
- 1,677
1
vote
0
answers
58
views
Are cells of 4-polytopes a convex polyhedron by definition?
I'm going by the Wikipedia definition for a 4-polytope.
Do by definition, cells of 4-polytopes have to be a convex polyhedra?
If not, then are there polyhedra with non-convex faces?
If yes, is it the ...
- 11
3
votes
1
answer
363
views
What do convergent sequences of rational functions look like?
Let us consider the projective line over $\mathbb C$ equipped with a nice metric $\eta$ (like the Fubini-Study metric). We can define a metric $\mu$ on rational functions $f: \mathbb P^1 \to \mathbb P^...
- 7,746
4
votes
0
answers
239
views
Example of a computation of the volume of a subvariety in projective space $\mathbb{P}^n$
Let us consider the projective space $\mathbb{P}^n$ with the standard Fubini Study metric.
I searched all over the internet but I can't find an example of a calculation of the volume for a projective ...
- 1,343
96
votes
4
answers
5k
views
A curious relation between angles and lengths of edges of a tetrahedron
Consider a Euclidean tetrahedron with lengths of edges
$$
l_{12}, l_{13}, l_{14}, l_{23}, l_{24}, l_{34}
$$
and dihedral angles
$$
\alpha_{12}, \alpha_{13}, \alpha_{14},
\alpha_{23}, \alpha_{24}, \...
- 4,316
4
votes
1
answer
369
views
Comparing two Riemannian metrics on Grassmannian
Let $G_r(n)$ be the real Grassmannian which is the collection of all $r$ dimensional subspace in $\mathbb{R}^n$ equipped with the usual invariant metric $g$.
Let $U_A\in\mathbb{R}^{n\times r}$ and $...
- 1,495
3
votes
0
answers
45
views
Lengths of edges of a triangulated surface
Consider a triangulated surface of genus $g,$ which is embedded in $\mathbb{R}^3$. A simple parameter counting shows that the lengths of edges of the surface satisfy $6g$ algebraic equations. Have ...
- 4,316
9
votes
1
answer
529
views
Ricci Curvature on Grassmannian
Suppose $G_r(n)$ is the Grassmannian, which is the collection of all $r$ dimensional subspace in $\mathbb{R}^{n}$ equipped with the usual invariant metric. Let $Ricc(G_r(n))$ be the Ricci curvature ...
- 1,495
1
vote
0
answers
34
views
Find a third circles that crosses two other circles at an angle [closed]
Given two circles at positions $P_0$ and $P_1$ of radius $R_0$ and $R_1$, respectively, is it possible to find the position $P$ and radius $R$ of a third circle that intersects a point $P_2$ and ...
- 11
12
votes
1
answer
559
views
Square lying on moving chord of a simple closed curve
Consider a simple closed curve $C$ in $\mathbb{R}^2$. For any points $a$ and $b$ on this curve we associate a point $c_1$ on the left and $c_2$ on the right side to the chord ab, such that $ac_1bc_2$ ...
- 415
2
votes
1
answer
226
views
Moving chord on the simple closed curve
Consider a simple closed curve $C$ in $\mathbb{R}^2$. For any points $a$ and $b$ on this curve we associate point $c$ on the left (or right) side to chord $ab$ such that $\angle acb = 90^{\circ}, ac=...
- 415
2
votes
1
answer
198
views
Question $B_5 \equiv B_1$ or $B_5 \ne B_1$?
Let $(C_1)$, $(C_2)$ be two conics on the same Ellipsoid, (or Hyperboloid, or Paraboloid). Let $A_1, A_2, A_3, A_4$ be four arbitrary points lie on $(C_1)$; $B_1$ be arbitrary point on $(C_2)$. The ...
- 477
5
votes
0
answers
333
views
Which equation of a Butterfly?
Let $A, B$ be two points and $L$ be a line on the Euclidean Plane. Take two points $J, G$ on the line $L$ such that $JG=constant$. Let $AJ$ meet $BG$ at $P$, $AG$ meet $BJ$ at $Q$, then the locus of ...
