# 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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### Region bounded by planes in terms of inequalities

Suppose I have 5 planes that intersect to give a bounded region in 4D. For example, the planes are x1==0; x2==0; x3==0; x4==0; x1+x2+x3+x4==1; ...
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### Convert ConvexHull in a array to Inequality

This question is in continuation with the previous post Convert ConvexHull to Inequality There are two problems that I faced. Suppose we have two or more convex hull defined using an array R[i], like ...
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### Convert ConvexHull to Inequality

I have the following convex hull. P0={0,0,0}; P1={1,0,0}; P2={0,1,0}; P3={0,0,1}; ConvexHullMesh[{P0,P1,P2,P3}] Now I want the region algebraically as equality ...
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### Surface area of a list of points

I would like to know how I can get the area of the following point list. I am using the following script to make a mesh region and then find the area. However, the area is much larger than the value ...
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1 vote
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### Image Size change with changing ViewPoint

I have the following code to show the intersection of two convex hulls. The plot has no problem when ViewPoint->Default. But when I try to change the viewpoint manually by holding and dragging the ...
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1 vote
108 views

### How to label vertex of a 3D convex hull?

I have the following code to plot a 3d convex hull with 4 corners only. ...
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### A DistanceFunction for use with a mouse in a DynamicModule

I need to have a mouse cursor that identifies points that are close to the mouse position on a graphics plot. The difficulty is that the scaling of the plot makes this difficult. I think I am missing ...
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### RandomPoint inside mesh for walk-on-spheres Monte Carlo PDE solver

I'm trying to understand a Monte-Carlo Laplace/Poisson PDE solver: http://www.cs.cmu.edu/~kmcrane/Projects/MonteCarloGeometryProcessing/paper.pdf This method inspired by random-walks and ray-tracing ...
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### DelauneyMesh in 4d and higher

Is there any way to find the DelauneyMesh in 4D (or higher)? Its documentation implies that it should be possible "A Delaunay mesh consists of intervals (in 1D), ...
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1 vote
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### Periodic Voronoi Mesh in a box

Is it possible to generate a periodic Voronoi Mesh inside a rectangular domain? I would also like to obtain the coordinates of each cell separately so that I can perform further operations on each ...
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### Find smallest distance from a point to a Bézier curve

I wish to find the smallest distance from a point to a curved defined via a Bézier function. I want to do this automatically. For particular cases it is not to difficult. Here is a minimum working ...
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### How to add 'border' to Voronoi diagram

Suppose I have a set of points pts, from which I generate a Voronoi diagram VD: ...
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### Finding shadow area of a 3D object

Given a 3D object, how could one find the area of the shadow? In effect, this is a measure of optical cross section. The ultimate goal would be a table showing latitude, longitude, and, optical cross ...
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### How to speed up finding region centroid and plot3d

I have the following code to make 3D of 9 regions. ...
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1 vote
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### How to find intersection of higher dimensional convex hull?

I have often used ConvexHullMesh to obtain a convex hull for 2-d and 3-d points. But when I try to find it for 4-D points I encounter error even for simple cases. <...
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1 vote
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### Represent a Concave Mesh Volume

I programmatically generate a shape, where the top surface is defined by a series of polygons. I generate all the points of the polygon. ...
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### Periodic boundaries for a 3D walk in a Bravais lattice?

I'm using a Bravais lattice, 'face-centered', which is implemented in Mathematica in LatticeData[]. I want to simulate a random walk over neighboring points using ...
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### Projecting a 3D mesh region to 2D

How do I project a 3D MeshRegion to a 2D MeshRegion (or equivalent for a BoundaryMeshRegion). This question is similar to ...
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### Convex Hull Mess Axis Label

P0={0,0}; P1={1,0}; P2={-1,0}; R[1]=ConvexHullMesh[{P0,P1,P3},Axes->True ] I use this code to draw the convex hull but I want to also label the ...
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### How can I calculate volume of such a MeshRegion Object

The Data is here: meshData === PolygonData is here ...
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1 vote
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### How to use Mathematica to find the coordinates of 7 points?

