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 ...

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47 views

Polyhedra from number fields

A question on the disnub mentions golden ($x^2-x-1=0$) gives the dodecahedron + much more. tribonacci ($x^3-x^2-x-1=0$) gives the snub cube. plastic ($x^3-x-1=0$) gives the snub ...
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2answers
38 views

Could tertiary computation negate the need for large memory?

In normal computers, pointers in code "point" to memory. On a lower level, linking and loading turns those "pointers" into numerical codes which really do point to specific bytes (or bits) of ...
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30 views

Distance Geometry Problem (DGP) Programming Language Recommendation

We have been studying DGPs in clinic recently and I was hoping I might be able to get recommendations for computing languages in the processing of large network solutions. Specific computations ...
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25 views

Bound on “width” of points in a plane

Suppose we define width $w(P)$ of point set $P$ in a plane to be the ratio of the maximum distance to the minimum distance between the points in $P$. (Assume unique coordinates so that $w(p)$ is ...
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1answer
122 views

Does a convex hull solution in 3 dimensions result in a minimum-area or maximum-volume solution?

The wikipedia entry for convex hull shows a 2-d example of a random set of points on x-y plane, and the "elastic band" solution that bounds the points with the convex hull solution. The definition of ...
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1answer
23 views

Largest Convex Region in a Star?

Suppose I have a star-shaped region in the plane with a particular point marked. The marked vertex is in the kernel of the star. In the image the left most point is marked in blue. Yes I drew this ...
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1answer
40 views

Computing the approximate or exact area of an isosurface

The isosurfaces I'm reading about are defined by a constant value v in a scalar field. The scalar field is defined by placing n vectors in k-space such that $iso(\vec{x})=\sum_{i=1}^{n}\sum_k(x_k-p_{...
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32 views

3D mesh segmentation simple algorithm

I am developing the algorithm reported in this article: Lest square conformal mapping Here is presented an algorithm to flat a 3d mesh on the parametric space, but i don't understand the ...
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1answer
32 views

Polygon Equal Edge Offsetting?

If I have a random polygon of any complexity, be it a square or an irregular 20 sided polygon, how can I scale this up? I know the coordinates of each point on the polygon, but that is all. Another ...
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17 views

Proof for non-tetrahydralizability of Schonhardt polyhedron

It is established that not all polyhedrons are tetrahydralizable. Schonhardt's polyhedron is the simplest example for it. I was reading the proof for this given in the book "Art Gallery Theorems and ...
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2answers
83 views

Reliable test for intersection of two Bezier curves

Is there a test which reliably decides whether two Bezier curves intersect or not? I don't need to know how many intersections there are or at what parameters they appear at. I just would like to ...
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20 views

Intersection Multiplicity of Rational Plane Curves

Suppose I have two rational curves in the complex projective plane. I know their parametrizations, $<x_1(t),y_1(t)>$ and $<x_2(t),y_2(t)>$ I know I can use Grobner bases to find an ...
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1answer
31 views

What is the monotonicity of a polygon? [closed]

What is the monotonicity of a polygon and why is it necessary to check the monotonicity?
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38 views

Bearing to X,Y, Z [closed]

I’m working on a small project where I’m testing the ability of a laser to accurately measure the angle of a wire. My control for the experiment gives me X,Y,Z coordinates for a point on the wire and ...
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0answers
22 views

Most efficient way of transforming from V-representation to H-representation

What is an efficient way to transform from the v-representation of a convex hull (in terms of vertices) to its h-representation ($Ax \leq b$)?
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1answer
71 views

transforming data from cartesian plane to isometric plane

Here is a brief description of what I'm trying to accomplish. I created a 2d top down game that operated on a Cartesian plane and used some custom polygons to determine collisions and perform actions ...
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1answer
60 views

Bisecting points on a circle

I was working on the following problem. Given n points on a circle, where a point can be specified by its angle from the vertical, how does one find a diameter of the circle such that the number of ...
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72 views

Solving general (dis)entanglement puzzles

What is the state of the art in (modelling and) solving a general (dis)entanglement puzzle? The following picture shows a nice example: There is a project called "The Untangler", which seems to be ...
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1answer
42 views

Use Complex Analysis for Finding Reflection of a Point in a Line

In the question Point reflection over a line complex analysis was used. Can anyone tell me if this method is superior to the standard method (find a perpendicular from the point to the line, find a ...
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2answers
275 views

“Surface Area” of $k$ simplex in $\mathbb{R}^{k}$?

