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 ...
- 3,984
10
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 ...
- 4,634
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 ...
- 2,169
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 ...
- 52.4k
6
votes
Minimum distance from point to surface
You could try with gradient projection, here is a quick implementation in python:
...
- 265
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 ...
- 516
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 ...
- 4,571
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) ...
- 52.4k
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 ...
- 3,596
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
$$ a < \...
- 11.4k
4
votes
How to sample points in hyperbolic space?
I'm in the middle of doing this for myself. I think the most appropriate analogue to the Gaussian would be the heat kernel in hyperbolic space. Fortunately, this has been figured out before: https://...
- 41
4
votes
Shape measure for C-shaped objects
How about something like radial integral channel features?
Affordable person detection in omnidirectional cameras using radial integral channel features, Barış Evrim Demiröz, Albert Ali Salah, Yalin ...
- 2,169
4
votes
Commonly-used metrics to quantify the irregularity of a triangular mesh
I do not think that there exists an answer to this question in general, because it all depends on the intended use for the mesh. For instance, if you are doing computational fluid dynamics, you may ...
- 2,285
4
votes
Accepted
Computing numerical derivatives
The unit tangent vector is $\hat t(x)=\frac{C'(x)}{\|C'(x)\|}$. The normal direction is the derivative of this unit tangent, not just the second derivative
$$
n(x)=\hat t'(x)=\frac{(\|C'(x)\|^2I-C'(x)...
- 4,829
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 ...
- 543
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 ...
- 336
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, ...
- 4,571
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 ...
- 2,435
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 ...
- 173
3
votes
Accepted
Algorithm to merge two polygons (using connectivities)?
Easy:
Decompose the polygons into (unoriented) line segments each of which is sorted by vertex index:
$$
A = \{[1,2], [2,3], [3,4], [4,5], [1,5]\},
\\
B = \{ [1,6], [6,7],[7,8],[3,8], [2,3], [1,2]...
- 52.4k
3
votes
Approximating the boundary between two sets of points (in 2D): Fitting a region
I don't know if the following is a good idea. But it is an idea, and I hope that it helps.
This problem can be recast to "find a function $f: \mathbb{R}^2\to \mathbb{R}$ and $z\in\mathbb{R}$ s.t. ...
- 2,321
2
votes
2
votes
Distance between points
The problem of finding the smallest $a$-square that can contain $n$ points whose distance is not less than $1$ is equivalent to the circle packing in a square problem, see e.g Equivalence between ...
- 3,809
2
votes
Fast comparison of line segments lengths
Generally speaking, the $\mathtt{sqrt}$ function is going to be the slowest part of that. Fortunately, if their lengths are the same, then the squares of their lengths are also the same. Therefore, ...
- 1,522
2
votes
Angle of rotation at a point in a deformed triangle
Given a small (infinitesimal) 2D displacement field $u = (u_x, u_y)$ the infinitesimal counterclockwise rotation angle $\theta$ is simply
$$
\theta = \frac 12 \left ( \frac{\partial u_y}{\partial x} ...
- 3,809
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 ...
- 2,169
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 ...
- 3,596
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 ...
- 489
Only top scored, non community-wiki answers of a minimum length are eligible