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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13 views

Sum of distances of points in unit closed disk

Let $D$ be the closed unit disk in the plane, and let $p_1, p_2, \dots, p_n$ be fixed points in $D$. My question is, does there necessarily exist a point $p$ in $D$ such that the sum of the distances ...
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29 views

Relation between farthest pair of points and closest pair of points in plane

I am writing program for obtaining distance between shortest and farthest pair of points among the given points in plane .I am able to calculate them both the shortest one using divide and conquer ...
2
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0answers
49 views

3D kinematic geometry problem motivated by chemistry

It is well known that six carbon atoms can form a ring called cyclohexane. Since the angle between bonds is $\cos^{-1}\left(\frac{-1}{3}\right)\approx 109^\circ$, the ring is not a planar hexagon. ...
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2answers
7k views

How to find the third coordinate of a right triangle given 2 coordinates and lengths of each side

p2 |\ |b\ | \ A| \C | \ |c___a\ p1 B p3 If given point p1 & p2, side A & B how would you find point p3? I know given this information you ...
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2answers
2k views

Two plane intersection and angle between 2 planes

I am trying to implement my problems in different ways. So, may be though this question has some relation to some other questions, please answer me. We know; Intersection of two planes will be given ...
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2answers
2k views

How to calculate volume of 3d convex hull?

Convex hull is defined by a set of planes (point on plane, plane normal). I also know the plane intersections points which form polygons on each face. How to calculate volume of convex hull?
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0answers
48 views

What are the techniques one can used for rule based plane generation?

I've asked the question here at gamedev SE, but the response wasn't too encouraging. So I try to reask again, from a slightly difference perspective. I have a terrain, which is defined by mesh. And ...
3
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2answers
1k views

Point closest to a set four of lines in 3D

Given four lines in $3D$ (represented as a couple of points), I want to find the point in space which minimizes the sum of distances between this point and every line. I'm trying to find a way to ...
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2answers
576 views

“Concave hull” - Possible? Feasible? Deterministic?

So there are several questions regarding how to compute the convex hull of a set of points. However, let's say that on inspection the set of points inscribed a star shape. A Convex hull algorithm ...
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1answer
31 views

Tetrahedron subdivision

What are all the possible subdivisions of the P3 tetrahedron (i.e. for each face, 3 vertices plus two points per edge, located at 1/3 and 2/3, and the centroïd of the face, so a total of 20 points for ...
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1answer
20 views

How to find the closest line to two segments?

I have two segments in 2D space, defined by their endpoint x and y coordinates. How can I find a best-fit line using vector algebra (formally, that minimizes the integral of square-distance from it to ...
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0answers
8 views

Rotational normalisation of a sequence using PCA

I have a 1D contour, defined as a sequence of points in 2D space. For arguments sake, lets say I want to achieve rotational normalisation by aligning the direction of the first principal component of ...
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0answers
37 views

Unique Cicum-Sphere in n-Dimensions

I want to know if there is a generic way to find the circumcenter of an (n-1)-simplex in n-dimensions. Does there a exist a unique sphere which passes through n-points in n-dimensions? For example say ...
6
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2answers
82 views

How to determine whether a point is inside a closed region or not?

Take the following parametric equation of an implicit curve as an example: $$ \left\{\quad \begin{array}{rl} x=& 9 \sin 2 t+5 \sin 3 t \\ y=& 9 \cos 2 t-5 \cos 3 t \\ \end{array} \right. $$ ...
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0answers
29 views

Relation between parallel transport and Jacobi field II

Before I asked a question here: Relation between parallel vector field along a geodesic and Jacobi field along that same geodesic The current question is related, and actually arise from numerical ...
2
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1answer
93 views

Algorithm to compute whether a stabbing line exists for a set of line segments

Let $S$ be a set of n segments in the plane. A line $L$ that intersects all segments of $S$ is called a traversal or stabber of $S$. Give an $\mathcal{O}(n^2)$ algorithm to decide if a stabber for $S$ ...
6
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3answers
70 views

Way to measure the similarity/difference of 2D point clouds

i need a way to measure the similarity or difference of two point clouds? The number of points in each point cloud can be different. The Point clouds are already aligned. By similarity I mean the ...
0
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1answer
24 views

finding volume of an n-dimensional pyramid numerically

In my experiment I need to compute hypervolume/area from a set of points, let's start with a base case -- Triangle: In this case, I have 3 points in a 2D space and they make a triangle, $p_1 = ...
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1answer
16 views