Consider the $k+1$ vertices $(x_1,\ldots,x_{k+1})$ with $x_i\in\mathbb{R}^k,i=1,\ldots,k+1$. I know that the "volume" of the $k$-dimensional simplex formed by these vertices is proportional to $$\...
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35 views

translate of a homothet of a convex body

Suppose we have given a convex body $K \subset \mathbb{R}^2$. How can we prove that it contains a translate of its homothet $-\frac{1}{2} K$? hint: take three vertices $A, B$ and $C$ of the convex ...
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3answers
38 views

Dual set of the unit ball is part of the unit ball.

Define the unit ball centered at the origin as $B=\{x\in\mathbb{R}^d\mid \|x\|\leq 1\}$. Define the dual set of set $X$ as $X^*=\{y\in\mathbb{R}^d\mid\langle x,y \rangle\leq 1\ \forall x\in X\}$. I'...
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2answers
54 views

How can we find the area of the triangle which covers a finite point set in $\mathbb{R}^2$ by using the interior triangles with specified area?

Suppose we have given a finite point set $X \subset \mathbb{R}^2$ in a way that any triangle made by vertices of $X$ has area at most 1. How can we prove that there is a triangle of area 4 which is ...
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2answers
43 views

do infinite family of lines in $\mathbb{R}^2$ have a common point by knowing that any three of them have common point?

Suppose we have given an infinite family of lines; say $\mathfrak{F}$, in the plane $\mathbb{R}^2$ such that any three of the lines in $\mathfrak{F}$ have a common point. How can we prove that all ...
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2answers
116 views

Check if a point lies in a circle defined by three other points.

I'm learning Computational Geometry, and need to check whether a point p lies inside a circle defined by a triangle(made by 3 points $a,b,c$, in counterclockwise order). A very convenient method is ...
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83 views
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1answer
34 views

Convex hull of union of polytopes in halfspace representation

Suppose I have two polytopes in $\mathbb{R}^n$ given in H-representation as $P_1: \{x | H_1 x\leq b_1 \}$ $P_2: \{x | H_2 x\leq b_2 \}$ My question is, if it is possible to efficiently (i.e., avoid ...
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0answers
46 views

Area swept out by a moving polygon

Say you've got a polygon (say a quadrilateral) that is moving along a certain known path in a plane. The polygon may be changing in shape as it moves, however you know the paths of each of the ...
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16 views

Reference request: quantifying qualities of a bunch of points using statistics derived from their Delaunay triangulations

I am interested in using Delaunay Triangulations (DTs) to explore the statistics of a cluster of points. Here's an example cluster of points $P$, with its $DT(P)$ (for now, ignore the difference in ...
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1answer
19 views

How to ensure that a point A(x,y) doesn't cross a imaginary line between two other points.

I'm programming an web app and I need some help with a problem, as my mathematical skills are not great. I have 4 points in an XOY system with the origin in the top left. I need to make sure that any ...
2
votes
1answer
29 views

Determine the locus

Let $0<a<b$. Consider two circles with radii $a$ and $b$ and centres $(a, 0)$ and $(b,0)$ respectively with $0<a<b$. Let $c$ be the center of any circle in the crescent shaped region $M$ ...
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29 views

Calculate the shortest continuous path between shapes without passing thru other shapes in a specific order?

