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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3
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0answers
559 views

Circle Packing Algorithm

I have question related to circle-packing. The problem is to find the circle of minimum radius enclosing four non-overlapping circles of arbitrary radius. I have to write a program in C for this ...
0
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0answers
49 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 ...
2
votes
1answer
461 views

Voronoi average number of vertices $< 6$

My text says "the average number of vertices of the Voronoi cells is less than six". Then it creates the vertex "at infinity", connects the half-infinite edges to this vertex and shows the equation: ...
1
vote
1answer
73 views

Determine if two polyhedrals are the same shape and if so, map their vertices

I have a polyhedron and want to determine whether it is combinatorially equivalent to another polyhedron. I know how many faces comprise each polyhedron and for each face, I know all of its vertices, ...
2
votes
3answers
74 views

Explanation of the following notation

I am having a hard time understanding the meaning of the union operation in this equation. $$C(A)=\bigcup_{x \in A}C(x)$$ For context, here is the sentence: The candidate set for $x$ is $S \cap ...
3
votes
1answer
123 views

Computing the free-part

I wanted to ask about some existing algorithms for computing points over elliptic curves. Background : We know that the famous theorem of Mordell and Weil says that " Group of rational points on an ...
1
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0answers
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....
0
votes
1answer
618 views

line projection on top of a plane

If I have a horizontal line (a 3d point and 3d vector with zero z component) and another plane (could be an oblique or a horizontal; i have normal vector of the plane); then how do we get the ...
1
vote
0answers
104 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 ...
3
votes
1answer
254 views

What is the complexity of computing the minimum distance between two convex polyhedra that both have $n$ faces?

EDIT: (in response to what deinst said) sometimes using a sledgehammer for some menial task is rather convenient - especially if it also has the complexity $O(n)$ (which is what my question is about) ...
1
vote
1answer
317 views

How to get a projected 3d line segment, lie on another 3d line parallel to that line segment.

I have a 3D line segment and another 3D position which locate slightly away from the line segment. I want to get the projected line segment (3D) which lies on imaginary 3D line which passes through ...
6
votes
1answer
173 views

Constructive algorithm for Minkowski's theorem.

There is a theorem of Minkowski that says that given $k$ unit vectors $x_i$ that span $\mathbb{R}^n$ and $k$ positive real numbers $a_i$ such that $\sum_{i=0}^k a_i x_i = 0$ then there exists a unique ...
0
votes
1answer
935 views

Bearing from one point to another on 2D number plane

Title says it all, I'm looking for the formula to get the bearing from one point to another on a number plane. I have found examples of this for lat/lon around the earth but that's not exactly what i ...
2
votes
4answers
2k views

How to test any 2 line segments (3D) are collinear or not?

if we have two line segments in 3D, what would be the way to test whether these two lines are collinear or not? (I fogot to mentioned that my line segments are 3D. So, I edited the original post. ...
0
votes
1answer
404 views

Computing the average coordinate for more than 2 points on a 3d line segment

Suppose, I have many 3d line segments which suppose to be intersected with another given line segment. So, I wish to take a line segment and the given line to get the intersection point. Again, I wish ...
2
votes
1answer
173 views

How do I calculate the unique k-dimensional hypersphere's center from k+1 points?

I'm working with the Bowyer-Watson algorithm to determine the Delaunay tessellation of stochastic points in k-dimensional space. This algorithm assumes that the center of a simplex can be used as the ...
1
vote
1answer
195 views

Computing surface normal, floating point arithmetic

If I have a $n$-gon in $\mathbb{R^3}$, and I want to compute the surface normal, how can I get a value that minimizes error in a floating-point system? For example: Would I gain accuracy by first ...
6
votes
1answer
208 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 ...
0
votes
2answers
272 views

Computing the minimum distance between two hollow tubes provided a method for computing the minimum distance between finite line segments

Say I have a method of calculating the minimum distance between two finite line segments in three-dimensional space. How might I adapt this method to provide the minimum distance between the surfaces ...
6
votes
1answer
164 views

Correlations between neighboring Voronoi cells

For a sequence $X_1,X_2,X_3,\ldots$ of random variables, what it means to say $X_1$ is correlated with $X_2$ is unambiguous. It may be that the bigger $X_1$ is, the bigger $X_2$ is likely to be. If, ...
0
votes
1answer
569 views

Area of Union of n circles

I am trying to calculate the area of union of n circles in a plane when it is known that all circles are of equal radii and their centers are also known(of all n circles). I was trying to follow the ...
3
votes
2answers
656 views

Computing a volume (area) of intersections

The task should be very common, what are the best and easiest to implement algorithms to compute the volume of union/intersection of given bodies? Or union/intersection area for 2D figures. I don't ...
2
votes
1answer
134 views

Fitting a cylinder of known length to the chord between its circular edges s.t. the cylinder's mass near a point is minimized

Let the pair of three-space coordinates $p_1$ & $p_2$ define a chord between the edges of a cylinder of radius $R_c$ and length $L$. The edges here represent the two circular borders between the ...
2
votes
2answers
309 views

Finding an appropriate axis of rotation for two points such that they can be rotated and translated to overlay a given line

I have two lines with known parametric equations and some number of distinct points along each line. I would like to rotate the points on $L_2$ some number of degrees $\theta$ along one and only one ...
1
vote
1answer
207 views

Ancient astronomers, planetary conjunctions, and epicycles

How did ancient astronomers predict planetary conjunctions? I know they used a system of epicycles to represent the path of planets, but finding the point and time of alignment of two planets still ...
1
vote
2answers
610 views

how can one calculate the minimum and maximum distance between two given circular arcs?

