# Questions tagged [computational-geometry]

The study of efficient algorithms and data structures to solve various problems involving point sets, line segments, polygons, polyhedra, simplices, etc.

187 questions
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### Closest interior point on integer grid to a vertex of a convex polyhedron

I have a 3 dimensional convex polyhedron whose vertex coordinates are rational. For one of these vertices, I would like to find the nearest integer grid point (under the Euclidean metric) that is ...
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### Dissipation and symplectic manifolds

I'm working on an API for simulation of port-Hamiltonian systems. As far as I understand it, a Hamiltonian system is symplectic if it is power conserving, and so including resistive elements would ...
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### How to calculate the area of intersection between a 3D volume and a 2D plane

Hello if anyone can offer insight on how to solve my problem that would be great! I am looking to calculate the area of intersection between a 3D volume and a 2D plane. 3D volume: defined by 6 points ...
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### How to treat hexahedral element with shifted hanging node?

When using the Hexpress grid generator one gets hexahedral cells, possibly with hanging nodes. Because of a smoothing step, the hanging nodes can be shifted: they are not necessarily on the straight ...
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### Backward stable projection and normalization of a vector

Given a machine precision unit vector $n$, and an arbitrary vector $v$, I want an unconditionally backward stable method to compute $$f(v) = \frac{v-nn'v}{\left|v-nn'v\right|}$$ In other words, ...
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### exact area resampling [closed]

I do image processing, and right now I need to resample some images taken from slightly different perspectives so I can match up features. The pixel intensities have scientific significance, so I want ...
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### Error in Maple's CellDecomposition Command

I have a simple system that I want to process with the CellDecomposition command of Maple. I don't know why Maple is giving an error here! The code is ...
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### Conservative field mapping between two topologically disconnected surface meshes

Some background: the Front-Tracking method uses a triangular surface mesh to describe the boundary between two immiscible fluids. To deal with the breakup and coalescence of the fluid interface, ...
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### Restrict Voronoï diagram to a polygon

I managed to build the Voronoï diagram of n points using Fortune's algorithm. This gives me a set of half-edges, some of which being infinite (no starting point and/or no end point). I'd like to ...
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### Ray casting algorithm for multiple disjoint polygons is still valid?

We're dealing with country borders, that is the set of multiple disjoint domains that is made of polygons. To extract the different point on the map by a given country we've been said to implement ...
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### Recovering coordinates by eigendecomposition without double-centering

Suppose an Euclidean distance $D\in\mathbb{R}^{n\times n}$ matrix between a set of $n$ objects is given. To obtain inner-products (which will be further be used to recover coordinates), entries of $D$ ...
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### Concave polygon 'hull' finding

I implemented an algorithm to find the alpha shape of a set of points. The alpha shape is a concave hull for a set of points, whose shape depends on a parameter alpha deciding which points make up the ...
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### Random placement of euclidean points with constrained inter-point distances in a fixed area

I'd like to place as many random points as possible in a 2D square $S=[0,1]x[0,1]$ such that the euclidean distance $d$ between any two points $d$ is greater than a given value $b$ (b is small). I'm ...
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### Is there an algorithm to find an almost-convex hull given a tolerance angle?

I'd like to know if there is an algorithm that given a set o points and an angle computes the convex-hull if the angle is $\alpha = 0$ and given an $\alpha > 0$ computes an envelope that follows ...
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### Some good reading on polygon algorithms

What are some good resources (books, articles, sites) about polygon intersection and union algorithms?
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### Why is my lower convex hull extraction algorithm not working?

Recently, I wrote an algorithm to obtain a delaunay triangulation of a random point set in $I=[-10,10]$x$[-10,10] \subset R^2$ by projecting these points onto the 3 dimensional paraboloid $z=x^2+y^2$, ...
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### Sort a cloud of points with respect to an unstructured mesh of hexahedral cells

Question How would you sort a cloud of points with respect to an unstructured mesh of hexahedral cells? Each cell has a centre and a unique label to represent it. There are two cloud points ...
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### Given values on a mesh, what algorithm can I use to construct efficiently level set contours?

I have a mesh, faces $F$, edges $E$, and vertices $V$, and I have a list of predefined level set contours. What algorithm can I use to construct contours in the most efficient manner? A plot of the ...
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### Is there any 2D shape repository?

As far as I see, there are many repositories for 3D shapes. But in FEM and many other applications, a planar mesh domain is also very common. However, I did not find a mesh repository specially ...
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### Shape regularity in higher dimensions

In Finite Element theory, and other methods in scientific computing for PDEs, one uses meshes which fulfill several regularity criteria, many of them being equivalent. It is of interest to have ...
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### How to efficiently determine the intersection of a vertical cutting plane with a mesh

I have a list of vertical cut planes, and I have a polygonal mesh ( it's a 2D+0.5D mesh, something like a 2D mesh with an extra dimension, $z$ attached to each point). One can assume that the mesh ...
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### How are the Voronoi Tesselation and Delaunay triangulation problems duals of each other?

I have always been told that the Voronoi diagram is the dual of the Delaunay triangulation problem. In what sense can they be duals of each other? I thought that dual problems (i.e. in linear ...
I know that delaunay triangulation can be extended into arbitrary dimensions by solving the convex hull problem in $(p+1)$ dimensions and projecting the lower hull into dimension $p$ to obtain a mesh ...