# Tagged Questions

Questions on the application of Mathematica to geometric problems. You might also consider adding the [graphics] tag, if appropriate.

6answers
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### Generating evenly spaced points on a curve

In the KnotData package a simple command such as points = Table[KnotData[{3, 1}, "SpaceCurve"][t], {t, 0, 2 Pi, 0.1}]; will ...
4answers
3k views

### Create a torus with a hexagonal mesh for 3D-printing

I am new to Mathematica, and I'm looking for a way to create patterns on the surface of 3D objects. One thing I have not been able to do is to create a hexagonal mesh on a torus. What I would like to ...
7answers
6k views

### Intersecting graphics

Does the Mathematica graphics system have any concept of intersecting graphics? I've not found much in the documents so far. For example, if I want to show the intersection of two shapes: ...
2answers
2k views

### Numerically solving Helmholtz equation in 3D for arbitrary shapes

Context While studying manifold Learning I got interested in finding the eigenvectors of the Laplacian. (also in connection to this problem of solving the heat equation) Following this and that ...
6answers
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### How to plot ternary density plots?

How can I get a ternary density plot just like the plots from OriginLab? ContourPlot and DensityPlot seemingly can accept the <...
3answers
1k views

### Can Mathematica solve Plateau's problem (finding a minimal surface with specified boundary)?

Given a closed curve $\mathcal C$ in three dimensions, is it possible to use Mathematica's built-in functionality to compute a minimal surface whose boundary is $\mathcal C$? For simplicity, let us ...
6answers
2k views

### Finding length of intersection of two surfaces

I would like to know how we find the length of the intersection of two surfaces. For instance, in the following example,a surface intersects with a plane: How do we find the length of intersection ...
5answers
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### 2D random walk within a bounded area

I want to simulate a random walk on two dimension in a bounded area such as a square or circle. I am thinking of using If statement to define a boundary. Is there a ...
1answer
787 views

### Proving the hairy ball theorem using xAct

I would like to formally prove the hairy ball theorem in Mathematica, initially just for $S^2$, and then see about generalizing. An approach I thought about to use the xAct package to define $S^2$ ...
2answers
2k views

### How can I pack circles of different sizes into a spiral?

Given a list of circles of different areas, I need to arrange them tangentially in order of increasing area and spiraling outward. An example of the type of packing I'm attempting is shown by the ...
2answers
358 views

### Bug in ArcLength?

fixed in 10.1 (windows) With Mathematica 10.0.2: ArcLength[Line[{{0, 0}, {1, 0}, {2, 0}}]] ArcLength[Line[{{0}, {1}, {2}}]] (* 2 *) (* 2 *) However, ...
4answers
11k views

### How to calculate scalar curvature Ricci tensor and Christoffel symbols in Mathematica?

I am seeking a convenient and effective way to calculate such geometric quantities. I've used packages like TensoriaCalc, but they don't work at all time. Sometimes,...
4answers
4k views

### How do I calculate the area of a polygon given its coordinates?

I have a polygon: Polygon[{{0, 200 }, {200, 100}, {500, 300}, {100, 700}}] How can I figure out its area? The docs page does not have any example. So far I've ...
6answers
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### Drawing a square root spiral

Here is a start. I'm looking for a nice way to draw it. ...
7answers
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### Distance between point and line segments

How would you determine the shortest distance between a point and one or more segments? For example, what is the shortest distance between the point and the two segments below? Clearly the point is ...
3answers
866 views

### Creating sculptural forms using graphics primitives

This is a question based on this answer by halirutan. Some amazing images can be created with this code, and I was wondering whether it was possible to extend the principle to different shapes. I ...
5answers
763 views

### Plot a partition of the sphere given vertices of polygons

I saw in this question that Mathematica can draw spherical triangles. I guess something similar can be done to plot a spherical polygon. I am interested in something similar: I have a set of ...
7answers
2k views

### Find intersection of pairs of straight lines

I have a list of 24 points, in which two consecutive points (1st and 2nd, 3rd and 4th, …) are supposed to form a line: ...
3answers
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### Approximating an ornamental curve

How do I go about approximating this ornamental curve? Note variable thickness typical in calligraphy. Handbook and Atlas of Curves by E.V. Shikin (1995) contains many directions, including curve ...
2answers
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### How do I split up a curve into chords of equal length?

