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Choose n random points in the unit square. Compute the minimum point-to-point distance, $D$. Re-scale the space by $1/D$. The minimum point-to-point distance is now equal to 1. Edit: The best method to do step 2. depends on many things. If don't have too many points you can just compute (using matlab syntax): dmin=1; for i=1:n for j=i+1:n dmin = ...


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An algorithm for this computation is described in Computational Geometry in C, Chapter 7, Section 6. Code is available at that link. Much of the tricky code is concerned with "degenerate" cases. Here is a nice Univ Montreal web page (Eric Plante) that describes a different algorithm: link.          


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How about something like radial integral channel features? Affordable person detection in omnidirectional cameras using radial integral channel features, Barış Evrim Demiröz, Albert Ali Salah, Yalin Bastanlar, Lale Akarun, Machine Vision and Appliactions 2019 https://www.cmpe.boun.edu.tr/~salah/demiroz19omnidirectional They would allow you to: Specify ...


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You could also rewrite the expression to avoid square roots entirely. Reorder $a,b$ so that $a>b>0$, and let the inequality be $$ \sqrt{a}-\sqrt{b}<\tau. $$ This is equivalent to $$ a < \tau^2 + 2\tau\sqrt{b} + b \qquad\Leftrightarrow\qquad 2\tau\sqrt{b} > a-\tau^2-b, $$ which is in turn equivalent to $$ \Big( a < \tau^2+b \quad\text{or}\...


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As @Nicoguaro and @Paul have said in the comments to the question post, there are a great many ways to do this kind of thing, and I'm not sure if there is a single "best" approach. From a review study of Jonathan Richard Shewchuck at Berkley, an answer is: Please refer to the original document (version 31/12/2002) for symbology, terminology, special ...


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I do not think that there exists an answer to this question in general, because it all depends on the intended use for the mesh. For instance, if you are doing computational fluid dynamics, you may want to have a mesh that is extremely anisotropic near the boundary layer. Now if you are doing computational electromagnetics, the best mesh will be probably ...


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If $n$ is too large and the side length of your squarer $a$ is fixed then there may not be a solution to your problem. If you are willing to increase the size of the square to fit the points then the solution by @DougLipinski will work. If the square size if fixed (which is why I assume you asked the question in the first place), then the re-scale step in ...


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If you multiply the length of every side by a constant then all the angles will be preserved, this just produces a similar polygon. Multiply by a very small constant and you get a very small length, as small as you want, all the way down to zero. You can do this easily if you have the $(x,y)$ coordinates of the vertices of the polygon with perimeter $L$ and ...


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The Haversine formula is really the best way to calculate distances on the Earth unless you have outstanding accuracy concerns. The next step passed the Haversine involves taking into account that the Earth is not a sphere but ellipsoidal and which involves a much more expensive computation. There's a good page with some embedded javascript at http://www....


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What you're describing is a very specific way to solve the Laplace equation: using the Jacobi iteration to solve the five-point stencil when using finite differences. But there are much more general methods that would be difficult to derive from your starting point. A more promising approach is to start with the Poisson equation and ask how it can be ...


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Actually, I found the answer here . Thanks for the help though...


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The boundary of the domain has exactly two types of corners: convex 90 degree corners and concave 90 degree corners. For filling the domain with non-overlapping rectangles, only the concave corners are important. (I don't know how to upload pictures, sorry.) At each concave corner, we must have a "cut" in x- or y-direction. Each "cut" adds one rectangle, ...


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(i) Think of $Q$ to be the first two columns of an orthogonal matrix $U$. The rows of $XU$ are the points viewed in a rotated and/or reflected coordinate system, (i.e., a rotation or a rotation followed by a reflection) and distances are preserved in this transformation. (As 0 is preserved, there is no translation involved.) Discarding the last column of ...


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VTK has many utilities for cutting meshes with planes and implicitly-defined surfaces. It's possible that would work for you. To use it, you would probably feed your whole mesh into a vtkUnstructuredGrid (or similar) and have VTK do all the cutting. I think you will have a harder time if you want control over how topology changes when a cell is cut. A lower ...


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Here is a suggestion. Partition your problem into two parts: (1) Construct the intersection, (2) Find the smallest cube. (1) The intersection is a polytope defined by the union of your inequalities for both shapes. This is known as an H-representation of the polytope. There is software for constructing a polytope from its H-representation. polymake is one ...


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I was surprised for not receiving a satisfactory answer to the question above and my investigations showed me that this, indeed is an unexplored area. Hence, I put some effort in developing solutions to this problem, and published the following manuscripts: T. Birdal, B. Busam, N. Navab, S. Ilic and P. Sturm. "A Minimalist Approach to Type-Agnostic ...


