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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88 views

Higher Order Voronoi Diagram of a Poisson Point Process: What do we know?

This question is looking for probabilistic results of the Voronoi diagrams of 2-D space when the points are distributed by a homogeneous Poisson point process. The results can be the distribution of ...
0
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2answers
351 views

Circumsphere of a tetrahedron undefined?

I am trying to find 3D alpha shapes from my data-set. In doing so, I am keeping only those tetrahedra that have circumradius below a certain threshold. However, while finding the circumradius of the ...
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0answers
45 views

Background required for Computational Geometry

I am hoping to enroll in Computational Geometry course this spring. This was the textbook used for the course in the past. I am trying to figure out if I have required math background for this ...
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1answer
90 views

Problem while constructing Delaunay triangulation

At the moment I'm implementing an algorithm to construct a Delaunay triangulation for a set of points. I'm using the algorithm described in Computational Geometry: Algorithms and Applications. The ...
2
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0answers
68 views

Generalization of Minkowski's theorem

I would like to prove a generalized version of the Minkowski's theorem, but I don't quite know how to do it. Here is what I would like to prove: Let $X\subset \mathbb{R}^d$ is convex, symmetric ...
2
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1answer
76 views

Reference for important results in linkage theory and their proofs

Are there books or lecture notes that comprehensively introduce the (geometric/topological) theory of mechanical linkages, as well as important results and their proofs? For instance, Kempe's ...
2
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0answers
82 views

Orthogonal 4-cut of a convex polygon

Given a convex polygon with N vertices I need to cut it into four equal area parts with two straight orthogonal cuts. I feel that I have all the necessary pieces to solve this puzzle, but I can't put ...
4
votes
1answer
213 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 ...
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1answer
59 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 ...
4
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2answers
117 views

Convex sets: a hint on how to solve a problem

Could anyone give me a hint on how to solve the following problem? Let $X_1, \dots, X_{d+1}$ be some finite sets in $\mathbb{R}^d$, such that the origin lies in ${\rm conv}(X_i)$ for all $i \in \{1, ...
1
vote
1answer
204 views

3D Convex Hull and The Gift Wrapping Principle

I am currently trying to implement a 3D convex hull algorithm that is based on the paper Convex Hulls of Finite Sets of Points in Two and Three Dimensions by F.P. Preparata and S.J. Hong, but I’ve run ...
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0answers
125 views

Find vertex of a parallelogram/parallelepiped/parallelotope with minimum distance to a point

Suppose you have a parallelogram and a point. It's easy to tell which of the parallelogram's vertices is closest to the point (Euclidean distance) by checking the distance for every vertex - but this ...
2
votes
1answer
117 views

How to extract the indeterminates from a set of polynomial?

I am a biologist and I am facing a huge problem. I would like to extract the indeterminates of a set of polynomials, for example, I have: $f_{1} = \{\\x_{3}^{2} + x_{1}*x_{2} + x_{1} + x_{1}*x_{3},\\ ...
0
votes
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 ...
0
votes
1answer
844 views

How to find whether the line is inside the polygon or outside.

I have a polygon How can i prove whether the black color line lies outside the polygon or inside the polygon . Given the coordinates of the black line and all the vertices of the polygon.
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1answer
92 views

When is a convex polygon inscribable?

Defining the diameter of a convex polygon as the maximum possible distance between all pairs of vertices, can we conclude that the convex polygon is inscribable (i.e has all its sides as chords of a ...
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0answers
31 views

sweeping edges till they get a given elevation on an oblique plane

I am constructing wireframe model of 3d objects (prisms,..etc.). from a triangular mesh, I have obtained boundary points and fit striaght lines in order to get polygon edges refering to prism ...
0
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1answer
83 views

Packingof Spheres in 3D

I am looking to find out the size of the largest sphere , that can fit in the voids created by packing spheres ( hcp) of radius R.
0
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1answer
177 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
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1answer
113 views

Correcting plane parameters with the fixed azimuth angles

I am trying to reconstruct specific 3d objects such as cubes, pyramids and so on. For this, i am using point cloud data and then fitting planar surfaces for the segmented point patches. Planes ...
0
votes
1answer
77 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: ...
1
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1answer
244 views

Ellipse arcs. Draw a tangent line in the end point or make arc longer?

I read this article: link It describes how to draw ellipse arcs at all from svg. Each ellipse is described with the following params (and I know them): x1, y1, x2, y2 - arc from point (x1, y1) to ...
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votes
2answers
227 views

Finding the largest circle that contains a single point in a set (and no other point)

Given a bounded $A \times B$ rectangle with a set of chosen coordinates, generated for example with the command: ...
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0answers
201 views

Formula for intersection of “power” curve and parabola.

EDIT I have edited this question to make it more clear. I have spent quite some time trying to find this on Google, but haven't succeeded. I need the formula(s) to determine the intersection ...
1
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0answers
34 views

Length of the voronoi diagram

Does there exist an algorithm for computing the length of the voronoi diagram of a set of points or just gives the intersection points of the voronoi diagram?
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0answers
61 views

largest polygon from segments

There is a set of segments. and I want to calculate the area of the largest polygon which can be build using these segments. I try to search it, but I can't find anything. thanks
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1answer
128 views

Graphics clipping: How can repeated half-space clipping fail?

