10 votes

Linear algebraic research direction that's not to do with differential equations and physics?

Randomized linear algebra might be something you'd like. It has direct applications in data analysis and is related to several branches of Mathematics such as Geometry (see the Johnson-Lindenstrauss ...
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9 votes
Accepted

Generate random smooth 2D closed curves

Since your figure is a closed loop, its parametric curves $x(t)$ and $y(t)$ must be periodic functions. This suggests one way to generate such figures, by constructing random smooth periodic functions ...
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  • 4,316
6 votes
Accepted

Fitting Implicit Surfaces to Oriented Point Sets

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 ...
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  • 2,139
6 votes

Linear algebraic research direction that's not to do with differential equations and physics?

Here's another interesting connection: There is a research area that looks into the complexity of computing matrix-matrix products from a geometric perspective. It's a totally bizarre connection, but ...
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5 votes
Accepted

Commonly-used metrics to quantify the irregularity of a triangular mesh

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 ...
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5 votes
Accepted

Distance between points

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 ...
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5 votes
Accepted

How to find the nearest point inside a list in a given direction

The question really boils down to how far two points $x,y$ are from each other in direction $u$. This is easily answered: You need to compute the component of $y-x$ onto $u$, i.e., their (signed) ...
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5 votes

Approximating the boundary between two sets of points (in 2D): Fitting a region

Theory Clustering is unlikely to work in this case because your red points are separated from each other by the green points. You could use more clusters, but this will require a lot of manual ...
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  • 3,131
4 votes
Accepted

Fast comparison of line segments lengths

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 < \...
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  • 11.4k
4 votes

How to sample points in hyperbolic space?

I'm in the middle of doing this for myself. I think the most appropriate analogue to the Gaussian would be the heat kernel in hyperbolic space. Fortunately, this has been figured out before: https://...
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4 votes

Shape measure for C-shaped objects

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 ...
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  • 2,139
4 votes

Commonly-used metrics to quantify the irregularity of a triangular mesh

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 ...
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  • 2,205
3 votes
Accepted

Finding smallest cube in $\mathbb R^n$ that contains intersection between two regions

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 ...
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3 votes

Distance between points

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 ...
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3 votes

Minimizing the edge length of a polygon preserving its angles

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, ...
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3 votes
Accepted

Convex Polygon Intersection

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 ...
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3 votes

Linear algebraic research direction that's not to do with differential equations and physics?

For me, a classic problem in linear algebra that got me climbing down a rabbit hole of complex analysis, conformal mappings, and polynomials, was in trying to prove converge rate bounds and iteration ...
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3 votes

Generate random smooth 2D closed curves

What I ended up doing is to download thousands of random images, smooth them with a Gaussian filter, and extract contours at different levels. I took 2-3 closed contours from each blurred image and ...
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  • 163
3 votes

Approximating the boundary between two sets of points (in 2D): Fitting a region

I don't know if the following is a good idea. But it is an idea, and I hope that it helps. This problem can be recast to "find a function $f: \mathbb{R}^2\to \mathbb{R}$ and $z\in\mathbb{R}$ s.t. ...
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2 votes

Minimal surface solution in Python

You can use FEniCS: ...
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2 votes

Distance between points

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 ...
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  • 3,789
2 votes

Fast comparison of line segments lengths

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, ...
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  • 1,522
2 votes

Angle of rotation at a point in a deformed triangle

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} ...
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  • 3,789
2 votes

Fitting orthogonal planes to a point set

Here I devise a novel strategy, based on only 3D points, that I think, would work. I will parametrize a 3D plane by a point $\mathbf{p}$ and its normal $\mathbf{n}$. Imaging you take a pair of ...
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  • 2,139
2 votes

How can I find a line segment with the most intersections along with the coordinates of the intersection points?

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 ...
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  • 3,131
2 votes
Accepted

Does some form of documentation of GMSH exist?

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 ...
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  • 489
2 votes

Fast comparison of line segments lengths

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....
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  • 11.4k
2 votes
Accepted

Distirbution of Points along a Line

You can uniformly distribute points and apply transformation, $$ N_s(\zeta)=1-\zeta\\ N_e(\zeta)=\zeta\\ \phi(\zeta)=x_sN_s(\zeta)+x_eN_e(\zeta)+ \sum_i \alpha_i L_i(2\zeta-1) N_s(\zeta)N_e(\zeta) $$ ...
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  • 906

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