All Questions
Tagged with dg.differential-geometry plane-geometry
23 questions
1
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The optimal embedded and enclosing cardioids for a triangle
Ref: https://en.wikipedia.org/wiki/Cardioid
Earlier posts with similar questions: Smallest 3-ellipses that contain triangles and Curves of constant width that contain triangles
Questions: Given any ...
- 5,979
14
votes
1
answer
499
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Interpolating between disks in the plane
Below, a "disk" means a compact subspace $D \subset \mathbb R^2$ whose boundary is a smooth simple closed curve.
Task: Find a procedure which takes as input a pairs
of disks
$
D_0 \subseteq ...
- 43.2k
19
votes
1
answer
819
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All saddles in the unit ball have area $<2\pi$?
Let $M$ be the saddle surface in $\mathbb R^3$ defined by $x^2-y^2+z=0$. For any $r\geq 0$ and $(x_0,y_0,z_0)\in\mathbb R^3$, let $rM+(x_0,y_0,z_0)$ denotes the surface obtained by scaling $M$ by $r$ ...
- 525
1
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94
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Constant width curves and inscribed/ circumscribed ellipses
It is known (see for example the Wikipedia entry on the Reuleaux triangle) that for every curve of constant width (CCW), the largest inscribed circle and the smallest circumscribed circle are ...
- 5,979
90
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5
answers
4k
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Does this property characterize straight lines in the plane?
Take a plane curve $\gamma$ and a disk of fixed radius whose center moves along $\gamma$. Suppose that $\gamma$ always cuts the disk in two simply connected regions of equal area. Is it true that $\...
- 4,459
2
votes
1
answer
164
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Conformal isomorphism uniquely determined by boundary identification?
Let $\Gamma$ be a smooth Jordan arc, and let $\Phi \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \Gamma$ be a conformal isomorphism that fixes the point at $\...
- 239
5
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2
answers
565
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Geometry of Level sets of elliptic polynomials in two real variables
Updated:
A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its last homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$.The two answers to this post provide a ...
- 356
3
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1
answer
234
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Large class of curves which only intersect each other finitely many times
I am trying to find a large subset of piecewise-differentiable plane curves of finite length (subsets of $\mathbb{R}^2$) with the following property:
For any pair $\gamma_1, \gamma_2$ of curves in ...
- 509
4
votes
1
answer
160
views
What curve of positive curvature minimizes distance from the origin, given length and total curvature?
Let $\textit{F}$ be the family of $C^1$ curves in $\mathbb{R}^2$ of fixed length $\bar{l}$ and fixed tangent's turning angle $\bar{k}$.
What are the curves of positive curvature in $\textit{F}$ ...
- 405
1
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1
answer
125
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An asymptotic version of the Isoperimetric inequality
Let $U$ be a simply connected bounded open set in $\mathbb{R}^2$. The area of $U$ is denoted by $A$.
(We do not assume any thing about its boundary).
Assume that $\gamma_n$,s are smooth simple ...
- 356
9
votes
1
answer
382
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Are two triangles with equal corresponding medians, congruent?
Is the hyperbolic or spherical analogy of the following Euclidean fact, true?
Two triangles with equal corresponding medians are congruent.
More precisely: Assume that $\Delta ABC$ and $ ...
- 356
16
votes
1
answer
667
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Can a shape rolling inside itself reproduce that shape?
Q. Is the circle the only shape that, when rolling inside itself,
has a point that draws out a scaled copy of itself?
Let $C$ be a simple, closed, smooth curve in the plane.
(Likely "smooth" can be ...
- 151k
5
votes
4
answers
495
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Integral of the distance function to the boundary of a planar set
I have been stuck for a few days in a seemingly harmless question.
Given a simply connected open set $\Sigma\subset\mathbb{R}^2$, with smooth boundary $\partial\Sigma$, I am interested in estimating
$...
- 165
8
votes
2
answers
378
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Curves embedding in plane
Given two closed simple(no self-intersection point) curves $C_1,C_2$ in the plane $\mathbb R^2$, is there a good way to judge whether one curve can be embedded inside the other one, here embedding ...
- 1,915
2
votes
1
answer
358
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Perimeter of ellipse: Combination of two geometries
Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ such that for every ellipse $\gamma$ in the plane we have:$$\text{The Euclidien perimeter of}\; \gamma=\lambda (g\text{-diameter of}\;\gamma)$$...
- 356
2
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1
answer
246
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The points of half area of a triangle
Let $S$ be a simply connected Riemannan surface . Suppose $\Delta ABC$ is a triangle on $S$. The Area of a triangle is denoted by $\mathcal{A}$. A point $P$ in the interior of $\Delta ABC$ is ...
- 356
5
votes
0
answers
1k
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Boundary of an open, bounded and convex set in $\mathbb{R} ^n$
Let $U$ be an open, bounded and convex set in $\mathbb{R} ^n$. Since $\partial U$ is a rectifiable set it follows that up to a set of $H^{n-1}$-measure zero $\partial U$ is contained in a countable ...
1
vote
1
answer
267
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Vertices of Curves and Eigenvectors of Hessian
This might be a trivial question, but I can't seem to figure it out. Suppose I have an implicitly defined curve in the plane given by $f(x,y) = t$.
This curve is strictly convex, and feel free to ...
- 115
19
votes
5
answers
1k
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Lightray trapped between two mirror disks: Computation formulation?
I would like to calculate the angle of a ray $r$ from a given
point $p$ such that it gets "stuck" reflecting between
two congruent mirror-disks.
For why there is such a ray, see the (amazing!) answer
...
- 151k
2
votes
1
answer
308
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Connecting tangents of convex curves: at some point orthogonal?
Let $a(t)$ and $b(t)$ be two smooth, nested convex curves in the plane, $t\in[0,1]$:
Suppose the parametrization of $a()$ and $b()$ is such that $\dot{a}(t)$ is ...
- 151k
23
votes
3
answers
3k
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Trapped rays bouncing between two convex bodies
At some point during my research I was confronted with this problem, but I did not dedicate serious time to it. Anyway it stayed in the back of my mind and I'm still interested in hints for it. ...
- 9,007
13
votes
2
answers
702
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'Eigenvectors' of evolute operation
The evolute of a curve is the locus of its centers of curvature.
The evolute of some plane curves is a scaled, or scaled and
reflected/rotated, version of that curve.
For example, the evolute of a ...
- 151k
1
vote
0
answers
618
views
Which plane convex arcs have the smallest maximum curvature?
Let $p$ and $q$ be positive real numbers with $p \leq q$. Suppose that $H(p,q)$ is the class of all convex arcs $c$ in the Cartesian $x-y$ plane which satisfy the following conditions:
(1)The $y$-...
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