All Questions
16
questions
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60
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Where will a plane intersecting a cone maximize the distance to inscribed spheres? (V. Arnold)
A cone is cut by a plane, making a closed curve. Two spheres, each inscribed into the cone, are tangent to the plane, at points $A$ and $B$ respectively.
Find the point $C$ on the cut line (i.e. the ...
- 4,450
4
votes
0
answers
63
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closed curve mapping problem
I have this interesting, seemingly difficult problem I came across while think about rendering. It seems simple enough that I suspect it might have a name already (and hopefully be solved), but I can'...
- 93
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97
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What is the connection between optimization and symmetry?
I have found that in many optimization problems, there is some sort of "local symmetry" phenomena going on. For example, suppose we have a 'nice' function which has $f'(x)=0$, then we will ...
1
vote
1
answer
164
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constrained differential equation
I'm struggeling with the derivation of the ode equations of forward kinematics of an oriented object. Assuming to be in $R^2$ and using the coordinates $(x_1,x_2,x_3):=(\phi,p_1,p_2)$, where the angle ...
- 165
1
vote
1
answer
137
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How to find geodesics in metric spaces
Let $(X,d)$ be a metric space. Let $x,y\in X$.
How do we generally formulate the problem of finding the shortest path $\gamma :[0,1] \rightarrow X
$ between $x,y$ ? Is it something like
$$\inf\...
- 2,294
3
votes
3
answers
180
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Existence of "checkerboard" grid whose vertices intersect all given points on plane
This is a question about identifying a theorem or maybe even a subject matter. I've asked myself a question, but I don't know where to look for answers. Typing a problem statement in a search engine ...
- 811
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How would you go about proving that the right cone is the most optimal pyramid
Im trying to figure this out. Like you could go one by one optimizing a square pyramid by minimizing its surface area for a given volume. Eventually when you get up to an decagon based right pyramid, ...
- 141
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2
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123
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Can we measure total "closeness" of two geometric objects with analysis?
Say I have one sphere and one plane
$$x^2+y^2+z^2 = 1$$
$$z = 2$$
We can easily calculate closest distance between these. It is easy exercise. Maybe first exercise in differential geometry or ...
- 26.1k
1
vote
2
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77
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Must any shortest line between two surfaces coincide with normals to both surfaces?
In differential geometry I know of a result which says something along the lines of
...
- 26.1k
2
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0
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54
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Identify a point in the disc that is as far away from each point of S as possible Ask Question
I have a question.The question is this.
Say you have a finite set S of n points in a circular disc. Think of the points in S as houses on a circular island. You want to locate a garbage incineration ...
- 121
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145
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How to compute the geometric center of a curved manifold defined by a set of point?
Suppose that I have a set of points E which belong to a manifold. Typically a surface mesh like that.
$$E=\{x_{0},x_1,...,x_n\}$$ where the $x_i$ are the points coordinates in 3D such that : $x_i = (...
- 135
6
votes
1
answer
150
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Minimizing the size of a cylinder sliding in a tube
Lets define a spine curve $\Gamma$ given in terms of its arc length parameter $s$, and a family of closed curves $\Omega = \Omega \left( s \right)$. Lets say $\Gamma$ has a finite length, from $s=0$ ...
- 563
16
votes
0
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2k
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Properties of the projection onto a nonconvex set
Consider a set $\Omega\subseteq\mathbb{R}^n$ being "sufficiently regular", for example being the image of a $C^1$ mapping from $\mathbb{R}^p$ for some $p\ge1$. We may then consider the mapping
$$
g:\...
- 3,146
6
votes
0
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141
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Do balls optimize the boundary area for a fixed volume?
I recently saw this question, asking if the circle is the planar figure which has the least perimeter given the area. As far as I know, this is a classical problem, and the answer is affirmative. I am ...
- 12.6k
5
votes
1
answer
629
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On minimizing the area of an enclosing surface subject to nonnegative Gaussian curvature
This is inspired by this previous question on physical processes that might give rise to convex hulls.
Consider the problem of gift-wrapping a three-dimensional object using an inextensible material, ...
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