# Questions tagged [computational-geometry]

Questions on constructing graphical objects using relatively complex computations relating to the mathematical structures defining those objects. Examples include convex hulls, Voronoi diagrams, Delaunay triangulations, mathematical constructions, symmetries, genuses of curves and graphs and programmatic constructions of polyhedra.

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### How can we divide an N-dimensional space? [closed]

I am new with geometry and mathematics problems so maybe I'm saying some wrong things. I have an $N$-dimensional space where there is a set of points ($2^N$ points) and each point has a coordinate ...
165 views

### MorphologicalEulerNumber misbehaving in 3D?

Context I would like to compute the MorphologicalEulerNumber of 3D GaussianRandomField as a function of height above a given threshold. Attempt I proceed as follows:...
69 views

### Table of surface Parametric draw

I'm trying to reproduce this kind of graph for a spherical cap, with different span ratios. I tried with this: ...
128 views

### How to use VectorAngle[] in AnglePath3D[]?

Suppose I have a specific angle that I calculate from two vectors, in 3D: ...
111 views

### Problem with RegionMember

I think there is potentially a bug in RegionMember function, particularly in the RegionMember[reg] form. Can you evaluate the ...
92 views

### How to align two ellipsoids along center and long axes?

I'm struggling to understand rotation/translation transforms as applied to things like spheres and ellipsoids. Take the following silly example. Given this pair of random ellipsoids, how do I find a ...
69 views

### Transfer distribution of points on a sphere to an Ellipsoid[]?

I've seen this cool function, which generates a given distribution of points on the unit sphere, known as Dimroth-Watson distribution: ...
69 views

### Computations on differential geometry

I have to do some really long computations of basic riemannian geometry. I am trying to use Mathematica to check them. This is the kind of computations I want to do. Let $M$ be a riemannian manifold ...
50 views

### Grouping vectors into groups of 3 by attribute

Three groups are given, each with eight $3 \times3$-vectors: ...
107 views

### Analogue of SpherePoints in higher dimensions?

I'm looking for an analogue of SpherePoints that works in dimensions higher than 3, has this been created already? A random sample from unit ball will be ...
85 views

### How to generate an heterogeneous mesh

I'd like to generate a mesh with a big variance in cell size. Something like I'd prefer not to rely on a vertex model and perhaps use something simpler, like ...
94 views

### Efficient way to pick up pairwise vertices of certain Euclidean distance in arbitrarily dimensional coordinates? [closed]

I have a grid graph with x and y vertices in each direction. The 2-dimensional coordinates of the graph ...
69 views

### Inner polygon approximation

Suppose you want to perform an inner approximation of a planar semialgebraic set by a finite set of polygons. The following is a quick way to yield an approximation but I am not convinced that it is ...
74 views

### Does DelaunayMesh use exact computations? What is the treatment of non-generic cases?

Computing Delaunay complexes can be sensitive to numerical instabilities, especially in higher dimensions. I would like to know how much I can rely on Mathematica's answers when using ...
289 views

### Generate Aztec triangle of size n automatically？

In the paper titled "Perfect Matchings of Cellular Graphs" by Mihai Ciucu, the Aztec triangle of size n (n= 1, 2, 3, 4, 5, ...) is equivalent to a ...
150 views

### Automatically get points in a given triangular lattice and vice versa？

Given a triangular lattice which grows with number n, as follows I want to list all the connected points in the lattice to form the ...
151 views

### How do you calculate the area of a surface with edges in mathematica

Like the title indicates, how could I achieve this. Normally what I did for a closed surface is as follows: ...
52 views

### Checking Membership of Conic Hull for Multiple Inputs

Suppose I have a given $10 \times 12$ matrix T, a $12 \times 1$ vector Subscript[\[Xi], 0], a $4 \times 12$ matrix ...
41 views

### Determine if intersection of Polygon and HalfLine is non-empty

I have a concrete example: ...
91 views

### "Mesh -> Full" Disables MeshFunction in ParametricPlot3D

Here's my code ...
382 views

### Efficiently compute Minkowski sum of a 2D Region and a Disc of radius r?

Given a 2D Mathematica Region, e.g. A = Region[RegionDifference[Disk[{0, 0}, 2], Disk[{2, 0}, 1]]], how can I grow the region by an arbitrary radius r? For example, ...
334 views

### Voronoi diagram of two data sets

My primary data is ...
49 views

### RandomInstance of GeometricScene producing unexpected result

Consider the following scene as an example: ...
132 views

### Bounded polygons in Voronoi diagram and calculating the areas and the number of sides of each polygon

I have some data as ...
199 views

### Digitizing a geometrical sensor

I want to quantify the flow from photos taken from a sensor such as this one: For a limited number of readings, I can use a plot digitization method. However, I need to do this many times for ...
95 views