Voronoi diagram of a set of vertices of a mesh.

i have a triangulated mesh. I have some vertices which are part of the vertices of the mesh. Is there any algorithm to compute the voronoi diagram of these set of vertices. The triangulated mesh ...
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2answers
53 views

Tetrahedra from it's inscribed sphere

I'm facing a geometrical problem: Given a sphere S, I want to calculate the vertices of the tetrahedra T whose inscribed sphere is S. In other words I want to calculate a tetrahedra from it's ...
1
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2answers
76 views

Circumcenter of Tetrahedron (in 4D)

I am trying to calculate the circumcenter of a tetrahedron in 4 dimensional space. Basically what I am looking for is the center of the smallest sphere which passes through all 4 vertices of the ...
2
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0answers
28 views

Compute volume of the tetrahedron from circumsphere test

I'm working on a computational geometry algorithm. In every iteration I solve the matrix below, where (a,b,c,d) are the vertices of a tetrahedron, and e is an arbitrary point. Solving the determinant ...
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0answers
6 views

Equalize length of 1-ring edges of vertex

My question is how to equalize the length of edges in 1-ring neighbours of while-circled vertex in the below figure Hope to see your answer!
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0answers
75 views

Find the intersection between two convex hulls, in this specific case

We work over $\mathbb{R}^K$. Let $V$ the set of vectors whose coordinates take values $0$ or $1$, or equivalently the corners of the unit cube $[0,1]^K$, . Let $d:\{0, \ldots, K\} \to \mathbb{R}_+$ ...
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2answers
363 views

How to get the third point coordinates in isosceles triangle?

Isosceles triangle $ABC$ $AB = AC = d_1$ $BC = d_2$ $A = (x_1, y_1)$ $B = (x_2, y_2)$ $C = (x_3, y_3)$ $\angle BAC = \phi$ $\angle ABC =\angle ACB = \theta$ I want an equation for $x_3$ and $y_3$ ...
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6answers
2k views

Is it possible to solve any Euclidean geometry problem using a computer?

By "problem", I mean a high-school type geometry problem. If no, is there other set of axioms that allows that? If yes, are there any software that does that? I did a search, but was not able to ...
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0answers
12 views

Prove Theorem with Groebner Basis

I'm trying to prove some theorems using Groebner Basis (as described in Cox, Little and O'Shea Link ) The mentioned book gives as an excercise to prove Pappus theorem using the given methodology, ...
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1answer
34 views

How do I calculate center of mass of a grid-type object?

I am making a game with 2D objects that are grid based. These objects are made out of tiles that are actually the cells of the grid. Each tile has a "mass" that is more than 0 and no upper limit. I ...
0
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1answer
31 views

Find all nearest points

I have two sets: $$P = \{p_1, p_2, ..., p_n\}$$ $$Q = \{q_1, q_2, ..., q_m\}$$ For each $p_i$ point I need to find all nearest points in $Q$. I.e., $$p_i \rightarrow \{ q_{i_1}, q_{i_2}, ..., ...
1
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1answer
70 views

T-shaped polygons

Is there any coefficient that can indicate T-shaped polygons ? Examples of T-shaped polygons:
0
votes
1answer
16 views

Given a known line segment and two known horizontal lines, how do I find the line subsegment between the two parallels?

I am trying to clip a line segment between two parallel horizontal lines. I know the location of the two vertices on the larger segment, but need to know the points at which it intersects the ...
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1answer
305 views

Determining the direct and transverse tangent lines for two non-overlapping ellipses

I am trying to determine the direct and transverse lines for two non-overlapping ellipses. I specifically mean that the two ellipses are totally separated from each other with no shared regions. I ...
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0answers
34 views

What is the mathematics behind the two animations?

I found two animated GIFs from a designer's website, which looks very impressive: My questions are: what is the mathematics behind them? How to obtain the mathematical formulas and equations of ...
0
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1answer
31 views

What does R^d in last lines refer to

The image above is snapshot in the journal Geometric Approximation http://sarielhp.org/papers/04/survey/survey.pdf via Coresets .I could not figure out what is ...
0
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1answer
55 views

Knowing only the coordinates of the North-East and South-West corners of a rectangle, how to check if a point is inside a rectangle?