- 477
3
votes
0
answers
141
views
Which is the number of independent components of a flat spin connection in a 4 dimension Weitzenböck spacetime?
A spin connection $A_{ab\mu}=-A_{ba\mu}$ has 24 components. The number of independent components for a flat spin connection can be counted by subtracting the constrains set by the condition of null ...
- 51
0
votes
1
answer
181
views
Convex planar curves and intersections [closed]
Given two planar regular convex not-closed curves C and C_1.
Let A the set of finite intersections between C and C-1.
Then what is the stricter upper bound of |A|?
I would say 2.
Thanks.
1
vote
1
answer
174
views
Need help maximizing distances to nearest neighbor in a cylinder
I have a cylinder and I want to maximize the number of points in the cylinder such that the distances to the nearest neighbors are maximally spaced. How do I find out how many points I can have so ...
- 11
4
votes
2
answers
1k
views
approaches to Apollonius circle problems
I've been looking for solutions to finding the set of circles tangent to two other circles. one circle can be inverted to a line, but two circles can be mapped to a line and a circle
or equivalently ...
- 22.8k
1
vote
0
answers
88
views
Hausdorff limits of fibers of affine maps
Let $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, and let
$$
F=(P_1,\ldots, P_m):\mathbb{K}^n\to \mathbb{K}^m
$$
be a polynomial map. I would like to know under what conditions the preimages $F^{-1}(y)$ of ...
- 1,452
16
votes
2
answers
1k
views
Algebraic surface of constant width?
Does there exist an irreducible polynomial $f \in \mathbb{R}[x, y, z]$ such that:
$$ V := \{ (x, y, z) \in \mathbb{R}^3 : f(x, y, z) \leq 0 \} $$
is a solid of constant width with a finite symmetry ...
- 12.4k
6
votes
2
answers
1k
views
Motivation for Hirzebruch-Jung Modified Euclidean Algorithm
Let $a,b \in \mathbb{N} \ \ s.t. \ \ a > b$ have $\gcd(a,b) =1$. We can define the Hirzebruch-Jung modified euclidean algorithm as follows:
Let $e_i \in \mathbb{N} >2$, and $ r_k \in \mathbb{N}$...
4
votes
1
answer
226
views
A conjecture for a curve cuts a curve - variant Cayley-Bacharach's theorem
I propose a conjecture variant of Cayley-Bacharach's theorem.
I'm an electrical engineer, I am not a mathematician. I don't know how to prove this result. Could you give a solution or let me know ...
- 1,044
5
votes
3
answers
572
views
set of centers of sphere inscribed in tetrahedron
Having a sphere and three diffrent point $A,B,C$ on this sphere. Find set of all centers of spheres inscribed in a tetrahedron $ABCD$, where $D$ is some point on the given sphere. The problem reduced ...
- 69
2
votes
1
answer
489
views
An identity for Futaki-Donaldson invariant
Let $(X,L)$ be a polarized projective variety
Given an ample line bundle $L\to X$, then a test configuration for the pair $(X,L)$ consists of :
a scheme $\mathfrak X$ with a $\mathbb C^*$-action
a ...
user21574
18
votes
2
answers
700
views
Can all unit-distance graphs have their vertices at algebraic integers?
A graph $G$ is described as a unit-distance graph if there exists a function $f:G \rightarrow \mathbb{C}$ such that for every edge $(u,v) \in E(G)$, we have $|f(u) - f(v)| = 1$.
Obviously, we can ...
- 12.4k
9
votes
2
answers
718
views
Generalization of Pascal's theorem to higher dimensions
Pascal's celebrated theorem in classical geometry gives a necessary and sufficient condition for the existence of a conic through six given points in the plane. Does there exists a similar statement ...
- 4,484
0
votes
0
answers
127
views
Geometric interpretation of table with permutations and inversions
Let $T(n,k)$ is the number of permutations of numbers $1, ..., n$ and each of the permutations has $k$ inversions. We can consider a table for $T(n,k)$ for some $n$ and $k$. For eg.
