All Questions
Tagged with plane-geometry curves
12 questions
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Divide angles by coefficients relate to Fibonacci sequence
In the left Figure, consider a right triangle $OPA$ with $\angle {AOP} = 90^\circ$. Let $\ell$ be the reflection of $PO$ in $PA$ and $\ell$ meets $OA$ at $A_1$. Let $O_1$ be the center of the circle $(...
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90
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5
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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 $\...
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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 ...
- 45k
7
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A problem of four conics
I found a remarkable theorem of four conics as follows some years ago. But it has no proof; I am looking for a proof:
Theorem: Take three conics. Suppose that each of them touch a fourth conic at two ...
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2
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305
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Connecting a compact subset by a simple curve
Let $K$ be a compact subset of $\mathbb R^n$ with $n\ge 2$ (say if you like $n=2$, which is possibly sufficiently representative).
Q: Does there exist a closed simple curve $u:\mathbb S^1\to\mathbb R^...
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2
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Interpolation splines of bounded curvature
Given $n$ points $p_i=(x_i,y_i)$ on the [Euclidean] plane, and a positive real number $\rho$. Can we have a polynomial spline (e.g., natural cubic spline) passing through all these points, such that: (...
- 63.9k
3
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1
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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 ...
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4
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1
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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 $\textit{F}$ ...
- 45k
2
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1
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123
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Segments on a closed convex plane curve
Is it true that there does not exist a closed convex plane curve containing an infinite number of segments, belonging to distinct lines each?
- 39.9k
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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 ...
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1
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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 ...
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9
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1
answer
1k
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A chain of six circles associated with a conic
I found this problems three years ago. But I never have been a proof. Recently I posted in math.stackexchange.com. I am looking for a solution of the following problems:
A chain of six circles ...
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