I have a curve that is defined as f[x] and what I'm attempting to do is to divide the curve into equal straight lengths for a number of segments of my choosing that I've defined as nSeg. I've created ...
4answers
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### Finding unit tangent, normal, and binormal vectors for a given r(t)

For my Calc III class, I need to find $T(t), N(t)$, and $B(t)$ for $t=1, 2$, and $-1$, given $r(t)=\{t,t^2,t^3\}$. I've got Mathematica, but I've never used it before and I'm not sure how to coerce ...
5answers
4k views

### How do I draw a triangle given the lengths of the sides?

I know, of course, how to draw a triangle in the plane given the vertices: Graphics[Polygon[{{1, 0}, {0, Sqrt[3]}, {-1, 0}}]] But I'm not sure how to simply draw ...
5answers
481 views

### Is there a numerical method/built-in to calculate the boundary of a set of graphs?

Recently, I encounted a geometry problem in my work. For a curve-family that owns the following parametric equation: $$E(t,\theta)= \begin{pmatrix} x_E(t,\theta) \\ y_E(t,\theta) \end{pmatrix}$$ ...
5answers
5k views

### How to draw a great circle on a sphere?

I apologize for the text description, but new users are not allowed to post images. I want to draw a circle that cuts through the center of a sphere and has an inclination of 15 degrees with the ...
2answers
713 views

### How to simulate the true reflective movement of a particle bouncing around in an ellipse?

Please help me to simulate the movement of a particle inside a region with elliptical walls such that particle is reflected from the walls and continues to move. A friend was able write code to ...
2answers
1k views

### Rebuild a polygon so it doesn't self intersect [duplicate]

If you consider the following Polygon: ...
0answers
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### Bug in RegionMember?

Bug persisting through 10.4.1 RegionMember (new in V10) appears to have a bug. For some polygons it finds points that are clearly outside to be "members" of the ...
6answers
14k views

### How to determine the center and radius of a circle given some points in 3D?

I was wondering if anyone could give me a hand with this problem I have. I have six points on a plane, and I am trying to determine if they form a circle or not. I know that any three points in 2D ...
3answers
575 views

### Solve a trig equation system

Background of the problem In the same plane, P is a fixed point, A,B,C are moving point, PA=a, PB=b, PC=c, find the maximize perimeter of △ABC. let ∠BPC=A, ∠CPA=B, ∠APB=2*Pi-A-B, then the perimeter ...
1answer
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### Uniformly distributed n-dimensional probability vectors over a simplex

What's the right way to generate a random probability vector $p={p_1,\ldots,p_n} \in {(0,1)}^n$ where $\sum_i p_i=1$, uniformly distributed over the $(n-1)$-dimensional simplex? What I have is ...
3answers
542 views

### How to manipulate a circle in GeoGebra style?

I want to emulate the GeoGebra app. I want to get an effect like this in Mathematica. I don't know how to move whole circle without changing the radius of the circle. My sample code： ...
2answers
375 views

### How to export textures?

Is there any object geometry format for which Mathematica supports the exporting of textures for external (non-Mathematica) use ? If not does anyone know of work arounds? All the texture primitives ...
5answers
2k views

### Plotting an epicycloid

I am fairly new to Mathematica, and I cannot figure out how to plot an epicycloid. I have plotted some neat looking things in my attempts, but I still can't make one. I am not looking to make an ...
3answers
2k views

