# Tag Info

## Hot answers tagged differential-geometry

### How to estimate geodesics on discrete surfaces?

Geodesics in Heat Algorithm At the suggestion of @user21 I am splitting up my answers to help make the code(s) for calculating geodesics distances easier to find for other people interested in these ...
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### How to speed up estimation of Mean and Gaussian curvatures on triangular meshes?

Note added 1/29/2020: the routines here have a bug where the mean curvature is sometimes computed with the opposite sign. I still need to work on how to fix this. I guess I should not have been ...

Definition ...
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### Geodesics on a torus

I don't know if there's a simple way to find geodesics on a torus, but I can give you a general way to find geodesics on any curved surface. First, I define the torus: ...
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### How to estimate geodesics on discrete surfaces?

Nothing really new from my side. But since I really like the heat method and because the authors of the Geodesics-in-Heat paper are good friends of mine (Max Wardetzky is even my doctor father), here ...
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### How to speed up estimation of Mean and Gaussian curvatures on triangular meshes?

It took me a while, but the suggestion of @Michael E2 was quite helpful, and especially the post (Optimize inner loops). For those of you (like me) who are new to this style of programming in ...
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### The time-like geodesics (orbits) in the Schwarzschild spacetime

Studying basic solutions at the theoretical physics it is advantageous when one can get an exact solution. At the first sight one can see that the solution can be given in terms of elliptic functions ...
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### How to estimate geodesics on discrete surfaces?

Here is an exact algorithm but heavier to implement and to optimise. I chose to implement the "Line of Sight Algorithm" from Balasubramanian, Polimeni and Schwartz (REF). Exact Line of Sight ...
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### How to find the magnitude of a vector?

Norm in general assumes complex arguments and uses Abs to provide for that: Norm[{x, y}] Sqrt[Abs[x]^2 + Abs[y]^2] For ...
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### Solving the Frenet Serret equations for non-constant curvature and torsion, obtaining parametric equations

Okay, this'll be a short answer just to show what you can do. What you are trying for here is essentially an inverse to FrenetSerretSystem, which will give the ...
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### Can Mathematica solve nonlinear, coupled differential equations?

There were syntactic and conceptual problems with your formulation. Conceptually, NDSolve is a numerical solver, so you need to specify boundary conditions as ...
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### How does a Pringle lose its curvature?

You need to delay the evaluation of the right-hand side of ScalarCurvature: ...
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### Computing Gaussian curvature

Note this parametric surface of unit sphere (S^2) should have constant Gaussian curvature: 1. Surface: x[u_, v_] := {Cos[u] Cos[v], Cos[u] Sin[v], Sin[u]} ...
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### RegionNearest and neighborhoods

If you don't mind using undocumented stuff, you can access lots of useful properties by converting the BoundaryMeshRegion to a ...
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### Estimating Principal Curvature Directions on Discrete Surfaces

For this answer, I shall be doing something slightly more ambitious. In particular, I will be computing the so-called curvature tensor, which encodes information on the normal vector $\mathbf n$, the ...
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### Plot a space curve and its curvature

ArcCurvature is already built-in to Mathematica, so there is no need to compute curvature manually: ...
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### Visualization of Gaussian Curvature

I finally got around to fixing the routine in the math.SE answer the OP linked to. To make this answer self-contained, I'll reproduce the definitions here: ...

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

Since Version 9, functions to do this have been built into Mathematica but not documented. They live in the SymbolicTensors package which underlies ...
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### How to estimate geodesics on discrete surfaces?

IGraph/M's IGMeshGraph function makes it easy to implement the graph-based solution. This function constructs a graph in which vertices correspond to mesh vertices ...
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### Expand wedge product

One idea is to use TensorReduce. I will assume that r is real, and that Dt[r] and ...
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### Coordinate-free derivative

Maybe you could use the following approach: ...
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### Finding unit tangent, normal, and binormal vectors for interpolated function

First we get you interpolating function: ...
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### Estimating Principal Curvature Directions on Discrete Surfaces

OK, at least here is an attempt to solve my problem. Hopefully these thoughts and code may be useful to others. It seems like there is no single ideal algorithm to solve this problem. Some work ...
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