All Questions
Tagged with mg.metric-geometry terminology
47 questions
2
votes
1
answer
140
views
Does this result above six points follow have a name?
Does this result above six points follow have a name?
Let $A$, $B$, $C$, $D$, $E$, $F$ be six points in the plane and $AB, CF, ED$ are concurrent and $BC, DA, FE$ are concurrent then $CD, EB, AF$ ...
- 1,776
2
votes
0
answers
114
views
Another Butterfly theorem — Conway like circle
Have You seen these result as follows before?
In Figure 1: $AA'=BB'=tAB$; $CC'=DD'=tCD$, where t is real number then $ABCD$ is a cyclic quadrilateral iff $A'B'C'D'$ is a cyclic quadrilateral.
In the ...
- 1,776
1
vote
0
answers
68
views
Name of the perspector of the orthic triangle and excentral triangle
The orthic triangle and tangential triangles of a given triangle are in perspective. What's the official kimberling center associated with this perspector?
- 1,109
2
votes
0
answers
162
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Root system terminology
Let $\Phi$ be a root system. In a paper I'm writing, I need to work with subsets $\Phi' \subset \Phi$ satisfying the following two conditions:
For all $\lambda_1,\lambda_2 \in \Phi'$ and $c_1,c_2 \...
- 21
4
votes
0
answers
222
views
What does it mean "parallel"?
I am thinking on a strict definition of the notion of parallel affine sets in a linear space and came to the following
Definition 1: An affine set $A$ is parallel to an affine set $B$ in a linear ...
- 41.8k
2
votes
0
answers
137
views
Name of this geometric point? [closed]
Draw a triangle. At one of the vertices, draw a line through it that bisects the angle. At each of the other two vertices, draw a line through it which is perpendicular to the line that bisects its ...
- 1,109
3
votes
2
answers
432
views
Metric space whose bounded subsets are totally bounded
Is there a name for a metric space in which any bounded subset is totally bounded, or equivalently, in which any bounded sequence contains a Cauchy subsequence?
I have seen the name Bolzano-Weierstraß ...
- 2,834
11
votes
3
answers
1k
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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
7
votes
2
answers
529
views
What is the name for a point that is periodic to within $\varepsilon$?
Let $X$ be a set and $f: X \to X$ a function. A point $x \in X$ is, of course, said to be periodic for $f$ if $x \in \{f(x), f^2(x), \ldots\}$.
Now suppose that $X$ is a topological space and $f$ is ...
- 27.7k
0
votes
0
answers
62
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Terminology: maps which are bi-Lipschitz on compact subsets
Let $X$ and $Y$ be metric spaces and let $f:X\rightarrow Y$ be such that: for every compact subset $K$ of $X$ the restricted map $f|_K:K\rightarrow Y$ defined by $f|_K(x)=f(x)$ is bi-Lipschitz (with ...
- 5,405
3
votes
1
answer
2k
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Does this hexagon theorem have a name?
Question : Do you know this property of a hexagon?
Consider the configuration: Six points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, $A_6$ in a plane and let six points $B_i \in A_iA_{i+1}$ for $i=1, 2,\dots, ...
- 1,776
0
votes
1
answer
189
views
Terminology "upper" Ahlfors regular measure
Let $(X,d)$ be a metric space and $m$ be a Borel measure on $(X,d)$. The measure $m$ is called Ahlors regular if $m(B(x,r))\asymp r^q$ for some $q>0$ and each $x\in X$. Is there a name for ...
- 5,405
2
votes
1
answer
124
views
Name of the "s" parameter in Ungar's theory of hyperbolic geometry
I have done a R package which implements Ungar's approach to hyperbolic geometry, for the hyperboloid model. In this theory, there is a parameter $s>0$ which controls the curvature of the ...
- 2,560
0
votes
0
answers
28
views
Name for function projecting a 3D point to the surface of aligned sphere
Let there be a 3D point $\mathbf{P} = \begin{bmatrix} X & Y & Z\end{bmatrix}^{\top}$ and a sphere $\mathcal{S}$ with radius $r$ and centroid at the origin, everything expressed w.r.t the same ...
- 1
2
votes
1
answer
182
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Is a line associated with antipodal points (the fact, it is the generalization of Simson line) known?
First time, I found a line associated with antipodal points, detail:
Let $ABC$ be a triangle, $(C)$ is circumconic of $ABC$. $P$ and $P'$ are two antipodal points. Construct three lines through $P'$ ...
- 1,776
0
votes
0
answers
109
views
What is the correct name of points with this property?
Let $(X, d)$ be a metric space and $x \in X$.
