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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2answers
667 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 ...
2
votes
2answers
37 views

How to make sure that a given set of points lie on the boundary of a possible square?

Given a set of integral coordinates , check whether all the points given lie on side of a possible square such that axis of the square so formed lie parallel to both X-axis and Y-axis . Suppose ...
2
votes
2answers
51 views

Polytope parametrization

How one could parametrize a convex polytope? By parametrization I mean something like in multiple integrals, when to integrate over an area one can integrate over one variable in an interval $[l,r]$ ...
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vote
2answers
62 views

What is the equation stands for in geometry(intuitively)?

I am writing a bilinear interpolation method. This method can be abstract by solve the equation A*x = b, A is a 4x4 matrix below: $A=\begin{pmatrix} 1 &x_1 &y_1 &x_1y_1\\ 1 ...
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2answers
34 views

Room for computational geometry in advanced algorithms course

I am currently putting together an independent study in advanced algorithms and because of my interest in (computational) geometry, wanted to include as many interesting algorithms from this field as ...
5
votes
1answer
200 views

How to determine surface from given normal vectors and their distance on that surface

Situation: We have a bendable, non-stretchable surface, like a piece of cloth, with a regular grid on it. Unknown manipulation of the surface is done while preserving it's structure We recieve 3 ...
3
votes
1answer
77 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 ...
2
votes
1answer
63 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 ...
2
votes
1answer
208 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 ...
1
vote
1answer
60 views

Find equation of line without using division

I need an algorithm to find equation of a line without using division. Given a line by two points on it, with coordinates: $(x_1, y_1),\ (x_2, y_2)$. We can simply get the line equation by the ...
1
vote
1answer
38 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 ...
1
vote
1answer
90 views

Convex Hull in Hierarchy Structure

As a beggining to convex hull algorithms lecturer introduced the structure which it's called "Hierarchy Structure". Hierarchy Structure: in every given convex hull there is a maximum size convex hull ...
0
votes
1answer
19 views

Using cartesian coordinates how to get the segment overlapping two lines/segments?

There must be an algorithm to find the coordinates of the segment overlapping (fully or partially) two lines/segments but my googling does not produce any significant result. Maybe I don't use the ...
0
votes
1answer
145 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
87 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
votes
1answer
54 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: ...
0
votes
1answer
66 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 ...
0
votes
1answer
104 views

Steepest slope gradient of a vertical plane

I know the steepest slope gradient (Azimuth) of a 3D plane can be obtained by projecting normal vector onto XY Plane. So, when the plane is slant, the steepest gradient will be a some value. ...
0
votes
1answer
127 views

Analytic Intersection of Objects Located on a 3D Grid's Vertices

I previously posted this question on stackoverflow, but it's really more of a mathematical question. I have reworked the question for presentation here. I have a regular 3D unit cubic grid of ...
1
vote
0answers
72 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 ...
1
vote
0answers
126 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. ...
1
vote
0answers
101 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 ...
0
votes
0answers
19 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
0
votes
0answers
29 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 ...
0
votes
0answers
34 views

Computationnal geometry: vector, basis, point and coordinate system?

I am trying to build a small geometrical library in C++, that is mathematically consistent (not so false). The goal here is to construct two concepts: vectors and points. I am not sure that the ...
0
votes
0answers
27 views

Surface comparison using the vertex information and normal vectors

I have two point clouds with normal vector information. How can I use the normal vector information to measure the surface similarity of these two point clouds?
0
votes
0answers
15 views

Is there a well known algorithm for efficiently computing the vertices of a convex polytope?

A convex polytope, the one I'm talking about, is the hull around all points $x$ satisfying the matrix equation $Ax = b$ and $x \geq 0$. I've been digging around old papers from the 60's and 70's ...
0
votes
0answers
53 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 ...
0
votes
0answers
16 views

Chain of transformations -> continuous

Transformation $A_t$ rotates point $p(t)$ for angle $d\phi(t)$ around the axis $n(t)$ anchored at point $r(t)$ and finally displaces it for $r'(t) dt$. Point is now $p(t + dt)$. More specifically in ...
0
votes
0answers
67 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 ...
0
votes
0answers
23 views

Level set of 2D gaussians

I'm looking for a numerical method to rapidly find the level set $\{x,y\} | f(x,y)=h$, where $f(x,y)$ is a sum of a 2d gaussians (forming a mixture model), and $h$ a constant. Here's an example ...
0
votes
0answers
34 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 ...
0
votes
0answers
142 views

how to get optimal vector, which is parallel to intersection line of many plane (Least Square way)

My idea is to construct the best optimal 3D line representing the intersection of many 3D planes. (As we know, due to fitting errors or data errors, the fitted planes might not intersect exactly ...
0
votes
0answers
50 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 $x_{N+1}$ ...
0
votes
0answers
21 views

H-representation of high dimensional convex hull

I want to obtain the H(yperplane)-representation of a convex hull corresponding to a set of given points (not all are extreme points). The points are in very high dimensions (about 30). Is there any ...
0
votes
0answers
47 views

common coverage length portion of two nearly parallel 3d line segments

I am having many pair of 3d line segments. the case is, in non of the pair, one line segment with respect to other one is not parallel but nearly parallel withing (5-10 degree threshold). if i take ...
0
votes
0answers
56 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 ...
0
votes
0answers
139 views

Maximum diameter of a 2D shape

What is the diameter of an arbitrary 2D figure? (Diameter=The longest distance between two points within the 2D figure). What is the most efficient algorithm? Is it an exact one? Specifically, could ...
0
votes
0answers
60 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 ...
0
votes
0answers
192 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 ...
0
votes
0answers
64 views

Sufficient conditions for “2-sphericity” of orientable triangulated 2d surface in 3d space

Let $T$ be finite set of tetrahedrons in $\mathbb{R}^3$. Let $T$ be tetrahedral complex in a sense that if two tetrahedrons intersect, the intersection is a face of both. Let $\partial T$ consist of ...
0
votes
0answers
129 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 ...
0
votes
0answers
109 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 ...
0
votes
0answers
87 views

Does a single Gauss-Seidel iteration lead to unique coordinates?

I managed to reduce certain computational problem to the Gauss-Seidel solution of the following linear system: $$Ax=Ly,$$ where $A, L\in\mathbb{R}^{n\times n}$, and $x,y\in\mathbb{R}^{n\times 2}$ are ...
0
votes
0answers
84 views

Can 2 parallel lines be discriminated as 'away', 'beside' with respect to 3rd parallel line?

I have nearly parallel several 3D line segments. some line segments locate (blue line) beside to a spefic line segment (black line) and some other (red line) locate away from that line segment. i want ...
0
votes
0answers
47 views

Maintaining the line with the 2D iterands

Suppose a linear system is given $$AX=B,$$ where $A\in\mathbb{R}^{n\times n}$ is a symmetric strictly diagonal matrix, and $X, B\in\mathbb{R}^{n\times 2}$. Therefore, the 2D Jacobi iterative solver is ...
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votes
0answers
83 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 ...
0
votes
0answers
59 views

Multidimensional simplex meshing

I'm trying to figure how to make a simplex mesh on orthogonal domain. Basically it comes to this: Make (2) triangles of a rectangle Make (5) tetrahedrons of orthogonal prism (cuboid) etc. I don't ...
0
votes
0answers
72 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 ...