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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17 views

intersection of an ellipsoid and cylindrical plane.

I need to understand if an ellipsoid and a cylindrical arc intersect, what will be the general equation of the cutted ellipse? How can I solve for that equation? I know in 3D, the equation of an ...
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36 views

Algorithm to compute wether 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 $S$ ...
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28 views

How to Compute the Torsion and Curvature of a Parametric Curve

So I have a parametric curve $\bf{r}=${$x(n),y(n),z(n)$} such that the functions $x(n)$, $y(n)$ and $z(n)$ are polynomials of $4$-th degree. I have several of these curves, and I want to calculate the ...
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23 views

Estimating the missing points of a 3D point cloud

Consider a cloud of N points (forming a smooth 3D object), in which n points are missing. Also, consider that there is no prior knowledge about the original shape of the point cloud. The only ...
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41 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 ...
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39 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, ...
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26 views

Find points that defines the intersection of an ellipse with a plane.

I want to test for the intersection of two ellipses $E_1$ and $E_2$ in $\mathbb{R}^3$ represented on a computer. In some sense, this isn't a hard problem: ...
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38 views

Algorithm to optimize rectangles filling by rectangles

I have a set of rectangles, all of the same size (W,H) (in fact paper sheets). I have another set of n rectangles of different sizes (Wi,Hi), i = 1..n such that Wi <= W and Hi <= H (in fact ...
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45 views

How to calculate volume and surface area of three dimensional figures given set of three dimensional coordinates?

I have set of three dimensional coordinates, and the shape is unknown. I would like to calculate the surface area and volume for these coordinates approximately. What is the right approach to solve ...
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18 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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45 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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43 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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15 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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30 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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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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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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54 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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14 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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23 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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44 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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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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64 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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35 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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28 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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153 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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42 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
78 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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27 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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22 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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56 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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24 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 ...
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123 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 ...
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3answers
54 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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1answer
83 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 ...
6
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1answer
80 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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49 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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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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46 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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2answers
36 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
16 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\ ...
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2answers
98 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 ...
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2answers
225 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
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
147 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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57 views

Why are rectangular cartograms hard to generate?

I was reading this paper on rectangular cartograms - ...
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1answer
53 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 ...
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24 views

Schwarz–Christoffel-like mapping on differentiable simple cubic spline boundary

For a concept of a computer game I have in mind I came to need that. I have a 2D pond, which has a boundary that is a simple differentiable cubic spline. There are ducks floating around, looking at ...
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1answer
35 views

Altitudes of Triangle

I have a triangle defined as 3 lines, each defined by two coordinate points A and B. I have the area of the triangle but need to calculate the 3 altitudes and their respective sides A and B points. ...
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1answer
72 views

how many unit balls are needed to cover a unit sphere (1-dense set on a unit sphere)

There is an exercise in a geometry textbook to prove that "any $1$-dense set in the unit sphere $S^{n-1}$ has at least $\frac{1}{2}e^{n/8}$ points". It is supposed to be easy. A set $T$ is ...