### Fitting ellipse to 5 given points on the plane

Five points are required to define a unique ellipse. An ellipse has five degrees of freedom: the $x$ and $y$ coordinates of each focus, and the sum of the distance from each focus to a point on the ...
3answers
362 views

### How to convert a surface into a solid

Context I am interested in converting surfaces into solids (so I can make a 3D mesh out of them using ToElementMesh) Say I have the following cool surface ...
4answers
267 views

### Administrative Divisions bordering a geographic region (e.g. an ocean)

Update 3 I've worked out a solution that is bearable for a small number of subregions in the neighbour and added it as an answer. I'm going to make a post for optimisation help. Update 2 For a ...
1answer
446 views

### How do I create a triangulated surface from points?

I have a set of points in a nx3 matrix and I would like to convert them into a surface, so that I may calculate its surface area. The function ListSurfacePlot3D creates the surface how I want it. ...
6answers
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### How to find lattice points on a line segment?

How do I find points on the line segment joining {-4, 11} and {16, -1} whose coordinates are positive integers?
5answers
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### To inscribe a circle in a given triangle

I am trying to to inscribe a circle in a given triangle but it isn't working. I've used GeoGebra with this construction and worked but as I'm new to Mathematica I am missing something. It can't be so ...
1answer
760 views

### Ellipse counting from image

I have hundreds of images similar to this image: I'm looking to count the number of ellipses as well as their major and minor diameters. The ellipses are touching, hence, ...
0answers
471 views

### Circle with negative radius?

Why does this work: Graphics[{Circle[{0, 0}, -Pi/4]}] But this doesn't? Graphics[{Circle[{0, 0}, N[-Pi/4]]}] Telling me: ...
3answers
920 views

### Finding the surface area of a 3D convex hull

I have noted several instances here and here where ConvexHullMesh, introduced in v10 has been used to greatly simplify some geometry problems. Determining the ...
2answers
884 views

### How to plot a barycentric line

I want to plot a barycentric function on an equilateral triangle (ternary plot). For example f1 = {Abs[Sin[x]], Mod[x, 2], Abs[Cos[x]]}; At the moment I evaluate ...
2answers
339 views

### How to make geometric intersection faster for geographic borders?

I have developed a solution to identify geographic regions bordering a region in this post (94723). Can this functionality be made to execute any faster? Find which entities in a set border other ...
5answers
774 views

### How to plot rectangles aligned by their center?

Supose I have a rectangle which area is $x^2$. In some cases I may not know what is the size of each side, for $x=12,$ we have several possibilites: ...
5answers
581 views

### How to choose three points on the circle so that the triangle is not a right triangle?

I want to choose three points $A$, $B$, $C$ has integer coordinates on the circle $$(x+2)^2 + (y+1)^2 = 25$$ so that the triangle is not a right triangle. But I can not. I tried ...
2answers
498 views

### Is there a triangle like this?

I want to find the numbers $a$, $b$, $c$, $d$ of the function $y = \dfrac{a x + b}{c x + d}$ so that the triangle $ABC$ with three points $A$, $B$, $C$ have integer coordinates and lies on the graph ...
6answers
303 views

### Probability of a random point inside the intersection of two triangles

I hope I can explain this problem clearly. It has been bothering me for sometime and I do not know how to solve it even with Mathematica's help. I finally stumbled upon an idea, see my answer below. ...
4answers
3k views

### Axis/Angle from rotation matrix

With r=RotationMatrix[a,{x,y,z}] I can compute a 3D rotation matrix from axis/angle representation. Given a 3D rotation matrix r...
1answer
317 views

### How to solve for the intersection points of two ellipses?

I have a parametric equation of two ellipses as follow $\begin{cases} x=a_1 \text{sin$\theta $}+b_1 \text{cos$\theta $}+c_1 \\ y=d_1 \text{sin$\theta $}+e_1 \text{cos$\theta \$}+f_1 \\ \end{cases}...