# 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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### 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}}]; ...
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### Create concentric circles with numbers inside

Hello there if you can give me a hand I want to create and export to pdf the following figure, which if you realize forms the multiplication table. The idea is to be able to control the central number ...
286 views

### Cube cutting problem

Show[Graphics3D[{Cuboid[]}], ContourPlot3D[{x==1/3,y==1/4,z==1/5},{x,-0.2,1.2},{y,-0.2,1.2},{z,-0.2,1.2}]] Three planes divide the unit cube into 8 small cubes, ...
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### Does Mathematica have a built-in n-dimensional hypervolume indicator?

Mathematica has so many functions hidden away in obscure packages that I thought I'd ask before starting to code one myself. Thanks in advance.
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### Speed up the intersection of the diagonals of a regular polygon

I need to calculate all the intersection points of the diagonals of a regular polygon, following code is really slow, when n=15, it take about 30sec. I also tried using ...
118 views

### Checking Membership in Boundary of Conical Hull

I want to check whether a given point is also a member of the edges of a given conical hull. For example, I have: ...
34 views

### Increase precision of ConvexHullMesh

I want to generate a set of equations for all planes in the convex hull as specified in this question. The problem is that the generated set of equations don't correctly cover all the points. I run a ...
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### Find intersection points of arbitrary BSpline curve

I'm researching some computational geometry and am trying to find a method of determining the {x,y} intersection of a circle and BSplineCurve for further processing. ...
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### How to find the shortest distance between two regions?

I need to find the shortest distance between the two regions and the two points of the shortest distance, but the following code cannot achieve this requirement: ...
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### Dump Curvature Information for 3d Models

Is anyone aware of any utility to generate and dump Principal, Gaussian and Mean Curvatures for a 3D Model or a set of points? I have reserched following tools - meshlab cloudcompare They are able to ...
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### For an enclosed surface region, how to find the furthest point on that surface to another point that is not on that surface?

I have an enclosed 3D surface (a distorted sphere surface region, which is got by finding an enclosed contour of a 3D volumetric data, and apply RegionBoundary[BoundaryDiscretizeGraphics[ ]] to get ...
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### Find the intersection coordinate of a line and a 3D surface defined by a contour plot

I have two points, for instance {1, 1, 1} and {10, 11, 15} that connected to form a line. I also have an enclosed surface region ...
202 views

### How to fill in interior in a mesh region

Say I have a mesh region which encloses a region. How do I get a mesh region where this region is filled? Take for example the meshregion below: ...
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### Finding plane equations of planes in a convex hull

I'm new to mathematica and I need to get the equations for the set of planes which are part of a convex hull that I have calculated using ConvexHullMesh. It is a ...
464 views

### Finding the intersection(s) of a 3D parametric surface and a line

I have some 3D parametric curve par and some line line. I want to find parameters u,v and t for the intersection points of par and line. I tried to do that with NSolve in many different ways and ...
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### Why does ConvexHullMesh give inconsistent output stuctures?

I know much of the stuff in ConvexHullMesh needs some TLC convexhullmesh-inconsistency But, can anyone see why this would produce inconsistent output: Case 1 (expected behavior): ...
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### How to modify a particular step length in a 3D random walk?

Suppose I simulate a random walk in 3D using AnglePath3D[]: ...
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### How to reorient a Sphere[] and associated point based on some arbitrary axis?

Suppose I have a unit Sphere[] sitting at the center {0,0,0}, with a point on its "north pole", like this one: ...
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### VoronoiMesh with boundary edges removed

I would like to create a Voronoi diagram without the boundary edges highlighted. This code gVD = VoronoiMesh[X, MeshCellStyle -> {{1, All} -> {Thick, Red}}] ...
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### Drawing a “realistic” VoronoiMesh

Consider a set of points generated via Lloyd's relaxation algorithm ...
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### An apparent bug in GeometryConvexHull2

I was replacing some usages of GeometryConvexHull2 with ConvexHullMesh and found one of my tests failing. It turned out the ...
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### Generating points inside convexhullmesh

How could I generate these points hexagon={{0, Sqrt/2}, {1/2, Sqrt}, {3/2, Sqrt}, {2, Sqrt/2}, {3/2, 0}, {1/2, 0}} following this rule ...
70 views

### How to Parallel submit nested do loop

This is the first time I'm trying to parallel compute something on Mathematica. So this question might be silly. I have a nested Do loop of the form: ...
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### Hyper Tetrahedron in terms of inequalities

The command Tetrahedron[] seems to work fine for the 3-d case. But how can we get hyper-tetrahedron for 4-d case in terms of inequality? Take for example the points: ...
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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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### 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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### 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 ...
121 views

### 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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### 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 ...
400 views

### How to add 'border' to Voronoi diagram

Suppose I have a set of points pts, from which I generate a Voronoi diagram VD: ...
322 views

### 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 ...
70 views

### How to speed up finding region centroid and plot3d

I have the following code to make 3D of 9 regions. ...
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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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### 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. ...
129 views

### 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 ...
14 views

### How to concatenate multiple CapsuleShape[] while changing their size?

I am trying to create a 'chain' of capsules so that they are concatenated at certain points. Then, I want to change the size (radius, etc) of a given capsule, while preserving the relative orientation ...
76 views

### 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=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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### 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 ...