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

Mapping optimal sensor placement problem to Art Gallery Problem

I am trying to design an algorithm for optimal sensor placements in a given area. I researched in this domain and found ...
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33 views

Geometric accuracy analysis of 2d rectangular models

I have reconstructed set of rectangular objects lie on a 2D plane (for ex. ABCD). All these objects are in a one coordinate system. On the other hand, I have reference models for all of them ...
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44 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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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 ...
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196 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 ...
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33 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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60 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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160 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 ...
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64 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 $x_{N+1}$ ...
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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 ...
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59 views

How to estimate error pattern of a set of line segments with respect to given reference segments (2D case)

I am having set of pair of line segments (2D). Though each pair should be coincided on top of each other they are not so. I have been the reference data and then I extracted other line segments ...
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43 views

problem in dimensionality reduction

I am using multidimensional scaling to plot my data in R. However there is a hierarchy in my dataset which i want to exploit and I am using the delaunay triangulation to visualize the plot. So now I ...
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63 views

Integration through a Rotated Square

I have a 2D square S. S is described by s, the side length, theta, the angle it is rotated by, and c, the position of S's center. There is an axis-aligned rectangle R that extends infinitely in the ...
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139 views

Approximating Bezier curves

I would like to approximate one cubic Bezier curve with two quadratic ones. In other words, I would like to split a cubic curve at some parameter t and approximate ...
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254 views

Algorithm for intersection between polyline and rectangle?

My problem is simple, and probably obvious from the title itself, but I'll still clarify it a bit: I have a rectangle and a polyline (array of N connected points). I need an optimal algorithm that ...
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121 views

polygon inside a polygon

i have several point patches lie on different planar faces. then, I obtained enclosing polygons to represent points so that i have several planar polygons (for example A,B,C,D). when i examine the ...
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395 views

differentiation of polygons, having cross borders

I have point data set and I segmented the data into different planar objects. after that, using contouring (convex hull), I obtained the boundary points. Please assume all points relevant to one ...
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36 views

How to discuss the maximum Area of Internal rectangular in an irregular region?

How to discuss the maximum Area of Internal rectangular in an irregular region? such as Fan-shape,or the region....
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110 views

The orientation of a closed discrete curve embedded in a triangle.

The two triangles $xyz$ and $x^{\prime}y^{\prime}z^{\prime}$, shown below, have opposite orientations. A closed curve $abcd$ is embedded in the first triangle ($abcd$). The vertices of the ...
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321 views

Line comparison algorithm advice

Line is array of points (2 or more). I have a plane full of lines. For a given line in plane I need a measure which will tell how much difference there is between this and any other line in plane. I ...
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80 views

Questions about interpolating translated points from a grid

I would like to do the following transformations on a very low resolution bitmap (64x64 pixels). I am doing this transformation on a computer images, but it has nothing to do with computers, you can ...
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151 views

Nonlinear least squares and polygon area

I found this paper that describes preserving the global area of a polygon given some deformation (section 5): http://www.kunzhou.net/publications/2DShape.pdf I'm trying to do something very similar. ...
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89 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 ...
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103 views

Complexity of Counting the number of inducing $n$-gons

Definition: A $n$-gon is simple if it has no self intersection and is in general position if no pair of its edges are parallel. It is clear that by extending the edges of each simple $n$-gon in ...
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5 views

Equalize length of 1-ring edges of vertex

My question is how to equalize the length of edges in 1-ring neighbours of while-circled vertex in the below figure Hope to see your answer!
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11 views

Prove Theorem with Groebner Basis

I'm trying to prove some theorems using Groebner Basis (as described in Cox, Little and O'Shea Link ) The mentioned book gives as an excercise to prove Pappus theorem using the given methodology, ...
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33 views

What is the mathematics behind the two animations?

I found two animated GIFs from a designer's website, which looks very impressive: My questions are: what is the mathematics behind them? How to obtain the mathematical formulas and equations of ...
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54 views

What is the shape of the set of integer sided acute triangles with largest side n?

I played around with Gauss circle problem and found that if you take a certain sum in reverse and "in forward" and subtract the resulting sequences you get the OEIS sequence: https://oeis.org/A247588 ...
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36 views

does any polyhedral partition admit a convex piecewise quadratic surface defined over?

Given a polyhedral partition, i learnt that there exist some conditions for the existence of a convex piecewise affine surface over this partition for example the following study. ...
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46 views

How to use CVX to solve this problem?

I have a function in the variables $x_{kl};\ k,l=1\ldots,m$, $$\sum_{i=1}^n \sum_{j=1,j<j'}^{N_i}\left( b_{ij} b_{ij'}- \sum_{k,l=1}^{m}x_{kl}f_k(a_{ij})f_l(a_{ij'})\right)^2$$ where ...
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24 views

What does “maximum geometric error of a chunk” mean?

In this paper, on the top of page 7 it says, Where $\delta$ is the maximum geometric error of a chunk ... What does that mean? Thanks :D
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41 views

How to navigate around a smooth surface?

Suppose I want to find the shortest path between two points in $\Bbb{R}^3$ with smooth obstacles in the way? I understand things like Dijkstra's algorithm for shortest paths on a graph. But what about ...
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17 views

Ways of partitioning n points into some cubes

Assume there're $n$ fixed points in $\mathbb{R}^d$ contained in a ball with radius $M$,and you can partition the space by cubic grid with cube's edge length $h>\epsilon$. How many different ways of ...
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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 ...
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60 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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41 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 ...
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63 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 ...
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28 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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36 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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157 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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50 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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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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44 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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36 views

Is there a way to compute the empty area between a group of touching polygons?

Given a bunch of convex polygons layed out like a house truss, is there a way to compute the empty area, or get a polygon for each of those "holes" between the polygons? I tried starting from any ...
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99 views

Estimating the geometric shape of a point cloud without using the vertex information

Consider a point cloud format that describes 3D point clouds by vertices, triangle labels and normal vectors. If we miss the vertex information, is it possible to retrieve the lost data by triangle ...
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122 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 ...
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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 ...
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69 views

transformation function using genetic programming

If I have a set of points in two spaces, say set $A$ contains 50 points and set $B$ contains 50 points. I have to find a transformation function such that if I transform the points in set $A$ using ...
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70 views

Plot randomly oriented gaussian kernel

I would like to plot with scipy randomly oriented gaussian kernels. For a gaussian kernel along x and y axis (with an angle 0 w.r.t. coordinate system), I simply plot function ...
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163 views

Decomposition of multidimensional point set

I am trying to use point sets to define the subdivisions of a multidimensional space and use a hash table to store the subvisions. This approach requires decomposing the multidimensional space into ...