0
votes
0answers
34 views

Farthest vector direction relative to other vectors

I wished to know the cheapest computational means (be it analytical or numerical) to find the vector from origin (normalised or not; I do not care about its magnitude) given any arbitrary set of ...
0
votes
0answers
18 views

Every polyhedron $P \ne \mathbb{K}^n$ equals an intersection of finitely many half spaces.

Currently, I am reading some lecture notes on linear optimisation. I cannot see why the following (seemingly trivial) proposition holds. (How could I understand/proove it?) Every polyhedron $P \ne ...
1
vote
1answer
97 views

The minimum number of circles in order to obtain a COVER of a specific square

Suppose a unit square $X$, with side length $l=1$ as below, which is COVERed by a set $Y$ of circles with the same constant radius of $r=\dfrac{\sqrt{2}}{10}$, where a ...
0
votes
2answers
37 views

Maximum Area of a Triangle when 1 Side, Perimeter Known

This is an example of a "quantitative comparison" question the GRE would test. Suppose the following information is known: one side of a triangle has length 12 the perimeter of the triangle is 40 ...
0
votes
1answer
17 views

Steepest descent direction with surface constraint (geometry problem)

Let say I have a function $f(x,y,z)$, defined on a surface by the level curve $g(x,y,z)=c$. I want to know what is the direction of the steepest descent at a given point, taking into account the ...
2
votes
2answers
51 views

Find an equation for a moving rod

The two endpoints of a 1-metre long rod have an initial position at $(0,0),(0,1).$ The rod slides continuously to the position $(1,0),(0,0)$ sweeping out a region in the positive quadrant. Determine ...
1
vote
1answer
25 views

Maximum area ellipse that does not include a set of points

I'm interested in finding the maximum area ellipse that does not cover some points $\mathbf{p}_i$ and that is centered at the origin. Hence, ideally I'd like to solve this optimization problem: $$ ...
1
vote
0answers
46 views

A point minimizing total great circle distance to a given set of points on a hemisphere

If you have a set of points on a hemisphere, how do you find a point on that hemisphere that has the minimum total great circle distance to the points in the set.
3
votes
2answers
46 views

Largest Equilateral Triangle in a Polygon

Is there an algorithm to determine the largest equilateral triangle in a convex polygon?
0
votes
1answer
30 views

Minimal volume of a tetrahedral

I'm unsure how to solve the following problem: Let $\textbf{p}=(a,b,c) \in \mathbb{R}^{3}$ with $a,b,c > 0$. For $\alpha , \beta > 0$ the equation $$\alpha (x-a)+ \beta(y-b) + (z-c) =0$$ ...
0
votes
1answer
27 views

What is $s$ in s-energy (eg. Riesz s-energy)

I'm trying to understand fekete problems. There is a variable $s$ and a related concept of 's-energy' [1] [2] [3] [4] that comes up repeatedly when borrowing the concept of potential energy to find ...
3
votes
1answer
28 views

Find a maximum triangle that lies on a polyline (with constraints)

If there's a polyline (a GPS track, actually) with a lot of points (could be several thousand), that looks like this 1) How can I find such a triangle with the biggest possible perimeter, that its ...
0
votes
1answer
22 views

Convex hulls for a finite amount of points

I'm trying to understand what a convex hull intuitively is, and say given for a set of points $(x,y)\in\mathbb{R}^2$ how is it generated from these points? I tried reading the wikipedia article and ...
13
votes
1answer
236 views
+50

How fat is a triangle?

The slimness factor of a geometric shape in 2 dimensions is the ratio between the side-length of its smallest containing square and its largest contained square. This is an important factor in ...
9
votes
1answer
178 views

Characterization of sphere.

I'm editing the question because I think the previous formulation was leaving a key element of the problem out and that was making it impossible to answer the question. I tried to update/improve the ...
4
votes
1answer
74 views

Division of plane into equal area regions

We divide a plane ($\mathbb{R}^2$) into infinite number of regions each of area equal $1$. We can use only (one-dimensional) curves which may meet at points. Fix a point $p$ on a plane and consider ...
0
votes
0answers
16 views

Solve for transform of rotating frame to fixed frame given points in rotating frame and a planar constraint

Say there are 2 coordinate systems, with one orbiting around the other. Call one fixed ƒ and the other rotating ρ. The goal is to find the transform between the two frames. What's known is A set of ...
0
votes
0answers
17 views

