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Results tagged with computational-geometry
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user 51
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.
13
votes
Find the nearest locations for multiple points
I will crib shamelessly from example and code for illustrating by @ybeltukov.
The example:
n = {5, 5};
holes = N@Tuples@Range@n;
balls = RandomReal[{0, # + 1}, Times @@ n] & /@ n // Transpose;
We …
- 60.3k
21
votes
Computing the genus of an algebraic curve
I will show a method that is conjectural, though i believe it is correct. It differs from the more common approach of using (quadratic) birational transformations to force singularities to be double p …
- 60.3k
16
votes
Insphere for Irregular Tetrahedron
This may not be as neat as the other methods posted. About the only things I can say are that it is derived from basic principles, and it is fortunate that I had my hair buzzed rather short a few days …
- 60.3k
17
votes
Accepted
Determine whether points lie within a cow
This is basically a rehash of code I posted in a prior thread on this topic. The underlying method is to shoot a ray from the point and see how many surface triangles it intersects.
elsie = ExampleDa …
- 60.3k
3
votes
How to efficiently implement k-FN (k-Furthest Neighbors)?
Not necessarily best of quality but maybe could be made better with a bit of tuning.
kDistant[pts_List, n_] := Module[
{objfun, len = Length[pts], ords, a, c1},
ords = Array[a, n];
c1 = Flatten …
- 60.3k
6
votes
Approximating Voronoi diagram without any distance checks
Here is a method that combines usage of "fast marching" and Nearest from responses here. It is not as pretty as the one shown here but is probably more efficient computationally. The idea is to work o …
- 60.3k
9
votes
Distances between points in periodic cube
You want something like
PeriodicDistance[pts_, size_: 1] :=
Outer[EuclideanDistance, size*FractionalPart[pts/size],
size*FractionalPart[pts/size], 1]
But what you did is a bit different in ter …
- 60.3k
0
votes
Accepted
Finding a set of characteristics of a system of polynomial equations
This is a bit of a wild guess, based on two articles sent by the poster (references below).
First rewrite the system using lower case names, making the N into a time-dependent "input" variable, and c …
- 60.3k
18
votes
How to check if a 2D point is in a polygon?
Since someone dragged in Canada...
Here is the code from a MathGroup post I had referenced. I have modified to compile to C and that speeds it further. The one-off preprocessing does take time but it …
- 60.3k
4
votes
Place spheres randomly in a box without collisions?
Okay, here goes.
Given a distro, box length, and number of desired points one can generate a bunch more, remove the ones that are too close to the edges to fit, then iterate through what remains to d …
- 60.3k
3
votes
Help finding the point(s) inside "non-star" closed shape with the highest average ray length?
One approach is to parametrize the boundary, and use that parametization for the innermost integral that defines the averaged radius. This might or might not correspond to the definition you have in m …
- 60.3k
4
votes
Simplifying an expression to a sensible conic section polynomial
This is not a general-purpose answer but it works in this case at least. First step is to obtain an implicit polynomial in {x,y} for that second expression. Often enough, GroebnerBasis can do this. It …
- 60.3k
3
votes
Generate convex-hull of a 15 dimensional space
Start with the definition (I use Together to make it somewhat shorter).
func = {1/225 (-15 x5^2 y1^2 - 15 x6^2 y1^2 - 15 x5^2 y2^2 - 15 x6^2 y2^2 -
5 Sqrt[15] x2 x5 y3^2 - 25 x5^2 y3^2 + 5 Sqrt[ …
- 60.3k
16
votes
Voronoi diagrams for generators other than points
While I cannot match @whuber's simple elegance, I will show a bit of brutishness by using Fast Marching from scratch. This finds distances from a specified boundary. I'll modify it so that, for each p …
- 60.3k
21
votes
Voronoi diagrams for generators other than points
Here is a Nearest-based method. This is quite similar to what @Mr. Wizard did for approximating 3D (ordinary) Voronoi.
comps = MorphologicalComponents[img];
cmap = Flatten[MapIndexed[#2 -> # …
- 60.3k