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11 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 ...
rchilton1980's user avatar
  • 4,906
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 ...
Juan M. Bello-Rivas's user avatar
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 ...
Tolga Birdal's user avatar
  • 2,229
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 ...
Wolfgang Bangerth's user avatar
6 votes

Minimum distance from point to surface

You could try with gradient projection, here is a quick implementation in python: ...
Marko Lalovic's user avatar
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) ...
Wolfgang Bangerth's user avatar
5 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 ...
BrunoLevy's user avatar
  • 2,315
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 ...
Darren Engwirda's user avatar
5 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 ...
gipouf's user avatar
  • 203
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 ...
Richard's user avatar
  • 3,971
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://...
xue2sheng1's user avatar
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 ...
Tolga Birdal's user avatar
  • 2,229
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)...
Lutz Lehmann's user avatar
  • 6,109
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 ...
Richard Zhang's user avatar
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 ...
Joseph O'Rourke's user avatar
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]...
Wolfgang Bangerth's user avatar
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. ...
Abdullah Ali Sivas's user avatar
3 votes
Accepted

Partial derivatives for triangular meshes (in 3D)

I will give an answer for the two dimensional case. The extension to three dimensions is straight forward. Consider a scalar field $f$ as a function of $f(x(\xi,\eta),y(\xi,\eta))$, where each element ...
ConvexHull's user avatar
  • 1,334
3 votes

Partial derivatives for triangular meshes (in 3D)

This may or may not help you, but with practice it is often easier to work discretely with integral rather than differential calculus. There is then no need to invoke arguments in which you approach ...
Philip Roe's user avatar
  • 1,154
3 votes
Accepted

Computing discrete laplacian matrix for mesh fairing

I checked your solution, it's wrong. I ran the following MATLAB code: ...
lightxbulb's user avatar
  • 2,162
2 votes
Accepted

Distirbution of Points along a Line

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

Gaussian geometry optimisation: molecule is getting dissociated into sub group?

I don't fully understand where the problem is. Where the bonds are supposed to be, S-Se bonds? Like this, those distances don't seem much longer than Cd-S. In any case, you should not be troubled by ...
Anon's user avatar
  • 121
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 ...
Richard's user avatar
  • 3,971
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 ...
philm's user avatar
  • 499
2 votes

Minimal surface solution in Python

You can use FEniCS: ...
Nico Schlömer's user avatar
2 votes

Rotate a vector by a randomly oriented angle

If you want to use a single random number, you first need to find "a" solution (one vector w that meets the criterion), then rotate that vector about $\mathbf{v}$ by a randomly generated angle. We ...
Floris's user avatar
  • 243
2 votes
Accepted

Vector characterization of cylinder displacements in a box

You're correct, if the orientational vector is unitary. Otherwise you must calculate the unitary vector in the direction of $\vec{O}_1$ and then perform the projection (just divide by the norm of the ...
The Doctor's user avatar
2 votes
Accepted

Is it valid to assume the center of a bounding sphere to be also the center of the bounding box?

The book "Real-Time Collision Detection" suggests that the AABB centroid is a decent approximation to the bounding sphere. In fact, the AABB centroid is much more accurate than the center of mass of ...
Tyler Olsen's user avatar
  • 1,512

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