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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365 views

Best fit for 'puzzle' shapes inside a frame

Consider a large rectangle frame, as we want to fill it with small rectangles with variable sizes. How to calculate the best match of inner objects to minimize empty spaces inside the main frame? ...
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2answers
57 views

Tetrahedra from it's inscribed sphere

I'm facing a geometrical problem: Given a sphere S, I want to calculate the vertices of the tetrahedra T whose inscribed sphere is S. In other words I want to calculate a tetrahedra from it's ...
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2answers
52 views

Point as an element of an affine space vs point as an element of a topological space?

I am searching for the "most natural" definition of a (geometrical/space) point as an element of "something" in mathematics (I am trying to design a small computational geometry library on strong ...
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2answers
890 views

Finding the tangents common to two rotated ellipses?

Is there a way to find the four tangents that two rotated ellipses share? I believe that if two ellipses do not intersect and do not contain one another, they will have four tangents in common and I ...
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3answers
104 views

Are there any Heron-like formulas for convex polygons?

Are there any Heron-like formulas for convex polygons ? By Heron-like I mean formulas without angles as arguments and which takes as arguments only lenghts of sides of polygon - that is - we know no ...
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1answer
187 views

Algorithm Design for Delaunay Triangulation?

I am very much happy after seeing some very good answers in this site. I am trying to design a algorithm for the construction of Delaunay Triangulation using Randomized Incremental Algorithm.(I wont ...
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2answers
99 views

How does one compute the minimal bounding sphere of a k-simplex?

Suppose I have a list of $k+1$ points in $\mathbb{R}^n$, and I let $\sigma^k$ be their convex hull. I want to know two things: How can I determine, for any $\varepsilon$, whether open balls of ...
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1answer
1k views

The dual graph of the triangulation

I study Polygon Triangulation and have an execise. Prove or disprove: The dual graph of the triangulation of a monotone polygon is always a chain, that is, any node in this graph has degree at ...
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2answers
834 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 ...
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1answer
70 views

Shortest polygonal line that connects three disjointing circles

Given three disjointing circles, how to find the shortest polygonal line (consisting of two line segments) that connects the three circles (a line segment connects circle A and B if it starts with ...
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2answers
30 views

Maximal-Orthogonal Convex Hull (or Maximal-Rectilinear Convex Hull)

In my research, I have come across a this paper from the Computational Geometry field and I am not able to understand the concept of Maximal-Rectilinear Convex Hull of a given point set in plane. In ...
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1answer
74 views

Determining the position of a polygon inside a circle from only the angle of opposing sides/edges.

For illustration click here I have a simple convex irregular polygon (octagon in example image) inside a circle (circle and polygon are not always concentric and never touching or intersecting) and I ...
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2answers
85 views

finding volume of an n-dimensional pyramid numerically

In my experiment I need to compute hypervolume/area from a set of points, let's start with a base case -- Triangle: In this case, I have 3 points in a 2D space and they make a triangle, $p_1 = ...
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1answer
73 views

T-shaped polygons

Is there any coefficient that can indicate T-shaped polygons ? Examples of T-shaped polygons:
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1answer
277 views

Ellipse arcs. Draw a tangent line in the end point or make arc longer?

I read this article: link It describes how to draw ellipse arcs at all from svg. Each ellipse is described with the following params (and I know them): x1, y1, x2, y2 - arc from point (x1, y1) to ...
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1answer
137 views

Graphics clipping: How can repeated half-space clipping fail?

Hi I am currently going through the past exam problems and I am stuck on this clipping problem. Could you give me some hint on how to solve it? If we clip a polygon to a window, it is inadequate ...
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1answer
68 views

Center of Distance

I am given $N$ points in a 2D plane($x$ and $y$ coordinates). I have to find a point in this plane with coordinates $X$ and $Y$ such that: $$\sum_{i=1}^N \max\{|X - A_i|, |Y - B_i|\}\text{ is ...
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2answers
148 views

Checking convexity from outside

Is there any method or algorithm to determine convex (or non-convexity) property of a region from outside (perimeter) ? One way is plotting tangent line in each point of perimeter and discuss how ...
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1answer
156 views

Winding a space curve

Can I find parametric equations for a curve that is winding another curve, which I know -- let's say it's $\vec{f}(t) = (x(t), y(t), z(t)) = \{\sin (t)+2 \sin (2 t), \cos (t)-2 \cos (2 t), -\sin (3 ...
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1answer
117 views

Predicting the size of epsilon-net in SU(2)

I'm writing an algorithm that takes as input a finite set of matrices in SU(2) and consequently tries to generate an '$\epsilon$-net' by computing all possible matrix products (up to a given depth). ...
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1answer
76 views

$2d$ line equations in polar coordinates

I know in polar coordinates, a $2d$ line equation is given in the form of $$r = x \cdot \cos(\theta) + y \cdot \sin(\theta),$$ where the parameters are defined as in this. I want to derive an ...
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1answer
836 views

How to find the intersection of the area of multiple triangles

I have a couple of questions regarding finding the intersection of triangles. I have a system of 16 projectors that all have slightly different color gamuts. The color gamuts are represented by a ...
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1answer
241 views

Algorithm for computing the plane that passes through three arbitrary points

I'm writing a computer graphics library, and I'd like to compute the plane that passes through three arbitrary points, $p$, $q$ and $r$. I'm defining the plane in the form of $Ax + By + Cz + D = 0$. ...
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1answer
227 views