2

I'm using OpenFOAM for a while now (for my PhD as well), and I can only recommend it: It has a huge amount of physical models, and it's written in C++, which means: modularity, encapsulation, automatic parallelism of top level codes, boundary conditions bound to the fields but implemented as a model library, multi physics [6DoF, two phase flow, combustion, ...


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I don't think that you can find the radius $r$ with the informations that you provide. For example, you can imagine having a circle with diameter $d=1$ centered in a square with side $l=1$ a circle with diameter $d=2$ centered in a square with side $l=2$ Both circles will be tangent to their respective squares and both yield $r_a=1$. So, if your data ...


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Suppose, that your circle have unitary radius: $r=1$, then length of square's side equal to $l = r_a \cdot d = 2\cdot r_a$. So, area of circle equals to $S_c = \pi$ and area of square equals to $S_s = 4 r_a^2$. If you "throw" $N_t$ random points inside of square, a part of them (say, $N_c$) will fall into the circle. Relation $\frac{N_c}{N_t}$ equals to ...


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For $d>1$, you need to determine all distances of a simplex consisting of $d+1$ points, and then the distances from each other point to these points. ($d$ points and distances already reduce the possibilities to a finite number, but then one needs additional disambiguation information). This makes a total of $d(d+1)/2+(n-d-1)(d+1) = n(d+1)-(d+2)(d+1)/2$ ...


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The formula from the paper is a factor of $(d+1)^2$ times smaller than the second formula you mention. Since the paper is only citing the equation as a well known fact I am going to have to argue that he simply mis-typed it, replacing the multiplication by $d+1$ with a division by $d+1$. Also, the fact that it says $1/3$ of a distance is required for the 3 ...


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I hadn't thought about it too closely, but I think the basic algorithm would work like this: Find the number of unique edges in either the $x$ or $y$ directions. Take the direction that has the smaller number of edges. Without loss of generality, we assume $x$ here. For each such edge, connecting $(x_{min},y_1)$ to $(x_{min},y_2)$, draw the largest ...


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The problem of finding the smallest $a$-square that can contain $n$ points whose distance is not less than $1$ is equivalent to the circle packing in a square problem, see e.g Equivalence between circle packing and point packing. Info about this problem can be found also in Circle Packing and a list of optimal or best known solution in The best known ...


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Given a small (infinitesimal) 2D displacement field $u = (u_x, u_y)$ the infinitesimal counterclockwise rotation angle $\theta$ is simply $$ \theta = \frac 12 \left ( \frac{\partial u_y}{\partial x} - \frac{\partial u_x}{\partial y} \right ) $$ In general, for $n$ dimensional displacement fields $u\in\mathbb{R}^n$ it can be shown that the infinitesimal ...


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Generally speaking, the $\mathtt{sqrt}$ function is going to be the slowest part of that. Fortunately, if their lengths are the same, then the squares of their lengths are also the same. Therefore, you can just omit that part and compare the squares of the norm. I don't know of any way to make that go too much faster using cross-platform code. However, if ...


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I want to submit a separate answer to this question, because there is a way to automatically simplify such formulas to remove square roots. I will use sage and QEPCAD (both free). The tutorial is here. The inequality $$ |\sqrt{a}-\sqrt{b}|<t $$ is equivalent to the statement that there exist $u, v$ such that $$ \exists u,v: \qquad (u-v)^2 < t^2, \...


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Currently there is a GMSH API in the works: https://gitlab.onelab.info/gmsh/gmsh/tree/master/api Also, there are rumors that there will be a fully documented API by version 4.0. In short, there is no documentation on the functions for GMSH (except for what is in the source all ready). But, they are planning on creating a documented version by GMSH 4.0. ...


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You can use the Bentley–Ottmann algorithm for this. Given a set of $n$ line segments with $k$ intersections, the algorithm can identify all intersections in $O((n+k)\log n)$ time and $O(n)$ space. In cases where $k=o(\frac{n^2}{\log n})$ (that is, cases in which $k$ has an appropriate upper bound) this offers time savings versus a naive $O(n^2)$ algorithm ...


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You can uniformly distribute points and apply transformation, $$ N_s(\zeta)=1-\zeta\\ N_e(\zeta)=\zeta\\ \phi(\zeta)=x_sN_s(\zeta)+x_eN_e(\zeta)+ \sum_i \alpha_i L_i(2\zeta-1) N_s(\zeta)N_e(\zeta) $$ where $L_i$ is Legendre polynomial. So if you set for example $\alpha_0=0$ and $\alpha_1=1$, you will get your distribution, when you apply function $\phi(\...


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