All Questions
Tagged with geometry optimization
855
questions
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Find Biggest Rectangle Inscribed in Overlap of 3 Equidistant Circles
I'm wondering if there's an optimization or geometric approach to find the maximum area of a rectangle inscribed in the overlap of 3 equidistant circles. The circles have centers $B_1$, $B_2$, $B_3$ ...
2
votes
0
answers
53
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Intersect polytopes defined by half-planes
Is there a quick way to find this region or its convex hull? $N$ is large, $M=d=5$ or so
$$
\bigcap_{n=1}^N \bigcup_{m=1}^M \{x \in \mathbb{R}^d :a_{nm}'x \geq 1\}
$$
The slow way to do it (the only ...
- 4,876
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1
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29
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Shape optimization for a given positive function
Consider a positive function $f(x,y,z)$ in $\mathbb{R}^3$. Consider also a bounded region in the space, say a ball $B(0,R)$. I want to find
$$ \min_{\Omega \subset B(0,R)} \int_\Omega f \text{ with }...
- 3,366
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100
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+200
Improving Ellipsoid Theorem $t+\mathcal{E} \subseteq K \subset t + c\cdot n \mathcal{E}$
Setting:
Definition (ellipsoid): An ellipsoid is $\mathcal{E} := \{y \in \mathbb{R}^n : ||A^{-1}(y-t)|| \leq 1\}$, i.e images under a given $A$ of full rank of the unit closed ball of $\mathbb{R}^n$.
...
- 5,164
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61
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optimization, geometry, integration, analysis [duplicate]
Given a sphere, a straight tube with circle cross section is bored out through the center. For a separate but same sphere, a straight tube with square cross section is bored out through the center. ...
- 37
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81
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Find the point $P$ on an ellipse such that $\overline{AP} + \overline{BP}$ is minimum for given points $A$ and $B$
An ellipse in $3D$ space is specified in parametric form as follows:
$ E(t) = C + V_1 \cos t + V_2 \sin t $
In addition, two points in space $A$ and $B$ are given. What I would like to find is the ...
- 24.2k
1
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0
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29
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Upper bound for $\sum \tan(\theta/2)$ subject to condition that $\sum(\theta)\leq 2\pi$
I wish to find an upper bound of $\sum \tan(\theta_i/2)$ subject to condition that $k=\sum \theta_i\leq 2\pi$. We may also suppose that each $\theta_i$ is within the range $(0,\pi)$. Or, even better ...
- 2,280
1
vote
1
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43
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Finding maximum Area of a convex quadrilateral inside a unit circle
Here's a question of my own:
• You have a unit circle.
• You want to place two points inside the circle.
• Now from each point you connect them to the nearest point on the circumference.
(Now you have ...
- 388
1
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0
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38
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Shortest path to connecting all vertices of triangle [closed]
Given a triangle $\Delta ABC$ on $\mathbb{R}^2$, what is the shortest path connecting $A$, $B$, and $C$ provided the path can pass through anywhere on the plane?
Is it guaranteed that such a path with ...
- 25
1
vote
2
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76
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What is the largest ellipse of given eccentricity that can be inscribed into square?
If eccentricity of ellipse is known, what is the position of the biggest such ellipse that can be inscribed into square? I could find the answer on the web – the biggest ellipse is positioned with its ...
- 139
3
votes
4
answers
165
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How to Minimize $PC + \frac{1}{2}PA$ for a Point on a Circle Geometrically?
Given the points $ A(1, 0) $, $ B(5, 0) $, and $ C(0, 5) $, a circle is drawn with center at point $ B $ and radius 2. Let $ P $ be a moving point on this circle. I need to find the minimum value of $ ...
- 381
2
votes
1
answer
57
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Shortest path on the surface of a cylinder between given points $A$ and $B$
Suppose you have the cylinder
$ x^2 + y^2 = R^2 $
And points $A = (R, 0, 0)$ and $ B = (0, R, h) $. Find the parametric equation of the curve of shortest length connecting $A$ and $B$.
My attempt:
If ...
- 24.2k
1
vote
1
answer
31
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What is the minimal triangle that can be inscribed into equilateral triangle, while maintaining minimal distance between vertices of triangles
In equilateral triangle $ABC$ with side length $1$ inscribed another triangle $DEF$, so that vertices of $DEF$ belong to different sides of $ABC$ and vertices of $DEF$ are not closer to vertices of $...
- 139
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1
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40
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Volume of a cylinder inscribed in an octahedron of edge 1cm
I am trying to find the maximum volume that a cylinder inscribed in an ooctahedron of edge 1 cm can have. Given that the cylinder is on a diagonal of the octahedron. With some calculus and geometry we ...
- 349
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1
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25
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Minimizing the resultant of two vectors where one vector is magnified by some $b$
Consider two vectors $\vec{r},\vec{s}$ in $ℝ^3$ with a constant $b$ chosen such that the length of the resultant $\vec{r}$ and $b\vec{s}$ is minimum. Prove that this occurs when $(\vec{r}+b\vec{s}),b\...
- 767