### Why the output is not in numerical form in Mathematica 12.2?

Consider the following code: ...
177 views

### How to find the MeshConnectivityGraph[] from a list of independent cell mesh regions?

I'm generating a bunch of points inside a BoundaryMeshRegion. Then I generate the Voronoi mesh of the points, and take the intersection of the Voronoi cells into ...
47 views

### Maximizing the interior angles of a triangle on the Earth [closed]

Gauss once surveyed a triangle in the Harz mountains formed by Inselberg, Brocken, and Hoher Hagen to check if the sum of the interior angles really added up to $\pi$ radians. With the precision ...
121 views

### Determine and plot major and minor axes of ellipse

I have an element (El) constructed from a list of nodal coordinates (Nodes). Note: The list of nodes can be arbitrarily long. I then create a minimal area ellipse that encloses all of the node using a ...
127 views

### problem with DelaunayMesh 3D coordinates

I found DelaunayMesh works fine for 2D. For example, ...
190 views

### colouring Voronoi cells by their properties

I put a couple of previous posts together How can find the 2D Voronoi cell area distribution? Color code Voronoi cell areas depending on number of vertices color voronoi cell based on area to make the ...
48 views

### Fortune's Algorithm (Conditional Step Size in Manipulate)

I am trying to implement Fortune's Algorithm. Here is my code so far. ...
570 views

### Faster "Closest Pair of Points Problem" implementation?

The closest pair of points problem is a common computational geometry problem: given n points, find a pair of points with the smallest distance between them. A naive algorithm of finding distances ...
144 views

### Property "Centroid" of result from ConvexHullMesh[] can not be extracted

As the title describes, specifically, simple code below ConvexHullMesh[RandomReal[1, {10, 2}]]["Centroid"] returns an error in V. 12.2, but it worked in ...
243 views

### Generate random lines that don't be too crowded between the intersections

It's easy to generate random lines, such as this ...
158 views

### How to obtain the adjacency matrix of morphological components?

I am wondering if there's a simple way to obtain the adjacency matrix of the morphological components of a segmented image. Consider the following example (originally, from this question): ...
67 views

### Get coordinates of Translated Polygon

I have many polygons of the following form that I wish to do more complicated processing and analysis based on it's updated location (such as relationships between geometries). ...
129 views

### How to unite polygons of the same color into a single polygon?

Let a partition of a planar polygon into colored polygons be given, i.e. something similar to We know the coordinates of the vertices and the color of each part, e. g. in such a way ...
73 views

### How to create an ImplicitRegion from a list of 2D coordinates [closed]

I have a set of (x,y) coordinates. How can I link them together to form an ImplicitRegion?
123 views

### Adopt Mathematica codes for circumscribed and inscribed ellipsoids to related sets

Users Daniel Huber and Dominic have provided an interesting answer each to the question JohnEllipsoids of constructing circumscribing and inscribing ellipsoids for a certain three-dimensional convex ...
119 views

### Estimation of the expected Euclidean distance between two random points on a unit $n$-hemisphere

What is the best approach to estimate, with Wolfram Mathematica, the expected Euclidean distance (in a $(n+1)$-dimensional space) between two points selected uniformly at random on a unit $n$-...
159 views

### Estimate the expected distance between two random points on the unit $n$-sphere [duplicate]

What is the best approach to estimate, with Wolfram Mathematica, the expected Euclidean distance in a $(n+1)$-dimensional space between two points selected uniformly at random on the unit $n$-sphere? ...
127 views

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### Graph/Construct (John) ellipsoids circumscribing and inscribing a certain 3D convex set

A famous theorem JohnEllipsoids of Fritz John informs us that associated with a convex body are circumscribed and inscribed ellipsoids of minimal and maximal volumes. Now, a body--argued to be convex ...
106 views

### Maximum area of projection of cuboid onto plane

For didactic purposes I solve with Mathematica the following problem (see Vasilyev's box on p. 17): when the area of the projection of a rotating cuboid in the three-dimensional space with coordinates ...
145 views

### Get 3D surface grown by constant distance

There is a nice example of how to generate an isometric visualisation of surfaces which are at a constant distance from a given region. All these examples are just working on one single region. If one ...
183 views

### Exporting a FEM mesh from a parametric surface

How is it possible to export a mesh for suitable finite element analysis (such as Abaqus) from a parametric surface expressions? I'm working on this kind of surface: ...
60 views

### Intersection Region of two Conic Hull

I have the following semi-infinite conic hull, r1=ConicHullRegion[{{0,0,0}},{{-1,0,0},{0,-1,0},{1,0,1}}]; r2=ConicHullRegion[{{0,0,0}},{{-1,0,0},{0,-1,0},{0,1,1}}]; ...