0
votes
2answers
185 views

Proof of the following: How many $(n-2)$ dimensional faces from a corner of a hypercube

I asked a question earlier regarding the number of $(n-2)$ dimensional faces exiting a corner of an $n$ dimensional hypercube. (For example the number of points in a corner of a square, or the number ...
4
votes
1answer
89 views

Covering all the edges of a hypercube?

Consider an arbitrary $n$- dimensional hypercube: If we select $n - 1$ corners of that hypercube and highlight all $(n - 2)$ dimensional elements that originate from each of the corners is it ...
4
votes
0answers
119 views

Biggest ball included in an intersection of balls

I would like to prove that for any family of balls $\{B(c_i,r_i)\}_i \subset \mathbb{R}^d$ such that $\{c_1, \dots, c_n\} \subset \bigcap_i B(c_i,r_i) $ and $\forall i, r_i \geq 1$, there exists a ...
1
vote
2answers
94 views

Approximating Euclidean geometry, restricted to $\mathbb{Q}$

I'm having trouble putting this into a fully coherent question, so I'll give the broad question, then a few bullet points to give you a better idea of what I'm asking. I'm looking for a line of ...
1
vote
3answers
86 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 ...
1
vote
1answer
591 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 ...
0
votes
0answers
145 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 ...
6
votes
4answers
3k views

Find the area of overlap of two triangles

Suppose we are given two triangles $ABC$ and $DEF$. We can assume nothing about them other than that they are in the same plane. The triangles may or may not overlap. I want to algorithmically ...
1
vote
0answers
101 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 ...
1
vote
2answers
559 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 ...
2
votes
0answers
602 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 ...
3
votes
2answers
1k views

How to calculate volume of 3d convex hull?

Convex hull is defined by a set of planes (point on plane, plane normal). I also know the plane intersections points which form polygons on each face. How to calculate volume of convex hull?
3
votes
2answers
189 views

How do I prove that the following method to find whether a point lies within a polygon is correct?

I came across the following method to determine whether a given point lies inside a convex polygon - however, I'm not sure how to prove it. Given any three points on the plane $(x_0,y_0)$, ...
2
votes
1answer
1k views

Solid body rotation around 2-axes

I am trying to understand how to describe the rotation of a solid body flying in 3D space. From physics forums, I understand that the rotation of any solid object in space, is around 2 axes ...
6
votes
1answer
202 views

Efficient algorithm for finding how many times a point is inside the triangles formed by given points

Given n 2D points and a special point p, what would be the best way to find how many times p is inside among those $^nC_3$ triangles formed by the n points.
15
votes
5answers
1k views

Is it possible to solve any Euclidean geometry problem using a computer?

By "problem", I mean a high-school type geometry problem. If no, is there other set of axioms that allows that? If yes, are there any software that does that? I did a search, but was not able to ...
1
vote
0answers
102 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 ...