Could anyone please help me with using Mathematica to find the coordinates of 7 equally spaced points on the unit sphere $S^3$ in $\mathbb{R}^4$? Thanks a lot!
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### Torus Hex Mesh - Need Unique Hex Dimensions

I'm looking to laser cut a bunch of hexagons and assemble them into a torus (using hot glue or tape). How do I get the dimensions of the individual unique hexagons? I've seen Create a torus with a ...
249 views

### Penalty function on discrete mesh using Laplace-Beltrami operator?

Context I am interested in extending to the ill-condionned regime the inversion of linear equations arising from inverting differential equations which have been solved via 0-splines over a mesh ...
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### Finding an instance of a parametric equation that intersects given points

So I have this equation for a circle centered at the origin that has some orientation in 3D space. ...
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### Mathematica Won't Recognize a MeshRegion

I've implemented the code in the top answer of this question on the Plateau problem in differential geometry, and it works with the example shown in Example 1, so there is nothing wrong with the code. ...
149 views

### Minimization problem and Voronoi Mesh

I have a random convex mesh $Q$, made of $n$ polygons, and I want to test how close it is to a Voronoi tessellation. In other words, I'm looking for generators (seeds) $\{(x_i,y_i)\}_{1\leq i\leq n}$. ...
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### Why can't an annulus be mapped as a rectangle

The following code is used to map an annulus to a rectangle: ...
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### Checking if a Graph forms a Mesh

Given a random graph, I want to define a function MeshGraphQ that tests if the graph is a lattice (or mesh) graph. Here, I don't need the tiling to be regular. ...
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### How to visualize a spherical mesh on a squared plane?

I need to visualize the points of a spherical mesh on a squared plane (this question is different than this one, I believe). For instance, let us first generate points ...
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### Efficiently finding spheres furthers apart along and perpendicular to a given direction

Setting I am dealing with a box of spheres, all having the same radius and inserted into a finite box size without any overlaps. Along a given direction $\mathbf v$, I am trying to find: (a) The two ...
243 views

### How to wrap any function/curver around a circle? [closed]

I have a function defined as $y=f(x)$ which i want to wrap around a circle, to get a transformed function $r=f(ϕ)$. It is important that the length of the wrapped curve stay's the same. For two ϕ ...
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### unbounded Voronoi iteration with finite and non-equal point radii

I would like to create a cool differential line growth animation based on this great tutorial. The only missing part is the relax node, but I couldn't find any pointers on the web how to implement ...
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### How to calculate border length between adjacent Voronoi regions?

I have a set of Voronoi cells that I partition into some defined regions. I would like to then calculate the length of the "borders" between these adjacent regions, using the edges of the underlying ...
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### Retrieving boundary points of concave hull (or alpha shape) in 3D

I am trying to find the boundary points of some concave 3D region which is described by a list of points. If the region were convex I would take the convex hull and simply extract the coordinates ...
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### Making a graph or network interactively over an image

Given an image, is it possible to manually draw a graph over it? For example, taking the following image: I want to manually pick the vertices and draw the edges in a "Paint-like" manner to get, for ...
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### Getting cell edges from a picture

I would like to write a program that, given a picture of an epithelium (2D cell array), for example automatically detects the cell edges and returns the corresponding lattice. Illustratively, ...
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### Algorithms used for DelaunayMesh

Is it possible to find what algorithms are used for Delanunay triangulations?
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### VoronoiMesh as a Graph

I want to draw the following graph using VoronoiMesh. For example, since ...
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### Updating a graph under ODE dynamics (vertex model)

Consider the following graph, which is obtained from a VoronoiMesh ...
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### How to plot parametric time dependent plot with different parameters taking average of time?

I am trying to plot the solution given in the code with respect to "delc". Now the problem is that it can be plot for a particular value of "t" like t=10,20,50,60,.. upto 100 but what i need is mean ...
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### Where to get the detailed code of MMA for grid mesh

I know DelaunayMesh can triangulate the specified area, but I can't see the code and partition idea of this built-in function. I want to know how to get the source ...
266 views

### Rectilinear polygon

It is possible to generate a random Rectilinear Polygon (without holes) in WL? I found this option, but without any possibility to ask for a Rectilinear Polygon. ...
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### How to get 3D building data in map functions?

Most modern mapping APIs have 3D model data for city buildings built-in. For example: Is there some way to get this functionality in the Geography related functions? I believe ...
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1 vote
147 views

### How to get the overlap's percentage that occurs between 2 circles or among more than 2 circles?

The following code draws the following circles; ...
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### Computing cross section area between cylinder and disk

I am creating a cylinder, whose center disk is placed at position vector v1 and whose orientation is given by the disk's normal vector ...
Closely related to this question about highlighting intersection of two disks, I am trying to figure out if one can do so similarly for disks embedded in $3D$ (e.g. in a bounding box). The difference ...