Questions tagged [geometry]
Geometry is a branch of mathematics. Geometry studies the spatial relationships and forms of objects, as well as other relationships and forms, similar to the spatial in its structure.
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How to correctly discretize volume elements in different geometries?
I am solving a reaction-diffusion problem in one dimension for a catalyst particle to get the internal effectiveness factor ($\eta$),as given below:
$$ \eta = \frac{\int_0^{V_p}{R_i\ dV}}{R_i^{surf}\...
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What exactly is a "unit-torus"?
I've seen references to the "unit torus" in papers such as this (Start of Sec 3.3, page 5). So, what exactly is a unit torus? Is it just a square or cube in d-dimensions with periodic ...
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How do you build a polyharmonic discrete system?
Polyharmonic equations, to my understanding, are defined as:
$$\Delta ^k u = 0$$
i.e. one repeatedly applies the laplace operator to the function a certain number of times and the result must be 0.
...
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Computing discrete laplacian matrix for mesh fairing
I asked this question on the math stack exchange and got an answer, but I am just as utterly confused as before. My fundamental goal is to actually construct the matrix, that is, a series of steps I ...
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Constructing generalized Laplacian matrix?
I am staring intently at this paper by Botsch and Kobbelt.
In particular, I want to make the matrix specified in equation 5. I am trying to understand the specific computations I must instruct a ...
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How to find the formula of a projected circle in a pencil of conics structure?
Hi this is my first question on the platform so feel free to comment if I have a mistake regarding the question.
I'm working on an ellipse detection scheme in which I have markers consisted of 3 ...
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Algorithm for 1-dimensional minimal surfaces
Consider a set of points. For simplicity, let's say that those are 2D points (although the problem works in higher dimensions as well). The goal is to find the minimum possible length of a connected 1-...
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Partial derivatives for triangular meshes (in 3D)
A grid offers an obvious definition for the partial derivatives at a grid point, given
$x$ the value of a point $p$ in an $n$ dimensional grid, the forward partial derivative that point for coordinate ...
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Finding maximums in mesh of graph?
I have a triangle mesh which is an approximation of a smooth graph. i.e. a scalar function of $xy$.
I am interested in finding extrema. One naive way I did it was to look at some number of points ...
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Suggestions for libraries that can numerically compute geodesics from a given Riemannian metric?
I am dealing with a non-trivial Riemannian metric $H$ defined on a particular subset of Euclidean space ($E \subset \mathbb{R}^n$). I was able to show the Riemannian manifold $(E,H)$ is geodesically ...
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Computing numerical derivatives
I am trying to create a sweeping surface, for which I need the frenet frame of a curve. I am trying to compute this for arbitrary curves but for testing I am just using the parametric unit half circle....
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Open source implementations of the medial axis transform for vector shapes
Are there any open source implementations of the medial axis transform for vector shapes?
I have searched without finding any useful results. It seems that CGAL library doesn't have it implemented nor ...
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Difference between Numeric, Combinatorial, and Geometric Computing
In the paper [1], author has discussed a distinction between the 3 types of computations: numeric, combinatorial, and geometric. The author says that Geometric computation is one that has elements of ...
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Min supporting line of a set of points
I am following along Rourke's book and I am trying to do the excercies mentioned in this SO post:
Min supporting line for a set of points
Design an algorithm to find a line 𝐿 that:
has all the ...
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Minimum distance from point to surface
I’m looking for code that is well-suited to solving a fairly simple minimization problem:
I have a reference point $\mathbf p$ in 3D space, and I want to minimize $\|\mathbf x - \mathbf p\|^2$ subject ...
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How to do a parametric study of arbitrary 2D surface?
I need to do a parametric study of the performance of a room heater on different rooms by simulating the temperature distribution in there. The problem here is that the rooms are not simple rectangles ...
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Finding weighted average of curves
This is related to my previous post here
I have a dataset with values of multiple curves. An example plot is shown below.
I want to scale the curves (move up/down) so that all curves overlap.
The ...
