# 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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145 views

### Finding Contours in Photo to Create a Topographical Map

I am setting up a special table to simulate the scenario in this problem in this problem, but instead of finding the geodesics, I want to create a topographical map using the contours. Here is a ...
1 vote
69 views

### Subtract ParametricRegions from each other

I have 2 surfaces defined parametrically that overlap. I want to subtract the helical surface from the cylindrical surface to obtain a mesh of just the blue surface in the image. The parameter bounds ...
277 views

### Minimal surface bounded between turns of helix

I'm curious to find the shape of a surface bounded between the rungs of a helix, ie the shape of the cloth stretched between the rungs of this child's play tunnel. I'm wondering if we could find it ...
82 views

### Speeding up RegionIntersection of multiple sets of rectangles

There is a set of disjoint rectangles shadow that change position over time. I want to find their intersection with another set of small rectangles ...
2k views

### Efficiently Mowing Grass with Mathematica

Recently, I was mowing my yard and was thinking about how to use the least amount of time/fuel and how to simulate. Mowing a yard that is shaped as a convex polygon is not difficult, you can just make ...
1 vote
119 views

### What are some quick methods for calculating hyperbolic equations?

Given that the hyperbola passes through three of the four points (m1, m2, m3, m4), find the equation for the hyperbola These four points are like this ...
63 views

### Finding the Complete Silhouette Area Formula given a Convex Polyhedron

Like always, I am probably over complicating this and there is a much simpler way to derive a general formula using built-in Mathematica functions, think about it differently, or a faster way to solve ...
255 views

### Packing unequal spheres into minimal cuboid

There are several non-overlapping spheres in 3D. How to find a cuboid Cuboid[{a,b,c}] containing these spheres with minimal ...
61 views

### Weird behaviour of TransformedRegion and GeometricTransformation

What I would like to do is to take some polyhedra (such as Cuboids, Prisms, and ...
84 views

### Solving the clothing problem with Mathematica

Given a surface $\mathbf{r}:\mathbb{R}\rightarrow \mathbb{R}^3$, the Chebyshev clothing problem consists in finding a parametrization $(u,v)$ such that \begin{align} \left|\frac{\partial \mathbf{r}}{\...
1 vote
128 views

### How would you use Mathematica to extract a metric tensor from a metric formula? [duplicate]

I'm trying to follow an explanation for extracting a metric tensor from a metric formula and having trouble following the mathematical explanation. I'm sure I would understand it better if written in ...
514 views

### Tomographic Reconstruction of a Convex Polyhedron from its Silhouettes

So, I can construct a random polyhedron and find its 3 silhouettes onto the 3 standard planes. For example, ...
36 views

### Transition to a new basis with rotation of one of the axes [closed]

The lengths of the basis vectors e1 and e2 of the general Cartesian coordinate system on the plane are equal to 4 and 2, respectively, and the angle between the basis vectors is 120°. Relative to this ...
1 vote
253 views

### How to find a densest configuration of several non-overlapping unit disks? [closed]

Let us consider three non-overlapping unit disks in the plane. I'd like to find a configuration of these disks s.t. the area of the convex hull of these disks is smallest. This is related to the ...
817 views

### Converting an Unicode's image into polygon data for an SVG

So, I have the the Chinese character 熙, which I can easily turn into an image: img=Rasterize[Text[Style[FromCharacterCode[{29081}], FontSize -> 300]]] which ...
81 views

### How to make a function that outputs a mesh region object or some other geometric object that represents a cake number?

The lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a disk (a pancake or pizza is usually used to describe the situation) that ...
65 views

### How do I find the circumsphere of any spatial geometry? [closed]

How do I find the circumscribed sphere of any spatial geometry? I want to: automatically generate its circumscribed sphere in the original three-dimensional graphics, label its center position, label ...
299 views

### How to create a random belt polygon or belt polyhedron?

A zonogon is a convex polygon that is made up of parallel sides. Generating a random zonogon in Mathematica can be found here. A natural generalization of zonogons is called belt polygons. A belt ...
1 vote
37 views

### Maintaining form of Manipulate to find the formula area of a projected cuboid

I am experimentally trying to solve this problem about the projected area of a cuboid. I sense patterns when I run the below code, but it is difficult for me to compare them. Here is the code: ...
40 views

### Inconsistence in calculating zero geometric area

First, let us see disk ...
369 views

### How can I calculate the volume of spatial geometry?

In the square prism ABCD A1B1C1D1, AB=2, A1B1=1, AA1=Sqrt , what is the volume of this prism? It is easy to calculate its volume using the volume formula: ...
61 views

### Define triple Integral tetrahedron truncated by oblique plane

I am struggling on defining the integration limits of a triple integral of an oblique truncated tetrahedron. Let's get more into the details: I have a standard tetrahedron in a ...
278 views

### Moment of Inertia for Hexahedron region

I've been going crazy about a problem that I think it has no solution, since I found nothing on the Internet about it. I would like to compute the tensor of inertia of a region. The region is the ...
191 views

### How do I calculate volume of a polyhedron knowing a list of vertices of polyhedron?

I have a list of vertices a = {1/2, 1/2, 0}; b = {-1/2, 1/2, 0}; c = {-1/2, -1/2, 0}; e = {-1/4, 1/4, Sqrt/4}; f = {1/4, -1/4, Sqrt/4}; I want to find ...
76 views

