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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2answers
57 views

Simplest graph that is not a segment intersection graph

Given a finite collection $S=\{s_1,s_2,\ldots,s_n\}$ of straight-line segments in the plane, their intersection graph $G(S)$ is a graph that contains a vertex $v_i$ for each segment $s_i\in S$, and an ...
2
votes
2answers
56 views

How to make sure that a given set of points lie on the boundary of a possible square?

Given a set of integral coordinates , check whether all the points given lie on side of a possible square such that axis of the square so formed lie parallel to both X-axis and Y-axis . Suppose ...
2
votes
2answers
90 views

Polytope parametrization

How one could parametrize a convex polytope? By parametrization I mean something like in multiple integrals, when to integrate over an area one can integrate over one variable in an interval $[l,r]$ ...
1
vote
2answers
64 views

Determinant (computational geometry)

Let p=(px,py),q=(qx,qy), and r=(rx,ry).Show that the sign of the determinant |1 px py| D=|1 qx qy| |1 rx ry| determines whether a point r lies to the ...
1
vote
2answers
66 views

What is the equation stands for in geometry(intuitively)?

I am writing a bilinear interpolation method. This method can be abstract by solve the equation A*x = b, A is a 4x4 matrix below: $A=\begin{pmatrix} 1 &x_1 &y_1 &x_1y_1\\ 1 ...
0
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2answers
38 views

Room for computational geometry in advanced algorithms course

I am currently putting together an independent study in advanced algorithms and because of my interest in (computational) geometry, wanted to include as many interesting algorithms from this field as ...
6
votes
1answer
208 views

How to determine surface from given normal vectors and their distance on that surface

Situation: We have a bendable, non-stretchable surface, like a piece of cloth, with a regular grid on it. Unknown manipulation of the surface is done while preserving it's structure We recieve 3 ...
4
votes
1answer
137 views

Dirichlet's approximation theorem (simultaneous version): proof via Minkowski's theorem

There is a proof of the Dirichlet's approximation theorem based on Minkowski's theorem. The proof is given on wikipedia (http://en.wikipedia.org/wiki/Dirichlet%27s_approximation_theorem) and it is ...
3
votes
1answer
88 views

Computing with graphs in surfaces

I am currently working on a research project involving a polynomial defined for graphs in surfaces, similar to the Tutte polynomial, except with terms accounting for the embedding. At the moment, it ...
2
votes
1answer
79 views

Modify the closest-pair algorithm to use the $L_\infty$ distance.

I'm trying to understand the closest pair of points problem. I am beginning to understand the two-dimensional case from a question a user posted some years ago. I'll link it in case someone wants to ...
2
votes
1answer
75 views

A method to test for uniform distribution over a convex polytope

Assuming I have a convex polytope defined as the intersection of $Ax=b$ and $x>0$ and I have a way to sample points from this object, is there a way I can test for uniformity of these sampled ...
1
vote
1answer
56 views

Prove that volume of a ball in a polytope is very small

An exercise in a book asks to prove that for a bounded convex polytope $P\subseteq\mathbb{R}^n$ defined as an intersection of $k$ closed halfspaces and for a unit ball $B^n$ contained in $P$ the ...
1
vote
1answer
77 views

Find equation of line without using division

I need an algorithm to find equation of a line without using division. Given a line by two points on it, with coordinates: $(x_1, y_1),\ (x_2, y_2)$. We can simply get the line equation by the ...
1
vote
1answer
56 views

Convex Combination of Disks

We can define a closed disk $D$ with center $c$ and radius $r$ as the set of points $x$ satisfying $f(x) \le 1$ where $f(x) = \frac1{r^2}\lVert x-c \rVert^2$. Now take two disks $D_0,\,D_1$ with ...
1
vote
1answer
111 views

Convex Hull in Hierarchy Structure

As a beggining to convex hull algorithms lecturer introduced the structure which it's called "Hierarchy Structure". Hierarchy Structure: in every given convex hull there is a maximum size convex hull ...
0
votes
1answer
21 views

How do I place two points on two axis-aligned line segments such that they have given Euclidean distance l?

This is my problem: I am given two axis aligned line segments $l_1$ and $l_2$ of finite length and a distance $l$. How do I find points $p_1 \in l_1$ and $p_2 \in l_2$, such that $||p_1-p_2||_2 = l$? ...
0
votes
1answer
17 views

Finding other two vertices of a triangle from centroid and one vertex?

I am working on some gesture recognition for my game and I want to find if a point is inside the triangle created by the user or not. For that I need three vertices. Currently I am using the '$1 ...
0
votes
1answer
23 views

Given a set of 2D vertices, how to create a minimum-area polygon which contains all the given vertices?

