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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30 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 ...
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9 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 ...
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2answers
411 views

Determing the distance from a line segment to a point in 3-space

Imagine I have a line segment defined by endpoints $p_1$ and $p_2$, and some 3-space coordinate $q$. Is there a robust (in the sense of never giving divide-by-zero errors) way to quickly determine ...
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35 views

Can a polygon with minimal perimeter self-intersect?

Recipe. Do the following. Throw $N$ random points $(x_0,y_0),(x_1,y_1),x_2,y_2),\cdots,(x_{N-1},y_{N-1})$ in the plane.Define $(x_N,y_N)=(x_0,y_0)$ : enumeration is $\mod N$ . These points are joined ...
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3answers
53 views

Where is the interior of the polygon?

There is an axis-parallel (orthogonal) simply-connected polygon given as a list of corners. How can I know whether a certain vertical segment has the interior of the polygon on its east or on its ...
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13 views

How get the new location of co-planar vertices oriented by the direction vector?

How am I supposed to get the new location of co-planar vertices, if I know the starting and ending point of direction vector, so the direction vector, in spatial space? The original vertices are in ...
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2answers
5k 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
226 views

Given two sets of vectors, how do I find a change of basis that will convert one set to another?

Given two sets of dimension $n$ vectors $\lbrace v_1 , v_2 , \ldots , v_m \rbrace$, $\lbrace u_1, u_2, \ldots , u_m \rbrace$, where $m > n$, is there a computational method (in particular, using ...
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1answer
25 views

Good data structure for hyperbolic tiling

Say you're doing something computational where each data point is a tile in a (not necessarily Euclidean) 2-dimensional tiling, for instance, a Life-like cellular automata. You might want a data ...
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1answer
12 views

Optimized collision check for all pairs using bounding volume hierarchy

I am working on broad-phase collision detection. I understood how to build a bounding volume hierarchy using AABB's as bounding volumes. I also understood how to check all collisions of a single AABB ...
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1answer
231 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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12 views

Link and its Intersection

Let $K$ be a simplicial complex in $R^2$ such that $|K|$ is a simple polygon with inside. An internal edge $ab \in K$ is an edge such that both of its two endpoints a and b are NOT on the boundary of ...
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1answer
59 views

What series of 'hyperpolyhedrons' do exist? Is there an effective way to derive their cross-sections by 3-d subspace?

There are two obvious series of 'hyperpolyhedrons'. 'Hyperoctahedron' with vertices $(\pm1,0...0), (0,\pm1,0,...0)...(0,...0,\pm1)$ and each vertex connected by an edge with each other vertex ...
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35 views

finding quadilaterals from given sides

I have a set of Points. I triangulated the Points as delaunay triangles but i only have edges for the triangles means edge array of the whole set of Points. Now i need to find the traingles and ...
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2answers
53 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 ...
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1answer
76 views

Finding triangulations on 2D space by projecting lower hull of 3D

So we know that we can get the Delauney triangulation of a polygon if we map all points to the 3D space such that $p'=(p.x,p.y,p.x^2+p.y^2)$, compute the lower hull of that polyhedron, and then ...
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0answers
18 views

Existence objective function given optimality regions

Let $I$ and $X$ be finite, nonempty sets, and denote by $\Delta(X)$ the set of probability measures on $(X,2^X)$. Suppose that for each $i \in I$, we are given a subset $M_i \subseteq \Delta(X)$ of ...
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1answer
22 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 ...
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2answers
781 views

Lat/Long grid points covered by projecting rectangle onto sphere

Before my question proper, a little background: I'm wanting to optimise some computer rendering by eliminating the drawing of things that aren't visible given the current view. Suppose we have a ...
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1answer
39 views

Cut the Cake into 4 parts

I'm facing the following problem: I'm given a set of coordinates on an integer grid that define the vertices of a polygon. The polygon is guaranteed to be convex. It's proven that such a polygon can ...
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2answers
143 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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1answer
26 views

What's wrong with this pseudocode for Forster-Overfelt's version of the Greiner-Horman polygon clipping algorithm?

The Problem I'm trying to understand and implement the Forster-Overfelt version of the Greiner-Horman polygon clipping algorithm. I've read the other Stackoverflow post about clarifying this ...
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1answer
54 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 ...
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26 views

Research scopes in Computational Geometry

I have taken a short a course on Computational Geometry and at present I want to do some research works of my own. Can anybody tell me about the research scopes on it? I mean, what are the active ...
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1answer
97 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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1answer
25 views

At what extent I can use trigonometric functions and properties with parametric curves?