Hi I am currently going through the past exam problems and I am stuck on this clipping problem. Could you give me some hint on how to solve it? If we clip a polygon to a window, it is inadequate ...
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1answer
67 views

Center of Distance

I am given $N$ points in a 2D plane($x$ and $y$ coordinates). I have to find a point in this plane with coordinates $X$ and $Y$ such that: $$\sum_{i=1}^N \max\{|X - A_i|, |Y - B_i|\}\text{ is ...
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2answers
137 views

Checking convexity from outside

Is there any method or algorithm to determine convex (or non-convexity) property of a region from outside (perimeter) ? One way is plotting tangent line in each point of perimeter and discuss how ...
2
votes
1answer
185 views

On finding the nondominated set of vectors. How to understand this algorithm?

L et us denote by $x_i(v)$ the $i$th coordinate of $v \in \mathbb{R}^d$. Then $v = \left [ x_1(v), x_2(v), \dots ,x_d(v) \right ]$ We say that a $v \in \mathbb{R}^d$ dominates another vector $w \in ...
8
votes
2answers
173 views

How to check if polylines can be untangled?

In a program I'm writing I need to be able to check whether a straight line between two points is homotopic to a polyline between them. For example in the below example the first one is equivalent to ...
0
votes
2answers
299 views

Proof of the following: How many $(n-2)$ dimensional faces from a corner of a hypercube

I asked a question earlier regarding the number of $(n-2)$ dimensional faces exiting a corner of an $n$ dimensional hypercube. (For example the number of points in a corner of a square, or the number ...
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vote
0answers
174 views

Optimal bounding boxes selection for $N$ rectangles

Suppose that I have $n$ straight rectangles on a plane $r_i = (x_i,y_i,w_i,h_i)$. Each rectangle has a cost function, its area $A(r_i) = w_i \cdot h_i $. I can also "merge" 2 or more rectangles into ...
0
votes
0answers
38 views

Polytime programming

Given a linear system of the form: $$x_r = a$$ $$x_j = b$$ $$c_1x_1 + c_2x_2 ... c_nx_n = n$$ $$x_1 + x_2 + x_3 ... x_n = k $$ $$0 \leq a,b,x_1, x_2, x_3 ... x_n \leq 1$$ $$k \geq 0$$ How quickly ...
3
votes
1answer
93 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 ...
4
votes
1answer
110 views

Covering all the edges of a hypercube?

Consider an arbitrary $n$- dimensional hypercube: If we select $n - 1$ corners of that hypercube and highlight all $(n - 2)$ dimensional elements that originate from each of the corners is it ...
2
votes
1answer
78 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 ...
0
votes
1answer
1k views

How to find rotation angles along X,Y,Z axes with a known vector to bring the axes to correct situation

I am working with 3d point data. When I checked the data I realized that there is some error on my data and need to do some kind of rotational rectification because the points which should be ...
0
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1answer
315 views

my plane is not vertical, How to update 3D coordinate of point cloud to lie on a 3D vertical plane

I have a bunch of points lying on a vertical plane. In reality this plane should be exactly vertical. But, when I visualize the point cloud, there is a slight inclination (nearly 2 degrees) ...
4
votes
1answer
55 views

How can I geometrically (or geographically) group items together?

I'm a programmer, and I'm working on a project that takes a bunch of photos and separates them into groups by their gps coordinates. I have no experience in things like geometric group theory so I'm ...
1
vote
1answer
165 views

What is the typical method for sampling uniformly in a convex polytope

The polytope in my case is the intersection of the k-plane $Ax=b$ and $\{x>0\}$ where $A$ is the constraint matrix and $b$ is some solution. I'd like to find a method that is fast and efficient for ...
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vote
0answers
67 views

Checking if a binary vector lies in the affine span of given binary vectors

Let $x_1, \ldots,x_N \in \{0,1\}^D$ be $N$ binary vectors, assumed affinely independent (in the field of reals). Is there an efficient algorithm for determining whether a new binary vector $x_{N+1}$ ...
2
votes
0answers
142 views

RANSAC line fitting (3d) by line segments (3d)

I am having many 3d line segments. some of them are nearly parallel and some are oriented in to different direction. I want to avoid outliers and to get the best line 3d to represent the given ...
3
votes
3answers
1k views

Finding the major and minor axis vertices for an ellipse given two conjugate diameters?

I've been googling, searching forums and looking in my old algebra/trig books to try to understand how to find the end points to the major and minor axis of an ellipse given the end points of two ...
0
votes
1answer
71 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 ...
2
votes
1answer
438 views

Distance between point and plane & orthogonal projection matrix

I am poor in mathematics and want to learn few fundamental ethics to understand some of advanced things; For plane $i$, denote $n_i\in\mathbb{R}^3$ and $o_i\in\mathbb{R}^3$ respectively as its normal ...
4
votes
1answer
191 views

Potential Division by zero in the construction of NURBS basis functions: how to handle?

Some background context In reading about NURBS I saw the definition that the B-Spline basis functions are defined by a recurrence relation. $N_{i,n} (u)= f_{i,n}(u) N_{i,n-1}(u) + g_{i+1,n}(u) ...
2
votes
1answer
576 views

Angle between different rays (3d line segments) and computing their angular relationships

I have several positions (say A,B,C,..) and I know their coordinates (3d). From each point, if a certain ray is passing in a way to converge them at a given (known) point (say O). This point O ...
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0answers
89 views

viewing ray geometry - with multiple aerial photographs

I am working with multiple aerial images. My idea is to model 3d objects (only upper parts). I am having known orientation parameters. As I am new to this field so that, I want to clarify few general ...
4
votes
1answer
192 views

Intersection of planes by forming 3d lines

If I have $n$ known planes (known normal vector and a point on a plane) that intersect each other in such a way so as to form closely located 3D lines, then (1). To get a common single 3D line to ...