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Approximating the boundary between two sets of points (in 2D): Fitting a region
Given two sets of points $p_{\text{in},i}$ and $p_{\text{out},j}$ inside and outside of what I intuitively call a "region", I would like to estimate and describe the boundary of this region. ...
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Computation of the tensor of curvature on surface mesh
Is there a formula which enables the computation the tensor of curvature knowing the following at each vertex and cell of a triangulated mesh:
Normal vector
Two arbitrary vectors in the tangent space
...
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Metric Space for Direction Cosine Matrix?
I work in the medical field. Sometime we receives MR images that have been acquired along the same direction, however when looking up the direction cosine matrix, the values are slightly different (up ...
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Calculating versors of a plane from the normal versor
I'm trying to calculate the 2 perpendicular versors (unit vectors), $\vec{n_1}$ and $\vec{n_2}$, that define a plane whose normal versor (unit vector) is $\vec{n_n}$.
For example, assuming that the ...
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Algorithm to merge two polygons (using connectivities)?
I am struggling with implementing an algorithm that does one simple thing:
Consider two polygons (one can just draw any two polygons and number their vertices), whose connectivities in a node list are:...
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Is there a source/cookbook of equations that approximate geometric shapes?
I'm numerically modelling flows around various geometric 2D shapes. Is there a good source/cookbook of equations that approximate these? Some examples are
Rectangle:
$(x-a)^n+(y-b)^n < r^n$ where ...
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Generate random smooth 2D closed curves
I would like to know how can I generate a collection of random 2D closed smooth curves.
I thought about generating a random 3D surface with random peaks, and then intersecting the Z=0 plane with it, ...
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What are some algorithms to calculate the width of an arbitrary polygon when a bounding box approximation is inaccurate
What are some alternative algorithms to creating a bounding box for finding the max width of a concave, simple winding polygon, like the one in the below image? I prefer solutions that are more ...
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Finding points inside cells of power (generalized Voronoi) diagram
Suppose we have a set of points $p_1,\ldots,p_n\in\mathbb R^d$ as well as a set of weights $w_1,\ldots,w_n\in\mathbb R$. Recall that the power cell associated to the pair $(p_k,w_k)$ is given by:
$$\...
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How do I find the portion of a cell/voxel lying within a defined surface?
We have a 3-dimensional grid of voxels (or cells), with individual voxels being of volume $dx\,dy\,dz$ where $dx=dy=dz=1$.
A cone-like surface is defined by some function, $z = f(x, y)$, which in ...
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Is the similar subdivision of a delaunay mesh still delaunay?
I have a delaunay triangulation for a 2d box with say an airfoil inside. If I uniformly refine this mesh by subdividing each triangle in the mesh into 4 triangles by halving each edge, is the ...
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Linear algebraic research direction that's not to do with differential equations and physics?
So I've found some interesting linear algebraic research areas that's both pure-ish, with a numerical bent to it, too -- e.g. inverse eigenvalue problems have both interesting theoretical and ...
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How to find the nearest point inside a list in a given direction
Being $\bar{\mathbf{x}} \in \mathbb{R}^3$ a point and $S =\{\mathbf{x}\}_{i=1}^N \in \mathbb{R}^3$ a sample of N points. I am looking for a simple algorithm to determine the nearest point in $S$ in ...
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Calculating depth mask from different lighting
I have a object which is static, the camera is static and light source is moving. How can the depth mask be calculated ?
Concept is to use - calculate height from shadow length
Lets imagine a have ...
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Ordering points from X Y coordinates
I have series of points extracted from a regular grid, with their X/Y coordinates. A previous algorithm (that I cannot modified!) output a list of these coordinates, but the ordering of these point is ...
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projective reconstruction from orthogonal views
This is a problem from projective geometry. Suppose I have a vector $z \in R^k$ of unit length $\| z \| =1$ inside a $k$-dimensional hypercube. I don't know its value but do know its projection upto ...
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Compute affine transformation between two sets of points
Consider two sets of points $P = (P_1, ...,P_n), \ Q = (Q_1, ..., Q_m) $ included in $\mathbb{R}^3$.