Suppose for all $x_1, x_2 \in X$ the following inequality holds:
$$
d(x_1, x_2) \le \max \bigl\{ d(x, x_1), d(x, x_2) \bigr\}.
$$
For example, singleton ...
- 1
5
votes
1
answer
433
views
Golden ratio as a property of conic section (is it known?)
I am looking for a proof of a discovery as follows:
Let $ABC$ be arbitrary triangle and $(\Omega)$ be an arbitrary circumconic of $ABC$ let $A'B'C'$ is its tangential triangle of $ABC$ respect to $(\...
- 1,776
1
vote
1
answer
352
views
Thirteen-point conic and four-point line, are they new?
We know that Five points determine a conic and Two Points Determine a Line. Here I found a simple construct of a conic through $7$ points (in PS I note that how the conic through thirteen points) and ...
- 1,776
13
votes
2
answers
2k
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Is it a new discovery on conic section?
I discovered a problem in plane geometry (there are some nice special cases) as follows:
Let $ABC$ be a triangle and $\Omega$ be arbitrary circumconic. Let two points $A_b, A_c \in BC$, $B_c, B_a \in ...
- 1,776
6
votes
0
answers
320
views
Does this plane geometry theorem have a name (well-known)?
Consider three circles $(O_1)$, $(O_2)$, $(O_3)$. Denote the homothetic center of $\{$$(O_1)$, $(O_2)$$\}$ by $A$, the homothetic center of $\{$$(O_2)$, $(O_3)$$\}$ by $B$. Let $C$, $D$ be two points ...
- 1,776
0
votes
0
answers
70
views
Looking for a name for a generalization of geometry to graphs
I am pursuing generalizations of planar Euclidean geometry to complete symmetric and weighted graphs, the guiding principle being applicability to the TSP.
The operations and tests that are available ...
- 13.2k
1
vote
1
answer
71
views
Terminology: Co-completion of Met?
In main-stream mathematical literature, the term metric space is reserved for $(X,d)$ where $X$ is a set and $d:X\times X\rightarrow [0,\infty)$ satisfies the usual properties of a metric. However, ...
- 5,405
3
votes
0
answers
104
views
Every partial isometry extends
I am interested in metric spaces $X$ where every isometry between two subsets of the space extends to a full isometry $X \to X$. Is there a name for this kind of space? Is there some paper which ...
- 31
2
votes
0
answers
601
views
Is there a name for a "convex hull with holes"?
If I have a (solid) 3d object, is there a name for the object created from it by taking the convex hull and subtracting from it all points that are on a straight line between any two points on the ...
- 121
3
votes
2
answers
2k
views
What is the name of the 65537-gon? [closed]
I know the name of the heptadecagon (17 sides) and the diacosipentacontaheptagon (257 sides). But what is the name of the polygon with 65537 sides? I am unable to figure it.
- 18.7k
60
votes
2
answers
4k
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Does this geometry theorem have a name?
Start with a circle and draw two tangent circles inside. The (black) inner tangent lines to the smaller circles intersect the large circle. The (red) lines through these intersection points are ...
- 509
5
votes
1
answer
180
views
$c$-coarsely connected space for every $c>0$
A metric space $(X,d)$ is called $c$-coarsely connected if for every two points $x,y\in X$ there exists a sequence $x=x_0,x_1,\ldots,x_n=y$ of points in $X$ such that $d(x_{i-1},x_i)\leq c$.
Question:...
- 1,782
1
vote
1
answer
305
views
Name of area between two parallel lines [closed]
Assume that there are two distinct parallel lines on a Euclidean plane. Is there a name for the zone between these two lines?
- 121
4
votes
0
answers
95
views
Name for metric spaces with useful unique-ball-intersection property?
When dealing with the problem of extending a Lipschitz function $f:A \to Y$ between metric spaces across an inclusion $A \hookrightarrow X$, one often imposes (conditions which imply) the following ...
- 15.5k
12
votes
2
answers
806
views
Term for a metric space for which the triangle inequality is strict?
Is there a standard term for a metric space in which $\rho(p,r) < \rho(p,q) + \rho(q,r)$ for any distinct $p$, $q$, $r$? Sort of the opposite of metric convexity.
For instance, a subset of ...
- 42.8k
5
votes
2
answers
453
views
What is this distance about?
For points $a,b\in \mathbb{R}^n\setminus \{0\}$ denote $$d(a,b)=\frac{\|a-b\|}{\|a\|+\|b\|}.$$
This question by Ritesh Ahuja (positive answered by Iosif Pinelis) says that $d$ is a metric. My ...
