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Results tagged with computational-geometry
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user 3056
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.
2
votes
How can I select all pair of two points A and B has integer coordinates and length of AB is ...
Find all integer-valued points on the sphere excluding $x y z \ne 0$, then choose pairs with no duplicate values and distance of 18:
SolveValues[
Element[{x, y, z}, Sphere[{1, 2, 3}, 9]] && x y z …
- 19.1k
2
votes
Simplifying an expression to a sensible conic section polynomial
Here's a hack around FindInstances returning complicated roots. simpleInstances finds three lines aligned with each axis which intersect with the region defined by expr and have a "simple" rational fo …
- 19.1k
4
votes
3
answers
189
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:
expr = x^2 + y^2 <= 1 &&
1/Sqrt[5] (2 (1 + Sqrt[5]) x^2 + (-1 + Sqrt[5]) x …
- 19.1k
40
votes
Accepted
How to exactly calculate the volume?
No numerics hacks here; this really computes the volume symbolically. It is a bit tedious and demands some tricks which may appear more obvious in this answer than they would really be on the first tr …
- 19.1k
23
votes
Accepted
Divide a geometric region by (many) lines
You can find symbolic connected components (which are those regions you are asking about) in this case using CylindricalDecomposition. This can be a bit of an overkill if your goal is only to visualiz …
- 19.1k
5
votes
2
answers
195
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:
Timing@With[{d = 50},
Wi …
- 12k
2
votes
How to find intersection points of $n$ $n$-spheres reliably and efficiently when $n$ is large
Well, speeding it up significantly (but not as much as @user293787...) was easier than I thought:
Timing@With[{d = 500},
With[{
p = RandomReal[{-1, 1}, d],
s = RandomReal[{-1, 1}, {d, d}]},
…
- 19.1k
12
votes
Mark all points in a triangle that have a certain property
If we phrase out the problem as "for each point {x, y} in the sought region there exists a line passing through it on which both points at the distance l/2 from {x, y} are inside the triangle", the pr …
- 19.1k
5
votes
How to generate approximately equally spaced points efficiently
Since Mma v12.2 spatial point processes have opened yet another possibility with HardcorePointProcess, which prevents processes from resulting points being pairwise closer than a specified distance fr …
- 19.1k
2
votes
Find duplicates in list of InfiniteLine
RegionEqual, like many region functions, is able to compute symbolic results as long as arguments are fully specified. This allows more efficient constructions of the following kind - where the symbol …
- 19.1k
13
votes
Accepted
How to visualize a spherical mesh on a squared plane?
I wouldn't try to do this on a square (assuming you want at least some sort of systematic correspondence between edges on the sphere and their projections), at least if you don't want crossings on squ …
- 19.1k
11
votes
RegionMember[ ] in polygon
This seems like a bug. It might even be a bug I have reported in the past.
In particular, your polygon is slightly degenerate: it has {-3, 16} twice in a row, creating a zero-length edge. (This doesn …
- 19.1k
11
votes
Unexpected behavior of the procedure `Area` on the object 'Polygon'
EDIT: Reported to WRI as CASE:4237867 - fixed on V12. :)
Seems like a bug in my opinion. Consider the following:
With[{poly0 = Polygon[{{0, 0}, {4, 0}, {2, 2}, {4, 4}, {0, 4}}]},
With[{poly = Tran …
- 19.1k
12
votes
Accepted
Decomposition of a semialgebraic set into connected components
EDIT: CylindricalDecomposition has been improved since I wrote this answer, probably in v11.2! Now it takes an optional topological operation argument. As a result, one can achieve the results describ …
- 19.1k
4
votes
Accepted
Simplify behavior: assumption as Interval versus assumption as bounds
Element treats Intervals as geometric regions, and members of those geometric regions are vectors, even when they are of single dimension. (I don't think this is really properly documented anywhere - …
- 19.1k