Create wind animation

I'm trying to visually illustrate forecast wind speed and direction, the programming is the easy part, the math, I'm fuzzy on. I have a grid of points (lat/lon) , the forecast wind speed and ...
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2answers
2k views

Polarity of the Surface Normal of a 3D triangle

I have a triangle (defined in 3D space) that has 3 points (p1, p2 and p3). Is it possible to work out what the polarity of the surface normal would be for the face knowing it lists each point in an ...
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1answer
529 views

Meaning of this 4x4 determinant

Let $p,q,r$ and $s$ be four points on the plane. Moreover, $p,q,r$ are given in clockwise order. My book said that the following determinant is positive if and only if $s$ lies inside the circle ...
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2answers
180 views

General Proof Of Intersection Of Two Segments

Sorry for a silly question, I am trying to prove the fact of intersection of two segments on the plane. For example, $(d_1,d_2)$ is the first segment, where $d_1$ and $d_2$ are endpoint of the ...
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1answer
489 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 ...
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1answer
103 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 ...
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1answer
116 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 ...
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2answers
178 views

Handling points to get closed Cycles

I have set of line segments, containing only 2 points. I know their point numbers. some point numbers are appeared in many lines according to their connections. So, when joining some end points, I can ...
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1answer
86 views

Cutting a d-simplex

Why is it possible to get any possible subset of nodes of a d+1 simplex in IR^d using halfspaces?
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1answer
233 views

Algorithm to Choose Consistent Normals for All Faces on a Polyhedron

I have a polyhedron $P$, in 3D, which consists of $f$ faces, each face consists of $V$ vertexes. My question is, how to choose a consistent normal orientation for all the faces? Consistent here means ...
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1answer
459 views

Using Chazelle's simplicity test to verify simple polygons intersection

Is there a way to verify whether a non-empty intersection exists between two simple polygons (not necessarily convex) using the Chazelle's simplicity test ?
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1answer
25 views

point inclusion in a half-plane 3D

I have a 3D half-plane defined using a line segment an a point (as shown in picture taken from here). I am wondering how I can detect if a point belongs to the half-plane. Is there any way to ...
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1answer
49 views

What is an offset bisector in 2D polygon skeletonization?

I'll be referring to the definition of the offset bisector from the definitions section of CGAL's 2D Straight Skeleton and Polygon Offsetting module. The halfplane to the bounded side of the line ...
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1answer
42 views

How many techniques are there to test collinearity of $n$ points?

How many techniques are there to test coliniariry of n points? For example, suppose we have 4 points A, B , C, D. How many ways can it be tested that they are collinear? This answer lists 03 ...
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1answer
28 views

What is the name for the image form you get you take a line segment and sweep it through a region of space?

For instance, if you were to take a line segment and translate it along a coplanar path, then you'd get a plane. If the path is cyclic and on that path you rotate the line segment on the axis ...
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1answer
111 views

Convex hull solving using a rubber band?

The convex hull can be found by stretching a rubber band so that it contains all the points and then releasing it. So my question is : lets assume that we have a robot (a theoretical robot) to solve ...
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1answer
40 views

Why simple polygons in plane have this property?

If we are given a simple polygon $P$ in the plane by the points $A_1, A_2, \dots, A_n$. How can we prove that there are $3$ consecutive points $A_i, A_{i+1}, A_{i+2}$ (if $i = n$, for $A_{i + 1}$ and ...
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1answer
85 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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2answers
42 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
491 views

Equation of hyperplane in Matlab

Given $n$ points in $n$-dimensions, using MatLab, how should we find the equation of the $(n-1)$-dimensional hyperplane passing through these $n$ points.
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1answer
22 views

find set of points for lots of triangulations

I should find a set of $n$ points $Q$ in a plane, so that $t(Q)$ (the number of possible triangulations) the following equation holds: $$t(Q) \ge 2^{n-2\sqrt{n}}$$ I solved the problem using the ...
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1answer
142 views

Calculate base and coefficient for power curve through 3 non-linear points

I have a formula that takes a 0-based bounded single dimensional input and transforms it to a specific power curve. EDIT This is single dimensional. There is no $y$. In the image, I'm showing how ...
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1answer
308 views

3D Convex Hull and The Gift Wrapping Principle

I am currently trying to implement a 3D convex hull algorithm that is based on the paper Convex Hulls of Finite Sets of Points in Two and Three Dimensions by F.P. Preparata and S.J. Hong, but I’ve run ...
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1answer
111 views

When is a convex polygon inscribable?

Defining the diameter of a convex polygon as the maximum possible distance between all pairs of vertices, can we conclude that the convex polygon is inscribable (i.e has all its sides as chords of a ...
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1answer
298 views

What is the typical method for sampling uniformly in a convex polytope

The polytope in my case is the intersection of the k-plane $Ax=b$ and $\{x>0\}$ where $A$ is the constraint matrix and $b$ is some solution. I'd like to find a method that is fast and efficient for ...
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1answer
67 views

What do we call the angular arcs between two edges of triangles?

I've been trying to find a geometry library for java which is as high level as describing angles between adjacent sides of triangles given 3 sides. So, what do we call such kind of arcs. In many ...
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1answer
139 views

angle between steepest gradient of two plane

IF I have two 3d planes such as Oab and Oa'b'. If these two planes intersect a horizontal plane and the intersection of each plane makes AB and A'B' lines. then, Does the angle between AB, A'B' ...