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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26 views

Closest pair algorithm in high dimension?

2D case is clear. But with dimensions higher than 2 I should choose a special partitioning hyperplane for the divide and conquer algorithm to get O(n log n). I am confused because to choose this ...
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1answer
26 views

Equality of polyhedra

Is a minimal representation for a polyhedron unique? And if so can we use this to prove that two polyhedra are equal (or maybe the same is a better definition).
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18 views

point-tuple orientation

Does something like point-tuple orientation exist? I read this book, pg 336, paragraph "One can even..." and I've bumped into this quite unfamiliar notation. Thanks for clarification. P.S. if someone ...
2
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1answer
145 views

points in general position

I'm really confused by definition of general position at wikipedia. I understand that the set of points/vectors in R^d is in general position iff every (d+1) points are not in any possible hyperplane ...
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2answers
256 views

Hexagonal Tessellation on a sphere

I want to detect collision of a sphere with another object and to find out(show) the deformation of the sphere. I have come to know that hexagon(regular)tessellation of a sphere is the most ...
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0answers
48 views

Find 3D concave hull based on original model and convex hull

I want to find the concave hull of a 3d model, with a threshold for the maximum edge size. Googling around let me to the following approach (mainly abstracting from 2d approaches): Determine the ...
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3answers
333 views

determine unit outward normal vector on a curve

It is necessary for me to find unit outward normal vector for the curve: $$\gamma=(x(t),y(t))$$ where $$x(t)=(0.6)\cos(t)-(0.3)\cos(3t)$$ and $$y(t)=(0.7)\sin(t)+(0.07)\sin(7t)+(0.1)\sin(3t)$$ I ...
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0answers
62 views

quadratic constraints representations

I have two surfaces of parameter $x \in \mathcal{X} \subset \mathbb{R}^n$ with $\mathcal{X}$ to be a polytope: $u(x) = ax + b \subset \mathbb{R}^m, z(x) = x^TAx + B^Tx + C \in \mathbb{R}$. If $m=1$, ...
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1answer
50 views

Convex hull of a set of points

Let $a_1,a_2...a_r \in R^n$ be points in $R^n$. Prove:$$CH(\{a_1,...,a_r\})=\left\{\sum_{i=1}^r\alpha_ia_i|\sum_{i=1}^r\alpha_i=1,\alpha_i\ge0\right\}=:K$$i.e. the convex hull of the $a_i$ is the set ...
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0answers
73 views

Similarity of Polyhedra: What is the measure?

When comparing two convex polyhedra, how can one determine if they are geometrically similar. Is there any algorithm to determine if one is the distorted or truncated version of the other? Vertex, ...
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1answer
44 views

Convex hull is convex

A set $C \in R^n$ is called convex if the line segment $L = \{ tp + (1-t)p | 0 \lt t \lt 1 \}$ between two arbitrary points $p,p' \in C$ is contained in $C$. The convex hull $CH(C)$ of a set $C ...
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0answers
42 views

Two convex polygon intersection from set of m convex polygons with total n vertices

I have a set of m convex polygons $(p_1,p_2, \ldots p_m)$. $n_i$ is the number of vertices in $p_i$. $\sum_{i=1}^{m} n_i = n$. Each polygon has vertices listed in anti-clockwise direction, starting ...
2
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3answers
64 views

Number of layers in nested convex hull

Find the maximum number of nested convex hull
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1answer
33 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 ...
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0answers
31 views

Geometric Median

Is there a relationship between Voronoi Diagrams and the geometric median? I know that it is impossible to find a closed expression for the geometric median, but the two concepts seem related.
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1answer
151 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. ...
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1answer
104 views

Find the length of each side of a square containing regular hexagons [duplicate]

I have to find the length of each side of a square such that all the regular hexagons of same length side and radius lying inside the square have centers either inside the square or on the boundaries ...
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0answers
66 views

Mathematical proof to find the length of each side of a square filled with Regular Hexagons

I have to prove or disprove that in a square box if there are full regular hexagons( whose distance from center to every corner is r) inside it, then the centers of those hexagons should lie inside ...
0
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1answer
80 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
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1answer
16 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
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2answers
308 views

Sorting a list of points in 2-D clockwise

I have number of points with co-ordinate (latitude, longitude) in 2-D: Here is a collection of some points: \begin{array}{ccc} \hline No.& lon & lat \\ \hline 1& 84.07921& 24.49703 ...
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0answers
16 views

ellipsoid and paraboloid relation

My task was to programme a paraboloid and an ellipsoid. I implemented paraboloid as a set of points that's distance from the focal point and the distance from the plane is the same. After running the ...
0
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1answer
124 views

Is k-means clustering guaranteed to converge if using Manhattan distance?

The k-means algorithm is an iterative clustering algorithm that partitions the data points into K clusters (with centroids {$\mu_1, ... , \mu_k$}, minimizing the Sum-of-Squared-Error: $$ SSE = ...
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0answers
31 views

Maximal area intersection of half-planes in $\mathbb{R}^2$

Suppose we have $m$ half-planes $H_1,...,H_m$ in $\mathbb{R}^2$ such that $H_1 \cap \dots \cap H_m = \emptyset$. Let $A$ be a set of subsets $S$ of $\{H_1,...,H_m\}$ with non empty intersection and ...
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1answer
135 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 ...
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 $S$ ...
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0answers
110 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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2answers
73 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 ...
2
votes
1answer
161 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
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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, ...
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0answers
40 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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0answers
178 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 ...
0
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1answer
203 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 ...
0
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1answer
30 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 ...
1
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2answers
150 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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0answers
51 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 ...
2
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1answer
65 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 ...
2
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0answers
14 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 ...
5
votes
1answer
63 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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2answers
69 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 ...
0
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1answer
26 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
43 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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1answer
110 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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0answers
20 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
117 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 ...
2
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1answer
57 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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2answers
361 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 $y_3$ ...
3
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1answer
95 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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1answer
35 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 ...
0
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1answer
27 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 ...