Questions tagged [computational-geometry]
The study of computer algorithms which admit geometric descriptions, and geometric problems arising in association with such algorithms. The two major classes of problems are (a) efficient design of algorithms and data classes using geometric concepts and (b) representation and modelling of curves and surfaces.
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I need help to understand the algorithm that tests if a point lies inside or outside a general 2D polygon
The algorithm in question is from this webpage.
The complete algorithm from this webpage is as follows:
...
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Is there an efficient algorithm to find the smallest circumscribed circle to a set of points?
I have a set of $n$ points in the plane and I want to find the smallest circumscribed circle that contains them.
By circumscribed I mean that the circle must pass through $3$ of the points. Brute ...
user1548405
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Is the adaptive jamming complexity for aperture identification $O(m \log m + \log N)$?
Let $D\subset\mathbb{R}^{2}$ be a bounded, open, connected polygonal domain whose boundary decomposes (unknown to the observer) as a disjoint union
$$
\partial D \;=\; U \,\sqcup\, W
$$
where $U$ is ...
user1675092
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Toric arrangements and system of polynomial equations
I am working on some problem on toric arrangements at the crossroad between topology, combinatorics and algebraic geometry.
$\textbf{Setting}$
Let $m,n\geq1$ and let
\begin{equation*}\mathcal{S}=\left\...
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How to partition a polygon into contiguous regions with given area ratios?
I am looking for a way to divide a 2D polygon (possibly with a complex shape, like an "H". These are actually floor layouts of buildings) into several connected sub-regions, where each ...
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Simple medial axis algorithm for convex polygons gets unexpectedly tangled up
When a polygon is convex,
its medial axis
a.k.a. topological skeleton is particularly simple:
there are no curved parts; it looks like an undirected tree whose internal nodes
are all those points in ...
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How to test if a cubic Bézier approximates a circular arc using only its control points?
Given a cubic Bézier curve with control points P0, P1, P2, P3, is there a way to determine whether it approximates a circular arc using only those four points — without evaluating the curve at ...
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How to test if a point lies inside a spherical quadrilateral?
I have four 3D points
A
,
B
,
C
,
D
that define a quadrilateral (assumed to be convex) on the surface of a unit sphere (i.e., all points lie on the sphere centered at the origin). These points define ...
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Finding faces from vertices in integer-valued, high-dimensional spaces
I am working on a very particular problem in which I have obtained a collection of points in a high-dimensional space. What I know a priori is that each point has integer-valued coordinates, these ...
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What is meant by a linear combination of homogeneous coordinates?
This is based on Förstner and Wrobel's Photogrammatic Computer Vision.
For inhomogeneous coordinates in $\mathbb{R}^2$ using $\vec{x}_0\ne\vec{0}$ and a direction vector $\vec{d}$ we can express a ...
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Subset maximising nearest neighbour distance
I'm interested in the following problem:
On the plane, given $n$ points in general position, select $k$ such that for each point selected, the (Euclidean) distance to the nearest point in the ...
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Finding the Envelope of a Family of Circles
I am trying to determine the envelope of a family of circles defined by:
$$\left(x-k\right)^2+\left(y-k^2\right)^2=4\left|k\right|,k\in\mathbb{Z}^*$$
I have searched MSE and found a similar discussion ...
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What is a "linear time transformation" for algorithms? By extension, what does "linear time transformable" mean?
I'm reading a slice of Preparata and Shamos's book on Computational Geometry. I'm enjoying it so far, and I feel it's pretty well-written. However, some of the language is unusual and can't be found ...
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Volume upper bound of the integer-point-free cube
In 3D space, we define the integer points as those whose three coordinates are all integers. What is the maximum possible volume of the cube which doesn't contain any integer point?
https://www....
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Finding generators of an ideal
I have a commutative $k$-algebra which is given by an explicit presentation $R = k[x_1,\dots,x_n]/(f)$. I have a subalgebra $S \subset R$ which I know generators for: let's call them $a_1,\dots,a_m \...
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Two typos in the book by Joseph O'Rourke.
The book by O'Rourke, titled: Computational Geometry in C, Second Edition; states on page #64:
A convex combination of points $x_1,\cdots, x_k$ is a sum of the form $\alpha_1x_1+\cdots+ \alpha_kx_k$ ...
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Solving Hamilton’s equations on curved surface
I wonder if there are analytical or numerical methods that can achieve the light ray tracing on curved surfaces filled with in-homogeneous medium. Specifically, I want to solve the following Hamilton’...
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Linear programming constraints for a line intersecting all line segments in set
I have a set of line segments that are defined as a connection between two arbitrary points (e.g. $(x_1,y_1)$ and $(x_2,y_2)$). I need to find a line $y=mx+b$ that intersects all the line segments in ...
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Finding the coordinates of the quadrilateral, derived from two intersecting line segements whose coordinates are given.
I want the coordinates of the quadrilateral formed from two intersection line segments, where the opposite sides of the quadrilateral should be parallel to one of the intersecting line segments. For ...
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What mathematical representation/technique can I use to verify the relationship between a convex polyhedron and its dual
From [1]
It is easily seen that the convex hull of the union of $A^D$ and $B^D$ is the dual of the intersection of $A$ and $B$.
where the definition of dual in [1] is:
The dual $A^D$ of $A$ is ...
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Primal and dual configuration of point-line
The primal configuration of point $p=(a, b)$ implies non-vertical line $p^*=\{(x,y)|y=ax-b\}$ is dual configuration.
And the primal configuration of non-vertical line $\ell=\{(x,y)|y=mx-n\}$ implies ...
user1326505
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Help troubleshooting ellipse perimeter calculation algorithm
I'm trying to troubleshoot my implementation of an Infinite Series algorithm to calculate the perimeter of an ellipse. I'm sorry I don't have the expertise to express it in, what appears to be a ...
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Relationship between major and minor axis of an ellipse's circumference
I'm not a mathematician. I'm just doing 3D modeling and can't find an equation to solve this problem. In openscad, an ellipse is created by scaling a circle. So this code creates an ellipse with a ...
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Placing triangles around a central triangle: Optimal Strategy?
Now cross-posted to MathOverflow (link).
Question: There is an equilateral triangle. Two players alternate turns placing non-overlapping equilateral triangles of the same size that touch the original ...
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Algorithm to find if intersection of convex sets is empty [closed]
Is there an algorithm to find if the intersection of two convex sets is empty or not. The projection onto convex method (POCS) and similar methods finds a point in the intersection, but will they ...
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