How to find iso function value points without exploring all points in 2D space

Consider a 2D graph with dim1 and dim2 represented as X and Y respectively. The range of X and Y are 1 to 100. Hence there are 10000 points in the 2D space. Each point in the space is some function of ...
0
votes
1answer
74 views

finding points with maximum distance between them on a circle

I'm a computer science student working on a problem in computer graphics and looking for a formula that can find the x and y positions of a set of N points on the surface of a circle so that the ...
-1
votes
2answers
47 views

Maximize are of rectangle with semicircles on left and right [closed]

There is a rectangle with semicircles on the left and right sides. You know that the perimeter is 100. Maximize the area of the entire shape.
1
vote
1answer
88 views

Find the ellipse inscribed in a triangle having the maximum area

If we have a triangle of sides $a,b,c$, there are infinite ellipsis inscribed in the triangle. How can I find that having the maximum area? Is this ellipse the circle, or in what cases the maximum ...
5
votes
2answers
210 views

Maximal area covered by two triangles in unit circle

What is the maximal area covered by two triangles in a unit circle? There are no restrictions other than that. They can overlap, touch the circle, not touch the circle etc. So far I have shown In ...
0
votes
0answers
31 views

Dual of a polyhedra vs. dual of an optimalization problem

There are lot of fields where the term duality appear. Is there any relationship between dual of an optimalization problem and dual of a polyhedra?
0
votes
0answers
56 views

Maximizing $x\left(\displaystyle\sum_{k=0}^n \sqrt{1-kx^2}\right)$

Here's the motivation for my problem: Consider a right-angled triangle with legs of length 1 and $x$ and on the leg of length 1, and construct another right-angled triangle whose hypotenuse is 1 ...
4
votes
1answer
95 views

10 points inside a square - minimum distance between any of them

A square of side 1 is given, and 10 points are inside the square. If we divide the square into 9 smaller squares, and apply Dirichlet principle, we can prove that there are 2 of these 10 points whose ...
6
votes
1answer
78 views

Platonic solids and charged particles

It is known that there are five Platonic solids: If, lets say, there are 4 particles with the same electricity charge and whose movement is constrained to be on a sphere, resulting forces will ...
7
votes
2answers
131 views

Prove That the Second Moment is Minimized with a Circle Packing

Graham and Sloane studied the problem of minimzing the second moment of disks on the plane, i.e. minimize $$ U = \frac{1}{d^2} \sum_{i=1}^{n} || \mathbf{p}_i - \bar{\mathbf{p}} ||^2 $$ s.t. ...
0
votes
2answers
130 views

Optimization of the surface area of a open rectangular box to find the cost of materials

A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of the box is twice its width. Material for the base costs ten dollars per square meter and for the ...
1
vote
2answers
73 views

Maximize the distance between a point and a bounding rectangle

There are $n$ random points in the $x-y$ plane, whose coordinates are known beforehand. We can use a minimum bounding rectangle (MBR) to bound these points. In this scenario, the MBR can be rotated, ...
3
votes
0answers
42 views

How to find the minimal path between points in a planar set with holes in it?

When I was a commuter student, I would park in a very large parking that that had a set of stairs in a corner that I had to climb. In general, I had to park far away from this corner in an almost full ...
0
votes
0answers
56 views

max and min values on symmetric polytope

Let $-N\leq t \leq N$. Let $A$ be regular $(N-1)$-dimensional simplex with vertices $(t,0, \ldots, 0)\ldots (0, 0,\ldots, t)$ and $B$ be regular $(N-1)$-dimensional simplex with vertices $(t-N+1,1, ...
2
votes
1answer
100 views

How to calculate the point on the sphere that is nearest to some given points on the sphere?