I have the following points, shapes and paths I would like to calculate how to go thru: We do not have to move in a diagonal direction if that poses a problem. Here would be the movement with just ...
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9 views

Find Maximum Density of Point Set

Suppose I have $n$ 3D points. Given any point $x$, radius $R$, and integer $k \leq n$, I can efficiently return a list of (up to) $k$ points nearest to $x$ and closer than $R$ (i.e. bounded k-nearest ...
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23 views

Intersection of line with polygon

We have a convex polygon P with n edges and a line L (not a line segment!).You are allowed to do some preprocessing .After preprocessing ,find whether the line intersects polygon in O(logn) time . ...
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24 views

number of points with distance $\ge \delta$ that can fit inside a square with edge length $\delta$

I want to prove the claim that if you have a square with edge length $\delta$ and you want to fit as many points, each pair has distance $\ge \delta$, inside that square, You can fit at most 4, and ...
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23 views

Tangent Plane of two polyhedron from below in 3D

The convex hull or convex envelope of a set X of points in the Euclidean plane or Euclidean space is the smallest convex set that contains X. A polyhedron (plural polyhedra or polyhedrons) is a solid ...
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1answer
47 views

equivalence of theory of reals and Rationals

Present a sentence φ that is in theory of reals but not in thoery of Rationals Following up from this question what is the approach to show that both the theories are equivalent Th(R, 0, 1, +, ≤) ...
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25 views

how to find a tetrahedron in $R^n$ to bound an ellipsoid (again in $R^n$)

Assume you are given the following ellipsoid in $R^n$: $E: (c+\sum_{i=1}^n \alpha_ix_i)^2$, where $x_i$ 's are the coordinate variables. c and $\alpha_i$'s are constant. now the question is how to ...
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2answers
55 views

Distance to a convex hull

I have $n$ points $x_1,\dots,x_n\in\Bbb R^d$. Given a new point $y$, how can I verify that distance from $y$ to the convex hull of $x_i$ is less than a given $\varepsilon$? It's not important for me ...
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0answers
20 views

Boundary points of a convex hull

I have $n$ points $x_1,\dots,x_n\in\Bbb R^d$. What is an efficient way to get (some) subset of boundary points out of them? Also, if I add a new point to this set, how can I efficiently update this ...
2
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2answers
58 views

Test if point is in convex hull of $n$ points

I have $n$ points $x_1,\dots,x_n\in\Bbb R^d$, and I would like to check that some other point $y$ lies in their convex hull. How can I do this in some efficient way? I think that there was an ...
2
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1answer
47 views

Fastest computation to find out if two vectors intersect (programming problem)

I'm trying to write a program that should solve a 12x12 rush hour problem: I won't go in the details of this program to much. The program already works for 6x6 puzzles, but for 12x12 puzzles, it is ...
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0answers
34 views

Rigid circles inside a square

There are $n$ circles each of radius $r$. They are needed to fit into a square with side length $t$ in a way such that, the circles can't move in any direction (each one is adjacent to some other ones ...
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97 views

How to measure the irregularity of a hexagon?

I need to evaluate the quality of a list of machine parts, which roughly has one center point surrounded by 6 exterior points. If the quality is good, then the 6 exterior points will form a regular ...
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2answers
32 views

Center of mass from shape boundary

It is possible to find a shape center of mass by only its boundary? Would the average coordinates of X and Y would approximate my center of mass? (If it would work how good the approximation is going ...
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0answers
13 views

What line-polygon clipping algorithm can I use to ensure that the resultant endpoints are always within the polygon?

I have a 2D plane, partitioned into n-sided, convex polygons. I'm using WRF's PNPOLY algorithm for polygon inclusion to ensure that a point belongs inside one and only one polygon. Is there an ...
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46 views

Constructing triangulations algorithmically

I am developing a Python package for computations in algebraic topology (namely: cohomology and Massey products on manifolds). Basically all the stuff I'm doing requires an explicit triangulation of ...
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27 views

Fencing $n$ points while keeping minimum distance $d$ from each point

Consider this problem **I have a land consisting of $n$ trees. Since the trees are favorites to cows, I have a big problem saving them. So, I have planned to make a fence around the trees. I want ...
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1answer
114 views

Calculate the area of an irregular cyclic convex polygon

I want to write a program in C++ to calculate the area of irregular cyclic convex polygons. However, the inputs are in the form corner point angles. I am just not sure what the inputs mean and what ...
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0answers
53 views

The number of facets of an affine image

I have a full dimensional polyhedron $P_1 \subseteq \mathbb{R}^d.$ Now i define another polyhedron as follows: $$P_2 = AP_1 \oplus B$$ with $A \in \mathbb{R}^{(d-1) \times d}, \,\, B \in \mathbb{R}^{...