$n=1,...,6$, $k=1,....
15
votes
1
answer
846
views
What is the longest algebraic curve?
Consider a convex body $\Omega\subset \mathbb{R}^2$. Let $L(d)$ be the maximum over all curves $C$ of degree $d$ of the length of $C\cap\Omega$.
Is $L(d)\leq d P(E)/2$, where $P(E)$ is the ...
- 7,836
-2
votes
1
answer
331
views
Polygon Problem [closed]
There are $N$ regions which are numbered from $1$ to $N$. Each region is represented by a single simple polygon on the 2D plane. Simple polygon means the boundary of the polygon does not cross itself. ...
8
votes
1
answer
573
views
Do elements of the fundamental group give rise to isometries
Let $X$ be a complex algebraic variety, and let $\tilde X\to X$ be its universal cover. Suppose that there exists a Kahler-Einstein metric on $\tilde X$. Note that $\pi_1(X) \subset Aut(\tilde X)$.
...
- 103
2
votes
1
answer
414
views
Difference between Kahler-Einstein and Bergman metric on a bounded symmetric domain
Let $H$ be a bounded symmetric domain.
What is the difference between the Bergman metric and the Kahler-Einstein metric on $H$?
- 103
5
votes
4
answers
1k
views
The Icosahedron Equation
$$1728 V^5 + F^3 = E^2 \;.$$
Can anyone point me to a concise, modern derivation and explanation of
the significance of the icosahedron equation, more modern and
concise than Klein's description in ...
- 151k
20
votes
5
answers
1k
views
Historical use of figures in geometry
I was surprised to learn from John Stillwell's comment in answer to the
question,
"Can the unsolvability of quintics be seen in the geometry of the icosahedron?",
that
There is not a single picture ...
- 151k
4
votes
1
answer
184
views
What are interesting 3-colorings of the plane without rainbow lines?
This question is about 3-colorings of the plane in which every line is bichromatic (or monochromatic), i.e., there are no three collinear points of different colors. Such colorings trivially exist, ...
- 18.8k
1
vote
1
answer
176
views
Helly's number from biconvex functions
Helly's Theorem states the following. Suppose $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq \...
- 135
2
votes
1
answer
171
views
Helly's Theorem for Biconvex Sets
Helly's Theorem states the following. Suppose that $X_1,X_2,...,X_N$ are convex sets in $\mathbb{R}^d$, such that for any index-set $I$ with $|I| \leq h(d) := d+1$, we have $\bigcap_{i \in I} X_i \neq ...
- 135
16
votes
2
answers
1k
views
Integer lattice points on a hypersphere
Is the following statement true?
For every integer $n\ge2$ and every integer $k\ge0$ there exists a hypersphere in $\mathbb{R}^n$ (circle, sphere etc) containing exactly $k$ integer lattice points ...
- 513
2
votes
0
answers
83
views
Largest subsets of quadrics consisting of "nonorthogonal" vectors
Assume we have an $A$-module $M$, and a quadratic form $q : M \to A$. Recall that it means that
1) $q (a m) = a^2 q (m)$ for all $a \in A$ and $m \in M$, and
2) $B_q (x, y) := q (x + y) - q (x) - q ...
- 311
1
vote
0
answers
142
views
Relationship between stabilizers of a general point and a boundary point
Let $V$ be an n-dimensional complex vector space, and $u\in S^nV$ be a polynomial, $G(u)$ be the stabilizer of $u$ in $GL(V)$. Let $[v]\in\overline{GL(V)\cdot[u]}\subset\mathbb{P}(S^nV)$, but $v\notin ...
- 105
6
votes
3
answers
539
views
Constructing a field from a spherical building
Tits proved that (sufficiently high rank) spherical buildings arise from an algebraic group and a field, so any building is some $\Delta(G, F)$. He also showed that a building isomorphism $\Delta(G,F)...
- 63