how can one calculate the minimum and maximum distance between two given circular arcs? I know everything of each arc: startangle, endangle, center, radius of arc. The only thing I don't know how to ...
4
votes
1answer
155 views

Shortest path in polygonal domain

The single shot query for the shortest path between two points in a plane environment with polygonal obstacles of complexity $O(n)$ can be solved in time $O(n \log n)$ using the continuous Dijkstra ...
16
votes
1answer
1k views

Geometry of nose in and nose out parking in parking lots

I would like some computational evidence in favor of my observation that one can park a car in tighter (parking lot) spaces by backing in rather than nose in. I have been doing this successfully for ...
3
votes
2answers
835 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 ...
-1
votes
1answer
112 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 ...
1
vote
1answer
98 views

Maximum number of points with a fixed minimum distance in a $d$-dimensional ball

Let $c \leq r$ be real numbers greater than $0$, $d \in \mathbb{N}$ and $B_r(0) = \lbrace x \in \mathbb{R}^d \mid \Vert x \Vert \leq r \rbrace$, the ball with radius $r$ at point $0$ ($\Vert \cdot ...
0
votes
1answer
935 views

Fast 2D Line Triangle Intersection test

In a 2D plane, I have a line segment ($P_0$ and $P_1$) and a triangle, defined by three points ($t_0$, $t_1$ and $t_2$). My goal is to test, as efficiently as possible ( in terms of computational ...
2
votes
2answers
5k views

How to multiply vector 3 with 4by4 matrix, more precisely position * transformation matrix

All geometry in computer graphics are transformed by position * transform matrix; The issue is the fact that position is a vector with 3 components (x,y,z); And transform matrix is a 4 by 4 with one ...
2
votes
1answer
109 views

Computing the point which is closest to many Planar surfaces

Suppose, i have been given different planes which orients to different direction (i.e. i know only the plane parameter of those planes). If i am able to find out planes (probably more than 3 planes) ...
1
vote
0answers
272 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 ...
1
vote
2answers
916 views

Check if point on circle is in between two other points (Java)

I am struggling with the following question. I'd like to check if a point on a circle is between two other points to check if the point is in the boundary. It is easy to calculate when the boundary ...
9
votes
4answers
1k views

Every polygon has an interior diagonal

How does one prove that in every polygon (with at least 4 sides, not necessarily convex), that it is possible to draw a segment from one vertex to another that lies entirely inside the polygon. In ...
2
votes
3answers
346 views

What does “identity map $id$” mean?

What does "identity map $id$" mean in this context? Two metrics $d_1$ and $d_2$ on $X$ are said to be Lipschitz equivalent if the identity map $id\colon (X,d_1)\to (X,d_2)$ is bilipschitz.
0
votes
2answers
124 views

Finding maximum value of smallest triangles over all triangulations

Given simple polygon we have to find maximum area of smallest triangle in all possible triangulations. I was trying to solve it by generating all possible triangulations, but for complex polygon it ...
3
votes
1answer
150 views

Generating all triangulations of simple polygon

Having simple polygon how can we generate all triangulations of this polygon? How can it be done ? What would be the approach ? I didn't find any paper explaining it, only about planar triconnected ...
9
votes
2answers
128 views

Efficient method for detecting a convex body in $\mathbb{R}^n$

Let $K_0$ be a bounded convex set in $\mathbf{R}^n$ within which lie two sets $K_1$ and $K_2$. Assume that, $K_1\cup K_2=K_0$ and $K_1\cap K_2=\emptyset$. The boundary between $K_1$ and $K_2$ is ...
3
votes
0answers
450 views

Detecting Planes through Point Cloud

Having a point cloud say (10000 points) which are randomly dispersed in 3D unit cube, the question is how to find planes within the cube that include more points with an acceptable tolerance (user ...
1
vote
0answers
76 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
4answers
321 views

What is the name for a maximal convex set of points contained in another set of points?

What is the name for a maximal convex set of points contained in another set of points X? Maximal in terms of inclusion. For the desired set to be unique, X can be restricted to be a simple polygon ...
3
votes
2answers
55 views

Finding how many points to which a certain point is connected

This may be a programming issue, not a mathematical one. If so, please let me know so that I can rewrite it specifically for that audience. Consider a shape with a random border. Each point on its ...
2
votes
0answers
651 views

Is there a formula for the solid angle at each vertex of tetrahedron?

A tetrahedron has four vertices as much as a triangle has three vertices. A tetrahedron therefore can have four solid angles as much as a triangle can have three angles. I am wondering: Is there a ...
1
vote
1answer
86 views

Line segment k-intersection

Could you please help me to design the following algorithm: I have a random-access list of line segments defined by a pair of points $[x^s_i; x^e_i]$. The list is initially unsorted, but of course ...
1
vote
1answer
92 views

Algorithm to compute mesh from intersection of infinite halfspaces

Is there a simple algorithm to compute the convex polyhedron (as a mesh with verticies, edges, and faces) resulting from the intersection of a set of infinite halfspaces? This is essentially a CSG ...
3
votes
2answers
157 views

How do I apply a digital filter to points on a sphere

Given a set of points on a sphere, how can I implement a higher order low pass filter on them? At the moment, I am just multiplying the vectors from the input and output set by their weights and ...
0
votes
1answer
158 views

Explanation of a formula for spherical 3d tag cloud

I am not an expert in mathematics, I am only a young programmer. I am trying to construct a spherical tag cloud and I've found this formula: ...