# Questions tagged [geometry]

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

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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 ...
• 423
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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 ...
9k views

### 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 <...
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### 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: ...
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### 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 ...
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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 ...
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### 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 ...
• 281
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### 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 ...
• 283
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### 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: ...
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### 2D random walk within a bounded area

I want to simulate a random walk in two dimensions within a bounded area, such as a square or a circle. I am thinking of using an If statement to define a boundary. ...
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### 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. ...
• 215
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### 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 ...
• 631
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### how to get $n$ equidistributed points on the unit sphere

We can get $n$ equidistributed points in the unit circle using CirclePoints. But how do you get $n$ equidistributed points on the unit sphere(surface of a ball)? ...
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### Rebuild a polygon so it doesn't self intersect [duplicate]

If you consider the following Polygon: ...
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### 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 ...
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### 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 ...
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### 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 ...
• 209
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### How to create this four-dimensional cube animation?

This is a tesseract, a four-dimensional cube, which contains two cubes. Here, each side length of the smaller one is 1, while the side length of the bigger one is 2. How do make I it? I am still ...
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### 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 ...
• 493
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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 ...
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### Axis/Angle from rotation matrix

With r = RotationMatrix[a, {x, y, z}] I can compute a 3D rotation matrix from its axis/angle representation. Given a 3D rotation matrix ...
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### Make an offset curve (parallel curve)

I have a polynomial curve that I got through interpolation. ...
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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?
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### 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 ...
• 311
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### Construct a polyhedron from the coordinates of its vertices and calculate the area of each face

I have the 3D coordinates for a set of points. I want to construct the convex polyhedron with those points as vertices. I know I can use functions like DelaunayMesh ...
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### Plotting the sum of two points on an elliptic curve

I am doing an experiment to prove the associativity of the addition of points on an elliptic curve. So far, I have produced a code which allows me to move points on my curve. To find their sum, I ...
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### Assumptions for RotationMatrix

I'm making C++ program, and in my program I need a rotation matrix around any vector. I wanted to extract RotationMatrix[fi,{x,y,z}] output and put it in my program....
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### generating randomly oriented non-intersecting cylinders

I could not find anything relevant with "googling" (only this) I can create a set of randomly oriented cylinders (code found here) ...
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### Finding the volume enclosed by two surfaces of revolution

I've come this far as to combine both functions in a graph: f = Plot[4 - x^2, {x, -10, 10}] g = Plot[-1 + 4 x, {x, -10, 10}] Show[g, f, PlotRange -> All] ...
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### 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 points ...
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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 ...
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### Divide a geometric region by (many) lines

Given a shape (e.g., a rectangle, a circle, etc.), how to divide it by $n$ randomly chosen lines. It is trivial to plot those lines (see figure below), using code like ...
• 519
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### Mathematica for teaching orthographic projection

Edit: All the four answers to this question are great, and if you're interested, you should take a look at all the answers. Nevertheless, belisarius' code was accepted since it was closest to what I ...
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### 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 ...
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### How to separate the regions enclosed by curves

There are three arbitrary curves that defined by three implicit functions. How to separate the 12 regions(one of them is the outside infinite region) Although we can use image methods to distinguish ...
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### How to generate 3D spherical sector

I looked but haven't found an answer to this one: I'd like to create a region that represents a sector of a ball, bounded between radii $r_1$ and $r_2$, polar angles $\theta_1$ and $\theta_2$, and ...
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### Unexpected behavior of GeometricTransformation

Bug introduced in 8.0 or earlier and persisting through 13.2 or later. Fixed in 13.3.1 on Windows 10 I have the following mapping on the complex plane: $$z \mapsto \tau \mu z-1,$$ where $\mu$ is ...
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### Finding volume of a segment

I'm still pretty new to Mathematica, so I would like to seek advice regarding a geometrical problem. I am currently trying to define that as an extra condition in the Mathematica code below. ...
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### Triangle mapped on a sphere in $\mathbb R^3$?

How can I map a triangle on a sphere? I want to visualize (plot or animate) it for my student in Non-Euclidean geometry. I have no restrictions on the triangle's kind or on the sphere in $\mathbb R^3$....
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### How to get a perpendicular segment inside of a triangle

I created a graphic to obtain a triangle with three vertices. The code is as follows: ...
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### Why is raytracing so slow?

I'm trying to get some rays to bounce in a circle. But I want to be able to control the reflections, i.e. the direction the rays bounce in the circle. I have a MWE below, and it is severely limited by ...
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### 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 ...
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### Drawing a square root spiral

Here is a start. I'm looking for a nice way to draw it. ...
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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 ...
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### Transform sphere to an ellipse in $\mathbb{R}^2$

In Lay's Linear Algebra and Its Applications textbook, he defines the matrix $$A=\begin{bmatrix} 4 & 11 & 14\\8 & 7 & -2 \end{bmatrix}$$ and claims that the transformation $T(x)=A x$ ...
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### How to roll a graph on the y-axis

I want to roll the function f(x)=sqrt(x), x∈[0,1] along the y-axis. I know how to rotate the graph around a point, but I'm not sure how to rotate along an axis in 2D. Rotating around a point e.g. (0,...
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### How to generate approximately equally spaced points efficiently

I don't very content with current method.So a better solution is expected still.I hope it meet two conditions in following. That space is approximately equivalence. We can control how many points to ...
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### 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: $$E(t,\theta)= \begin{pmatrix} x_E(t,\theta) \\ y_E(t,\theta) \end{pmatrix}$$ ...
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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 ...
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