# 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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459 views

### 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: ...
221 views

### 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 ...
460 views

### How to sample points in hyperbolic space?

Hyperbolic space in the Poincaré upper half space model looks like ordinary $\Bbb R^n$ but with the notion of angle and distance distorted in a relatively simple way. In Euclidean space I can sample a ...
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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 ...
210 views

### 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 ...
634 views

### Surface Mesh Library

I'm thinking a bit about the Front Tracking method used for simulation of Two phase flow with sharp interfaces. The literature tells me that the main issue is the surface mesh representation (...
612 views

### 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 $\beta$...
321 views

### Monte Carlo approximation of PI

I'm trying to understand how to compute the value of Pi by means of the Monte Carlo simulation. I have a circle inside a square where the sides of the square are tangent to the circle. As data I have ...
287 views

### How to fill a 2D set over a cartesian lattice with as few rectangles as possible?

Suppose I have a black and white image (composed of binary pixel values in a 2D cartesian array) that contains an irregular, nonconvex shape. Let's further suppose that the shape is one connected ...
243 views

### 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, ...
1k views

### Minimal surface solution in Python

Note: this question was also posted in StackOverflow and math.stackexchange. I have a set of 3D points defining a 3D contour, as shown below. The points in this contour lie in their best-fit plane ...
205 views

### Distance between points

I am wondering how can I solve following problem. Arrange randomly $n$ points inside a square of side $a$ under the condition that the distance between any two points may not be smaller than 1. I ...
1k views

### Looking for a library or algorithms to perfom clipping 3D unstructured meshes by a set of surfaces

We have a 3D (volume) unstructured, possibly hybrid, degenerative irregular mesh data structure that we are capable of generating (mostly composed of hexahedra and general polyhedra, using a mix of ...
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### Application of an orthogonal matrix to a 3D configuration of point

Suppose a 3D configuration of points is given, $X\in\mathbb{R}^{n\times 3}$, and a matrix $Q\in\mathbb{3\times 2}$, with orthonormal columns. Now, suppose a mapping to 2D is obtained as $$Y=XQ.$$ ...
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### Testing a simple polygon for monotonicity in linear time question

I'm looking for the algorithm of Preparata and Supowit for testing a simple polygon for monotonicity in linear time. I've found it referenced in many textbooks but I can't find the algorithm itself. ...
48 views

### 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 ...
1k views

### Algorithm to determine if two polygons intersect

I'm working on an algorithm which should check if two polygons, described by their vertex coordinates, are: one inside the other, are intersecting or are separated image below describe this three ...
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### Shape measure for C-shaped objects

There are many well defined measures for many basic geometrical objects such as rectangularity (area coverage of minimum bounding rectangle), triangularity (area coverage of minimum enclosing triangle)...
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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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### Geographic distance between two regions

I am currently trying to calculate the geographic distance between two regions as I want to correlate it with their similarity of another aspect (e.g., similarity in word usage). Currently, I have ...
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### Inclined plate capacitor grid/ mesh

You can calculate the electric potential over every point in a defined space by solving Laplace's equation. To do this in a computer program you set up an 2-d array/ matrix and loop the internal ...
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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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### 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 ...
477 views

### Fitting orthogonal planes to a point set

I have a set of 3d points to which I want to fit two planes. I know the assignment of points to the planes so I don't need any RANSAC or similar. Currently, I'm using a PCA-based approach to fit two ...
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### The closed form solution of a point on a plane

Given a plane in 3D Euclidean space is $\pi$: $ax+by+cz+d=0$ and a point $P$:$(X,Y,Z)\in \mathbb{R}^3$. Find a point $Q:(X^*,Y^*,Z^*)\in \pi$ such that: Q= \arg\min\limits_{Q^*\in\pi}\left\|P-Q\...
245 views

### Algorithm to compute the intersection of meshlines with a boundary

I need a program or an algorithm that computes the intersection of a mesh and a boundary. The mesh is structured orthogonal in nature and the boundary is a circle (for example). This will be used ...
67 views

### 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 ...
56 views

### Geometric interpretation of lemma

I am currently studying eigenvalue problems. I already worked through the Minimax-principles, seen why $\lambda_{h, m} \geq \lambda_m$, when comparing the eigenvalues of a discretization and the ...
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### Convex Polygon Intersection

Determining the intersection of two convex polygons is one of the fundamental problems in computational geometry . I'm asking for an algorithm having: INPUT: Given two convex polygons P and Q in 2D (...
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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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### Ideas on how to search nearby geospatial data fast

I am looking at a very simple problem, but can't quite find the best solution. I need to accept a lat/lon coordinate and based on that coordinate find all the points within roughly ~1km (accuracy is ...
144 views

### 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 ...
64 views

### Layer on which ball belongs in tetrahedron

What is the most computationally efficient way to find the layer on which a ball (i) belongs when arranged in a tetrahedron or 3 dimensional triangle with a triangular base. The ball on the top layer ...
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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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### Closed-form Jacobian of se3 element w.r.t. 6-dof motion

Let $A$ and $B$ be two rigid transformations in 3D space that transform things from global to local coordinates. Let their relative transformation be expressed by $W=A*B^{-1}$. $W$ can also be ...
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### Detecting and joining series of line segments that run along each other

Given: Several circular series of map GPS coordinates for several bus routes. The GPS coordinates are not all equal when they run along the same road. The number of GPS coordinates for a single bus ...
85 views

### Principal Components Analysis Not Behaving as I Expect it to

I have a bunch of points in $\mathbb{R}^3$ that I would like to translate and rotate so that their center is at the origin and the variance along the $x$ and $y$ axes are maximal (greedy, and in that ...
143 views

### Is it possible to generalize the two view Sampson error to multiple view cases in computer vision?

In multiple view geometry of computer vision, there is a geometric error called Sampson error which is very useful in the nonlinear estimation of fundamental matrix....
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### Environment for implementing/testing Computer Graphics algorithms [closed]

I need to code up a computer graphics algorithm for Surface Registration. Briefly surface registration is the process of finding "optimal" one-one correspondence between surfaces, where the meaning ...
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### Pixel-To-Angle Transformation in Camera Image

I'm trying to localize points I see in a camera image in terms of azimuth and elevation and match points between shots. Individual shots should differ only in rotation around the camera's center (...
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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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### 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 ...
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 ...