- 109k
2
votes
0
answers
811
views
Products between metrics in a product of manifolds
In the "Einstein Manifold" book written by Arthur Besse, chapter 16, there is a notation of a manifold composed by the Cartesian product between two others:
$(M_1\times M_2, f^p(g_1 \times g_2))$
...
- 21
4
votes
2
answers
399
views
Terminology for metrics?
For some reason, I'm currently interested in the following relation - let $d,\delta$ be two metrics on some space $X$. We call the metrics _______ if there are some constants $C,E>0$ such that for ...
- 689
2
votes
1
answer
198
views
Is there a name for the level-sets of the signed distance function to a set in a metric space?
$\newcommand \X {\mathcal{X}}$
$\newcommand \sd {d_{\rm sign}}$
Let $(\X, d)$ be a metric space and define the distance between a point $x \in \X$ and a set $S \subset X$ by $d(x,S) = \inf_{y \in S} d(...
- 75
32
votes
2
answers
1k
views
Term for "uncheckable constructions"
Is there a term for "uncheckable geometric constructions"?
Say, Angle Trisection and Doubling the Cube are checkable;
i.e., if the answer is given one can do finite Compass-and-straightedge ...
- 45k
5
votes
0
answers
152
views
Star shaped sets with a midpoint
Suppose $U$ is an open subset of $\mathbb{R}^n$ which is star shaped with respect to $p\in U$. I'll call $p$ a midpoint of $U$ if for any line $\ell$ through $p$, the point $p$ is the midpoint of the ...
- 3,020
3
votes
0
answers
148
views
term for a rectangle with a bounded aspect ratio
I am writing a peper about dividing a shape into rectangles, where the main issue is to make sure that the rectangles have a limited aspect ratio. I am looking for a clear, unambiguous term for such ...
- 3,549
2
votes
3
answers
188
views
Mapping in reference to a metric space (terminology question)
Let the pair $( S, d \, )$ be a metric space, i.e.
$d\!: S^2 \rightarrow R$, where for any three distinct elements $k$, $p$, $q$ $\in S$:
$d[ \, p, q \, ] = d[ \, q, p \, ] > 0$,
$d[ \, p, q \, ] + ...
1
vote
0
answers
385
views
General setting for triangle inequalities (terminology question)
Regarding the "mathematical object $(X, s)$" described below (in general, or under some more specific conditions) I'd like to know whether it is called a particular name, for reference in the ...
6
votes
1
answer
1k
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Löwner-John Ellipsoid: incribed and circumscribed
I have two questions about the
Löwner-John ellipsoid, one just terminology, the other
more substantive.
Let $K$ be a convex body in $\mathbb{R}^d$.
Q1.
Is "the
Löwner-John ellipsoid"
the ...
- 151k
6
votes
1
answer
555
views
Is there a name for the class of metric spaces such that the closure of the open ball of radius $r$ around each point $x$ is the set of elements $y$ such that $d(x,y)\leq r$ ?
Let $(X,d)$ be a metric space, let $B(x,r)$ be the open ball of radius $r$ about $x$ and $N(x,r)$ be the set of elements $y\in X$ such that $d(x,y)\leq r$. It is well-known that it is not always true ...
- 6,034
1
vote
0
answers
139
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Terminology: metric space with product and unit, and the opposite of a nonexpansive map
Someone I know is trying to figure out if the following concepts already have an established name in the literature, and MO is a great place to ask around.
1) Suppose $X$ is a metric space equipped ...
4
votes
1
answer
246
views
Name for an inequality of isoperimetric type
I want to know if the following fact has a standard name and/or reference
Let $X$ be a subset of $\mathbb R^2$ and $B$ be a disc of the same area as $X$.
Set $X_\epsilon$ to be the $\epsilon$-...
- 43
9
votes
4
answers
672
views
Is there a common name for the complement of a metric space in its completion?
Is there a common name for the complement $\widehat{X} \setminus X$ of a metric space $X$ in its metric completion $\widehat{X}$? Since $X$ is not necessarily open in $\widehat{X}$, the term boundary ...
- 44.4k
0
votes
1
answer
2k
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Categories of Geometry [closed]
I learn that Geometry has several categories/subfields from Wikipedia. But I am still not clear about the standards according to which they are classified.
It seems Euclidean Geometry, Affine ...
- 357
16
votes
2
answers
3k
views
Metric on one-point compactification
Is there a standard construction of a metric on one-point compactification of a proper metric space?
Comments:
A metric space is proper if all bounded closed sets are compact.
Standard means found in ...
- 45k
5
votes
2
answers
484
views
Better term for a (simplicial) contractible plane continuum
In this joint paper that I should be working on, we make significant use of a certain generalization of a triangulated disk. Many of our important examples are triangulated disks, but we are also ...
- 56.6k