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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.
3
votes
0
answers
91
views
Better discretization for intersections of algebraic surfaces
I am looking for better, preferably analytical approaches to discretize intersections of algebraic surfaces. When surfaces are not identical, these solutions are either curves or points.
An extremely …
- 19.1k
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
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
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
2
votes
0
answers
134
views
Computing symbolic surface normal of a surface point on a semialgebraic set
Consider a semialgebraic set; such as reg below:
With[{reg =
x^2 + y^2 + z^2 <= 1 && x^2 y^2 z^2 <= 1/1000 && -x - y + z <= 0},
RegionPlot3D[ImplicitRegion[reg, {x, y, z}], PlotPoints -> 200]]
…
- 19.1k
5
votes
Inflate and unite a list of 0D to 2D regions
Somewhat dumb method (for instance, every line has both two Disks and a StadiumShape overlapping), but it's not at least very complicated:
hulls0 = ConvexHullMesh /@ RandomReal[{-10, 10}, {3, 1, 2}]; …
- 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 …
- 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
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
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
2
votes
0
answers
71
views
How to speed up geometric Resolve query involving $\exists$ and $\forall$?
I would want to test connectedness of semialgebraic sets with naive code like this:
With[
{r1 = y > -1 && x <= Sqrt[1 - y^2] && Sqrt[2] + 2 y <= 1 &&
x + Sqrt[1 - y^2] >= 0,
r2 = Sqrt[2] + 2 …
- 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
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
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
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