I have a know-how and a library about trigonometry and trigonometric operations, I would like to know if I can possibly rely on trigonometry for parametric curves too and how the trigonometry from the ...
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0answers
37 views

A continuous centerpoint of a convex spherical polygon

In discrete geometry, a centerpoint $c$ of a discrete set $S$ of $n$ points in the plane is such that any half plane containing $c$ contains (roughly) $n/3$ points of $S$. (Such a centerpoint always ...
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1answer
21 views

What is and what are the use for an “ AINV preconditioner ” or “ SAINV ”?

In an article that I'm reading there is a mention to this "thing" and I absolutely don't know anything about it, for me it could be anything. I noticed that this thing is somehow related to the math ...
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0answers
51 views

How to determine if a convex polytope is contained in a union of convex polytopes?

Given that we are in a Euclidean space of dimension d, that we have a bounded convex H-defined polytope P, and N possibly unbounded convex H-defined polytopes, I am looking for an "efficient" ...
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23 views

Data structure issues with incremental Delaunay triangulation

I am implementing the incremental algorithm of Delaunay triangulation with a data structure based on Faces (triangles): 3 vertex indices and 3 Neighbor indices. The issue I have is that the structure ...
6
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1answer
75 views

Putting fence around sheep

Convex hull algorithms are well known. However, in my case, the goal is slightly modified: Given $N$ points in a plane, construct convex polygon with minimal area so that it contains all points, and ...
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0answers
99 views

How can I define a measure of similarity between two line segments in $\mathbb{R}^2$?

I have two segments in $\mathbb{R}^2$ and I would like to define a measure of similarity between the two segments. My idea is that I can apply: a scale transformation $s$ in order to equate the ...
5
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1answer
77 views

filling an occluded plane with the smallest number of rectangles

I've got a specific problem which I'll try to describe as clearly as possible. I have a defined rectangular region on a cartesian plane, and within that region there are other given rectangular ...
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1answer
96 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 ...
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1answer
998 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 ...
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1answer
44 views

Formula of signed distance from hyperplane to point

Let $H$ be a hyperplane defined by the points $p_1, p_2, ..., p_n$ and single point $x$ generally out of the hyperplane. Is there any formula to calculate the signed distance between $x$ and $H$? I ...
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0answers
19 views

Traveling Salesperson tour monotonicity

Prove that the travelling salesperson tour TSP of a set S of points is monotone, that is, if S ⊆ S′, then the length of TSP(S) is less than or equal to the length of TSP(S′).
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1answer
44 views

Avoiding gimbal lock

I am not really sure if I understand the phenomenon of gimbal lock correctly. Say I have a vector $\begin{pmatrix} x\\ y\\ z \end{pmatrix}$. And I want to keep the vector's length fixed but move it ...
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1answer
365 views

Radius of circle coverage of n circles in square packing configuration

Is there a reference about determining the minimum radius of a circle that would cover n circles of radius 1 that are in a square packing configuration ( see Wolfram's MathWorld packing packing ...
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2answers
35 views

Determining points on a circle in a particular plane

This is more of a computer graphics question really, but I was just wondering the efficient way to determine n equally spaced points on a circle, given a normal vector to the circle and the radius of ...
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1answer
14 views

Applications of logarithmic functions in shapes and geometries

As I understand this logarithmic functions are a family of functions where the equation for $f(x)$ is written like so $$\begin{align} f(x) = & \log_{a} x \\ & \mathtt{where\ }a\mathtt{\ is\ ...
2
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2answers
76 views

Given four points, verify if they form a Tetrahedron

As per the title, I've been requested to build a program that -given 4 points in space- determine if they form a 3D shape and if this is case to present it's volume. For the volume part of the ...
8
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2answers
208 views

Visual illustrations of circle packing theorem?

Circle packing theorem states: For every connected simple planar graph G there is a circle packing in the plane whose intersection graph is (isomorphic to) G. Paper Collins, Stephenson: A circle ...
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1answer
33 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
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1answer
81 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.
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1answer
53 views

What does $E^d$ mean?

I was reading the paper "Cutting Hyperplanes for Divide-and-Conquer" by B. Chazelle and in the introduction I came across the following: "Let $H$ be a set of $n$ hyperplanes in $E^d$." What does $E^d$ ...
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2answers
128 views

What is inside and outside of complex polygon?

I am reading this paper http://arxiv.org/pdf/1207.3502.pdf Given a complex polygon. Its edges may intersect. The algorithm finds out if given point is inside of polygon or not. It draws a line from ...
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0answers
59 views

How do I most efficiently find the perpendicular distance from a point to the convex hull of a collection of circles?

I have a collection of one or more line segments for which I know the (x,y) coordinates of the endpoints. The segments may or may not be parallel and may or may not intersect. Each segment endpoint ...
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2answers
51 views

Why are rectangular cartograms hard to generate?

I was reading this paper on rectangular cartograms - ...
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1answer
51 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 ...