All Questions
16 questions
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Ellipse of least perimeter that contains a given triangle
This post is related to Smallest 3-ellipses that contain triangles and tries to clarify a basic issue.
Question: Given a general triangle T, How does one find and characterize the ellipse of least ...
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1
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0
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124
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A center of convex planar regions based on chords
This is based on Chapter 6 of 'Convex figures' by Yaglom and Boltyanskii. This post also continues On two centers of convex regions.
A point $P$ in the interior of a planar convex region $C$ divides ...
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1
vote
1
answer
208
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On a possible variant of Monsky's theorem
See Wikipedia for Monsky's theorem which states: it is not possible to dissect a square into an odd number of triangles all of equal area.
Questions: Are there quadrilaterals that allow partition into ...
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2
votes
1
answer
273
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Triangulations of point sets — obtuse and acute triangles
Given a planar configuration of points in general position. It is known that the Delaunay triangulation is the 'fattest' triangulation possible. It is also easily seen that given 7 points with 6 of ...
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3
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Cutting convex polygons into triangles of same diameter
This question continues from: Cutting convex regions into equal diameter and equal least width pieces
Definitions: The diameter of a convex region is the greatest distance between any pair of points ...
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5
votes
1
answer
491
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Check if a polygon has an axis of symmetry in $O(n)$ time
Question: Is it possible to check if an $n$-gon has an axis of symmetry in $O(n)$ time?
Note: An $O(n^2)$ algorithm is easy to see: it is easy to check if any given line is an axis of symmetry of the $...
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3
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1
answer
190
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On some centers of convex regions based on partitions
These questions are inspired by Yaglom and Boltyanskii's 'Convex figures'.
Winternitz Theorem: If a 2D convex figure is divided into 2 parts by a line $l$ that passes through its center of gravity, ...
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5
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1
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On folding a polygonal sheet
Consider a polygonal sheet $P$ of area $A$ with $N$ vertices (it material is not stretchable or tearable). Let $n$ be a positive integer >=2.
Question: Let $P$ lie on a flat plane. We need to fold ...
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1
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Fast way to generate random points in 2D according to a density function
I'm looking for a fast way to generate random points in 2D according to a given 2D density function.
For instance something like this:
Right now I'm using a modified version of "Poisson disc&...
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4
votes
2
answers
213
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Algorithm for reporting all triangles with unique interior point
What is known about the complexity of and/or practical algorithms for reporting all triplets of points from finite set of at least four points of which no three are collinear in the Euclidean plane, ...
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2
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1
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Partitioning polygons into acute isosceles triangles
Question: Given an $N$-vertex polygon (not necessarily convex). It is to be cut into the least number of acute isosceles triangles.
Based on this MathSE discussion, one can think of a method to get $\...
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7
votes
1
answer
768
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To minimize the Hausdorff distance between convex polygonal regions
Definition: The Hausdorff distance is the greatest of all the distances from a point in one set to the closest point in the other set.
Question: Given two convex polygonal regions P1 and P2 on the ...
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2
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0
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126
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Intersecting balls with convex regions and a bisector thereof
This question is related to my previous posting
Angle subtended by the shortest segment that bisects the area of a convex polygon
Let $C$ be a convex region in the plane and let $s$ be the shortest ...
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2
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Determine the boundary points of a set of points [closed]
I have a set of points $S=\{(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)\}$. Then how to find the boundary points (which is a subset of $S$) of $S$?
There are methods like convex hull, concave hull and $\...
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5
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Get a point inside a polygon
I have a 2D polygon of arbitrary geometry. I need to find any point that is inside of that polygon. Taking the center won't work, because the polygon might not be convex. Is there a way to quickly ...
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3
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3
answers
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Is there a simple criterion to determine if two parallelograms intersect?
Assume we are given two parallelograms in the plane. How can I check if their intersection is nonempty?
Note that I do not need to actually find the intersection.
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