This is similar to this question. What's different is that only the coordinates of the North-East and South-West (or North-West and South-East) corners are known. My question is, can you directly ...
2
votes
1answer
102 views

Minkowski sum of two polytopes via the halfspace representation

If i have two polytopes denoted by $P_1, P_2 \subset \mathbb{R}^d$, suppose their halfspace representations are respectively $H_1x \leq K_1$ and $H_2x \leq K_2$. Now, considering their Minkowski sum, ...
0
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1answer
162 views

VC dimension for Rotatable Rectangles

It can be shown that VC dimension of rotatable rectangles is 7. The problem is I cannot understand how to approach the solution. So far I used bruteforce to solve this kind of problem, I was drawing ...
0
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0answers
58 views

What is the shape of the set of integer sided acute triangles with largest side n?

I played around with Gauss circle problem and found that if you take a certain sum in reverse and "in forward" and subtract the resulting sequences you get the OEIS sequence: https://oeis.org/A247588 ...
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1answer
83 views

(x,y) coordinates from gluing together a sequence of right triangles with arbitrary angles [duplicate]

I have been scratching my head all day over this question for one of my assignments. I haven't made any progress and I'm at the point of giving up. Here's what I need help with. Start by gluing ...
2
votes
1answer
45 views

Bounding Sphere for Two Hyperrectangles

Please see the image for best illustration of the task. I have two hyperrectangles, $\text{R1}$ and $\text{R2}$, whose exact location and size is arbitrary. Now, my task is to construct a bounding ...
1
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1answer
89 views

What is a composition of two binary relations geometrically?

the composition was defined as follow: (a,b) \in (R;S) <=> there is c | (a,c) \in R and (c,b) \in S . If our two relations R and S are two convex polygon ...
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0answers
42 views

does any polyhedral partition admit a convex piecewise quadratic surface defined over?

Given a polyhedral partition, i learnt that there exist some conditions for the existence of a convex piecewise affine surface over this partition for example the following study. ...
0
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0answers
57 views

How to use CVX to solve this problem?

I have a function in the variables $x_{kl};\ k,l=1\ldots,m$, $$\sum_{i=1}^n \sum_{j=1,j<j'}^{N_i}\left( b_{ij} b_{ij'}- \sum_{k,l=1}^{m}x_{kl}f_k(a_{ij})f_l(a_{ij'})\right)^2$$ where ...
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0answers
22 views

Find largest regions bounded by a set of planes

Suppose we are given a set of planes that partition the unit cube into a large number of regions. Is there a computationally efficient way to find the region with the largest volume?
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0answers
43 views

Understanding BlowUp Computation in Singular

Many of us might know that "Singular" is a computer algebra system for Algebraic Geometry, Commutative Algebra and Non-commutative algebra. This is a procedure in "Singular" for computing blowups. ...
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1answer
34 views

How to compute Convex hull of set points from voronoi diagram

Assume $n$ points in the plane and their Voronoi diagram are given, prove that the convex hull of the points can be computed in linear time.
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79 views

Fast search of local positive quadruples on the sphere

Let $U = \{u_{1}, u_{2}, \ldots, u_{n}\} \subset \mathbb{R}^{3}$ be the finite set of points on the unit sphere in $\mathbb{R}^{3}$: $||u_{i}||_{2} = 1$ Definition: Quadruple of points $(u_{i}, ...
1
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1answer
51 views

Convex hull solving using a rubber band?

The convex hull can be found by stretching a rubber band so that it contains all the points and then releasing it. So my question is : lets assume that we have a robot (a theoretical robot) to solve ...
2
votes
2answers
97 views

Find polygon with smallest perimeter that encompasses all points

Given a random set of points in 2D space such as: How would one go about finding the smallest perimeter polygon that encompasses all points and has a point as each one of its vertices? For the ...
1
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0answers
36 views

population of dots with normal distribution of pitch

I want to generate a plot that shows a rectangle populated with dots, where the dot-to-dot distance (pitch) distribution is a lognormal (or a gaussian). I want to be able to change the mean dot-to-dot ...