Given some points $X=\{x_i:||x_i||=1,i=1,\ldots,n\} $ located on the sphere, how to calculate the point $\tilde{x}$ on the sphere that is nearest to these given points. That is to say ...
3
votes
1answer
170 views

Finding the shortest path length on a curved surface(hyperboloid)

I wish to find the minimum path length between two points $P_1(\sqrt2,0,-1)$ and $P_2(0,\sqrt2,1)$ on a hyperbolic surface $S =\{(x,y,z)\in R^3\ |\ x^2+y^2-z^2=1\}$ I faintly recall studying ...
2
votes
1answer
143 views

Minimum distance to points in plane

Someone told me that the the following problem is elementary. Given three points $a=(-5,0)$, $b=(0,5)$ and $c=(5,0)$ in $\mathbb R^2$ with Euclidean norm: $$\mbox{minimize}\;\; \; f(x)=\|x-a\| + ...
2
votes
1answer
38 views

Chord Maximisation

I am currently going back through all the "Challenge" questions in preparation for exams, and for this I do not know where or how to start, any hints would be appreciated. Now the way I have ...
2
votes
2answers
184 views

Closest distance between two quadratic curves

I'm having trouble with the following problem : "find the closest distance between $x^2+4y^2=4$ and $xy=4$" I tried to solve using the properties of ellipse and hyperbola, but the relatively tilted ...
1
vote
0answers
30 views

Finding coordinates of nodes in a graph

I have a complete graph in which the edges represent the euclidean distance between the nodes which is known. Assuming a node to be (0,0), I want to find (approximately) the coordinates of other ...
0
votes
0answers
90 views

Maximum area of quadrilateral of given perimeter.

Let $0\lt a\lt b$ (i) Show that among the triangles with base $a$ and perimeter $a + b$, the maximum area is obtained when the other two sides have equal length $b/2$. (ii) Using the ...
17
votes
3answers
301 views

What is the largest circle that fits in $\sin(x)?$

Imagine dropping a circle into the trough of $\sin(x)$. Would it reach the bottom or get wedged between two points on the curve? Depends on the size of the circle. So, what is the radius of the ...
0
votes
1answer
40 views

$|\langle a_i, a_j\rangle|$ for $p$ points on a unit circle.

Is it true that given any $p$ points $a_1, .., a_p$ on a unit [euclidean] circle, there is always a pair $i \ne j$ such that $|\langle a_i, a_j\rangle| \ge \cos{\pi/p}$?
0
votes
0answers
17 views

Dimension of face of a polytope in terms of linear functionals

A face of a polytope is a subset consisting of the points for which some linear functional is maximized. Can one determine the dimension of the face in terms of the linear functionals which are ...
0
votes
1answer
37 views

Highest Volume/Area Ratio

Given a fixed volume of a solid, what would be the shape of such solid that would minimize the its surface area? How to determine it? I thought about it, but I cannot find an algorithm that doesn't ...
2
votes
3answers
82 views

Find out minimize volume (V) of tetrahedral

I have this problem: On space $ (Oxyz)$ given point $M(1,2,3)$. Plane ($\alpha$) contain point $M$ and ($\alpha$) cross $Ox$ at $A(a,0,0)$; $Oy$ at $B(0,b,0)$; $C(0,0,c)$. Where a,b,c>0 Write the ...
1
vote
1answer
16 views

Placing a shape on a grid

I am interested in a certain kind of geometrical optimisation problems. I will illustrate it on a semi-concrete example: You are given a two-dimensional shape, say a polygon, and a rectangular ...
3
votes
2answers
88 views

The smallest quadrangle inscribed in a rectangle

I'm supposed to find a quadrangle of the smallest perimeter possible inscribed in a rectangle. The inscribed quadrangle has each of its four vertices on another side of the rectangle. Let's call the ...
2
votes
1answer
79 views

Shortest polygonal path inside a rectangle

I hope you could tell me if my reasoning is correct. We are given two points $A$ and $B$ inside a rectangle $PQRS$. We create a path $AXYB$ such that $X$ and $Y$ lie on different sides of this ...
1
vote
1answer
50 views

T-shaped polygons

Is there any coefficient that can indicate T-shaped polygons ? Examples of T-shaped polygons:
3
votes
1answer
67 views

Minimizing distance of circles from points without overlapping

I am designing a user interface, and I have encountered the following problem: I have $p_1 ... p_n$ points in $\mathbb{R}^2$, and $c_1 ... c_n$ circles with constant $r$ radius. I want to minimize ...
1
vote
3answers
147 views

Max perimeter of triangle inscribed in a circle

What is the maximum perimeter of a triangle inscibed in a circle of radius $1$? I can't seem to find a proper equation to calculate the derivative.
2
votes
1answer
54 views

How to minimally move circles so that they don't overlap?

You're given a set of circles, all the same radius, residing at different locations in a 2d space. Some circles are in fixed positions. How do you make sure none of them overlap, minimizing the ...