I'm looking to compute an optimal affine transformation that "maps" $Q$ to $P$, although the sets ...
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Simultaneous update to barycenters
Suppose a tiling is given in 2D (an embedding of a planar triangulated graph), with all faces convex.
Now suppose one moves each point, one by one, to the barycenter of its neighbors. I think that ...
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Space covering optimization
I have the following problem:
In the space $E=\{1, 2, \dots, N_x\} \times \{1, 2, \dots, N_y\}$, I want to define $N_R$ rectangles $R_k=\{x_k^0, \dots, x_k^1\}\times\{y_k^0, \dots, y_k^1\}$ which ...
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Expanding Winding Number algorithm to arcs
I have a problem that I have been attempting to solve for a few days now. I was wondering if I would be able to get some assistance from the community.
In order to detect if a point is in a polygon, ...
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Vector characterization of cylinder displacements in a box
We have a cylinder of length $l$ (in units of its radius $d,$ as basic unit of length set to $d=1.$) in a box, and we consider an orthonormal Cartesian coordinate system with its origin placed at the ...
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Is it valid to assume the center of a bounding sphere to be also the center of the bounding box?
Computing an axis aligned bounding box of a point set is trivial. Computing a bounding sphere of a point set is also trivial when the center is known. Computing the center of the bounding sphere is ...
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Rotate a vector by a randomly oriented angle
We start off with a unit vector $\mathbf{v}$ randomly oriented in 3D space and we want to generate another unit vector $\mathbf{w}$ so that
$$ \mathbf{w}\cdot \mathbf{v} = \cos \beta $$
where $\beta$...
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Commonly-used metrics to quantify the irregularity of a triangular mesh
Say you have a triangular mesh on a flat plane.
This has been drawn to eventually solve some problem in mechanics, for example.
A mesh of equilateral triangles is the best inasmuch as the distances ...
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Matrix Decomposition of Conics
I was reading about ellipse-ellipse intersection and I came across this article:
https://math.stackexchange.com/questions/679622/intersection-between-conic-and-line-in-homogeneous-space/867428#...
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Distirbution of Points along a Line
I am facing the following problem:
Given is a line of length $L$ which I want to split into $N$ segments. The lengths of the first $(s_1)$ and last segment $(s_N)$ are given.
You can assume that the ...
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Gaussian geometry optimisation: molecule is getting dissociated into sub group?
I was trying to optimise CdSe (Cysteine) molecule using a semi-empirical method in Gaussian 09 (and gaussView) for a preliminary study of quantum dots. But it seems as the number of iterations ...
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Fitting Implicit Surfaces to Oriented Point Sets
I have a question regarding quadric fit to a set of points and corresponding normals (or equivalently, tangents). Fitting quadric surfaces to point data is well explored. Some works are as follows:
...
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Finding smallest cube in $\mathbb R^n$ that contains intersection between two regions
I don't think this is a pure math problem, so I post it here.
Assume we have two regions in $\mathbb R^n$:
$$
\lbrace x : a \leq x \leq b \rbrace\\
$$
and
$$
\left\lbrace x : \sum_{i=1}^n |x_i | \...
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How can I find a line segment with the most intersections along with the coordinates of the intersection points?
There are $n$ points in a 2-D plane and each is given by its $x$ and $y$ coordinates. They are stored in an array in an ascending order with respect to $x$.
All points are connected together by line ...
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Does some form of documentation of GMSH exist?
I am looking to implement GMSh into a simualtor that I am going to create. I am looking to integrate the geo, mesh, and post processor modules.
However, looking online, it appears the documentation ...
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Fast comparison of line segments lengths
I have two line segments given by their endpoints $(a_1,a_2)$, $(b_1,b_2)$ in $R^3$ and want to know if they have the same length (up to some error), so that the naive test looks like
$$|\, \Vert a_1-...
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Angle of rotation at a point in a deformed triangle
I have a 2D triangle which deforms with each vertex moving by some small ($\sin(x) \approx \tan(x) \approx x$) displacement vector. The displacement of any point in the triangle is linearly ...
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