### Incorrect result of ConvexRegionQ

Studying that important command of Mathematica in 13.2 on Windows 10, I obtain for both ...
224 views

### How to determine whether a point is within a multidimensional degenerate convex hull? [duplicate]

There are two set of points ...
249 views

### How to avoid poles in contour integration?

I have two sets of points and need to construct a contour (for contour integration) that encloses all singularities given in points2 , but avoids enclosing ...
191 views

### Why is SignedRegionDistance always returning zeroes?

I have a simple parabolic region: reg = ImplicitRegion[ x <= 1/2 && y <= Sqrt[1/2 - x]/2, {{x, 0, 1}, {y, 0, 1}}] It appears to be behaving ...
1 vote
202 views

### How to get the centroid of a geographical region? (or: the center of mass of Croatia is in Bosnia and Herzegovina, but where exactly?) [duplicate]

How can I get Mathematica to calculate the centroid of an extended geographical region (say, a country) in a way which (i) fully respects the spherical nature of the Earth, and (ii) involves a minimal ...
1 vote
62 views

### Angle sum at center of a skewed quadrilateral

The center of a discrete quadrilateral on a positive, flat or negative curved surface should make angles sum at center point 2 less than, equal to or more than $360^{\circ}$ respectively, making them ...
175 views

### Simplifying an expression to a sensible conic section polynomial

I have an expression which represents an intersection of the unit sphere and a cone, projected to two-dimensional plane: ...
2k views

### How to exactly calculate the volume?

Let us consider the intersection of four cylinders of the unit radius along the big diagonals of the cube $[-10,10]^3$ and the cylinder of the unit radius with the $z$-axis as its axis. More exactly, <...
403 views

### How can I detect the number of crossings in a layout of a graph?

This question was inspired by the game Planarity in which a player tries to position the vertices so that no two lines cross. As I move the vertices, the change in the number of crossings is detected ...
104 views

### Placing shapes along a square spiral

Proposed code example is inspired by the main idea of this challenge. Wolfram Language provides great opportunities in computational geometry. But not all of them are equal in terms of speed and ...
188 views

### Bugs with computational geometry: versions 12.2 vs 13.2

Updated System Info: HONOR laptop AMD Ryzen 5 4600H 3Hz 6 cores 16GB Memory, Windows 10 Pro 64 bit with updates. Mathematica Info: Versions 12.2 and 13.2, both on the same laptop. Examples: ...
116 views

### Problem with WindingPolygon

Bug introduced in 12.2 or earlier and fixed in 13.1 or earlier. I would like to slightly modify the standard example: ...
123 views

### RegionIntersection runs for a very long time and then aborts without returning a result

Is there a good strategy to speed up RegionIntersection computations like the one below: ...
88 views

### 3D BoundaryMeshRegion Minkowski difference via RegionErosion Failure

I need to be able to produce Minkowski difference of an arbitrary BoundaryMeshRegion and a cylinder. ...
1 vote
79 views

### Another way to find coordinates of two remain vertices knowing given two vertices?

Let be given the polyhedron ABCD with $AB = 4 \sqrt{3}$, $AC=1$, $BC=\sqrt{41}$, $AD=\sqrt{17}$, $CD=4$, and $BD=5$. I am trying to find coordinates of two vertices ...
315 views

### Two squares in the unit square

A square of side one contains two squares of sides $a$ and $b$ having non-overlapping interiors. How to prove the inequality $a+b≤1$ with Mathematica? The same question in three dimensions and higher ...
142 views

### How to find intersection points of $n$ $n$-spheres reliably and efficiently when $n$ is large

I have positions and radii of $n$ $n$-dimensional hyperspheres and want to find their intersection points efficiently. A very-straight-forward solution seems quite reliable: ...
126 views

### Plot3D plotting serrated edge

I want to plot the upper casket of a sphere in 3D using its parametrisation as $s(x,y) = \sqrt{1-x^2-y^2}$. Although this show be straight forward with Plot3D, I am running into an issue I find very ...
376 views

### Mark all Points in a triangle that have a certain property

I want to mark all points inside of a triangle having the following property: I can center a line segment of length $c$ on the point so that the line segment is entirely contained inside the triangle. ...
237 views

### What is the volume of the intersection of four cylinders of equally radius equally spaced?

I saw that question at the certain forum and answered it with help of Mathematica 13.1 in such a way. The angles between the unit vectors ...
178 views

### Parellel-To-Plane Version of RegionDistance

RegionDistance basically calculates distance between points P and Q=RegionNearest[Reg,P]: I need to find a way to find function ...
377 views

### Extract 2D quad mesh from 3D hexahedral mesh

I would like to extract 2D mesh of outer surface of 3D meshed object. Let's say I have 3D mesh data from here and I import the data into Mathematica. ...
119 views

### How to make a density plot on Ellipsoid[] surface?

Suppose I have the ellipsoid below and a set of points that I know rest on its boundary, which we can visualize: ...
261 views

### Generating a non-convex polyhedron from polygons and normals

I am working on developing "directional offset" module, which requires pretty tricky mesh generation: offset values vary in different directions. Suppose I have list of points and polygons, ...