Not sure whether this question belongs here or mathoverflow. You can assume that all the vertices are unique. The given vertices can be the vertices of the polygon, thus they do NOT have to be inside ...
0
votes
1answer
26 views

Visible objects from a point in a polygon with holes in 2d

The problem is the following: Given a polygon P with h holes/objects and a point c inside P but outside the holes/objects. P has n given vertices and each hole/object h has 4 vertices (the ...
0
votes
1answer
69 views

Use of Delaunay Triangulation and Voronoi Diagram to find alpha shape using Edelsbrunner's algorithm

I am learning how to find the shape of a set of points in 2-D. I understand that Alpha Shape method is a good way to find the shape of a set of points. Alpha Shape was originally introduced by H. ...
0
votes
1answer
25 views

Point location( planar subdivision)

Show that, given a planar subdivision S with n vertices and edges and a query point q, the face of S containing q can be computed in time O(n). Assume that S is given in a doubly-connected edge list
0
votes
1answer
14 views

Computing the Center of Gravity of the Unit Cube + Halfspace

Consider the unit hypercube in $\mathbb{R}^N$. $\mathcal{P} = \{\mathbf{x} ~| ~x_i \in [0,1] \text{ for } i=1,\ldots,N \}$ and a half-space which intersects the unit cube: $\mathcal{H} = ...
0
votes
1answer
23 views

Geometry (Convex Polygons)

Let P be a set of points in the plane. Let P1 be the convex polygon whose vertices are points from P and that contains all points in P. Prove that this polygon P1 is uniquely defined, and that it is ...
0
votes
1answer
32 views

Divide set of points by a plane so sum of distances of points on either side of plane is equal

I have a finite set of points A and another point C. I would like to compute a vector N so that the plane defined by C (point on plane) and N (normal of plane) divides all points in A with the sum of ...
0
votes
1answer
78 views

Term For Rotating 3d Vectors About a Pivot Point

What is the term for Rotating a 3d Vector about another 3d Vector (Pivot Point)? For example; if I want to move X distance from one point towards another point - the mathematical term for this ...
0
votes
1answer
42 views

How to test for a polygon witn n vertices if it's nonintersecting polygon or not?

How can you design an algorithm to know if an n-vertex polygon nonintersecting ? On what criteria is the test going to be
0
votes
1answer
75 views

Using cartesian coordinates how to get the segment overlapping two lines/segments?

There must be an algorithm to find the coordinates of the segment overlapping (fully or partially) two lines/segments but my googling does not produce any significant result. Maybe I don't use the ...
0
votes
1answer
269 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 ...
0
votes
1answer
125 views

Convex hull questions

Suppose I am in D-dimensional space, e.g. D=2. What is the minimum number of points to create a valid convex hull? If D = 2, would I need 3 points to create a convex hull (i.e. to form a triangle)? ...
0
votes
1answer
70 views

Understanding the Wolfram Demonstration “Distance of a Point to a Polygon”

I've recently came across a neat Wolfram Demonstration script by Jaime Rangel-Mondragon that calculates the minimum distance from a point to an arbitrary convex or non-convex polygon: ...
0
votes
1answer
69 views

Generating Vectors under Constraints on 1 and 2 norm

Update: I left out some important information in my previous description... I am actually dealing with a special problem, which is better described as follows: Given user-specified parameters ...
0
votes
1answer
134 views

Steepest slope gradient of a vertical plane

I know the steepest slope gradient (Azimuth) of a 3D plane can be obtained by projecting normal vector onto XY Plane. So, when the plane is slant, the steepest gradient will be a some value. ...
0
votes
1answer
145 views

Analytic Intersection of Objects Located on a 3D Grid's Vertices

I previously posted this question on stackoverflow, but it's really more of a mathematical question. I have reworked the question for presentation here. I have a regular 3D unit cubic grid of ...
-1
votes
1answer
245 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 $y_3$ ...
-1
votes
1answer
117 views

What are algorithms or approaches to find a convex hull on higher dimensions?

I have some background in 2D computational geometry and understand how to find a convex hull in 2D. Now I'm looking at a set of vectors with 20-some components and want to find the convex hull on ...
0
votes
0answers
94 views

Does a single Gauss-Seidel iteration lead to unique coordinates?

I managed to reduce certain computational problem to the Gauss-Seidel solution of the following linear system: $$Ax=Ly,$$ where $A, L\in\mathbb{R}^{n\times n}$, and $x,y\in\mathbb{R}^{n\times 2}$ are ...
0
votes
0answers
49 views

Maintaining the line with the 2D iterands

Suppose a linear system is given $$AX=B,$$ where $A\in\mathbb{R}^{n\times n}$ is a symmetric strictly diagonal matrix, and $X, B\in\mathbb{R}^{n\times 2}$. Therefore, the 2D Jacobi iterative solver is ...
0
votes
0answers
83 views

How to interpolate sequential points to obtain functions and/or vectors?

I would like to know how I can interpolate a sequence (time) of points in order to obtain curves as some kind of mathematical functions. Unfortunately math is not my area so I don't really know the ...