1
vote
2answers
82 views

Retrieve the initial cubic Bézier curve subdivided in two Bézier curves

I have a cubic Bezier curve subdivided to two cubic Bezier: Assuming that "t_cut" is the t value where this initial Bezier is cut: example of function subdivision(BezierCurve initialCurve, ...
2
votes
2answers
43 views

What are a geometric system and a finite geometry?

Wikipedia says A finite geometry is any geometric system that has only a finite number of points. I wonder what a geometric system is? Is it some set system $(E, F)$, where $E$ is a set and $F ...
1
vote
1answer
41 views

Finite family of subtori in the torus $(S^{1})^{n}$

Working on a problem on matroids, I've already ask a question about some subtori. Here's the link to a previous problem: Topological subspace in $(S^{1})^{n}$ Anyway, here's another problem related ...
0
votes
0answers
21 views

How could I calculate displacement along a 2D polyline by integrating each dimension separately?

This question's field of application is GPS trajectory analysis, but I'll try to give it a more abstract mathematical treatment. Suppose the trajectory of an object in 2D plane is described by a ...
0
votes
0answers
11 views

Generalization of $k$-sets with convexity

For a collection of points $X = x_1,\ldots,x_n$ in $\mathbb{R}^d$, $k$-sets are defined as the subsets of $k$ points of $X$ that lie strictly on the same side of a hyperplane. There is a large ...
3
votes
1answer
87 views

Placing n points in a MxM square grid

I am facing an apparently well-known problem: placing $n$ points in a discrete grid so that the points are 'evenly' distributed. By evenly I mean that I would like the density of points to be nearly ...
1
vote
2answers
43 views

How to prove this Mathematical Induction problem?

We got $n \geq 3$ lines drawn on a surface with conditions below: No two lines are parallel. No three lines make a conjunction in a specific point. Prove that one of the areas created by these ...
12
votes
2answers
309 views

What problems are easier to solve in a higher dimension, i.e. 3D vs 2D?

I'd be interested in knowing if there are any problems that are easier to solve in a higher dimension, i.e. using solutions in a higher dimension that don't have an equally optimal counterpart in a ...
1
vote
4answers
913 views

What is the maximum number of pieces that a pizza can be cut into by 7 knife cuts? (NBHM 2005)

I am seeing this question very first time and do not know any formal way to solve it. Which part of mathematics it is related to? What is the maximum number of pieces that a pizza can be cut into by ...
2
votes
3answers
148 views

Moscow Math Olympiad 1973

In every polyhedron there is at least one pair of faces with the same number of sides. Solution: Let $N$ be the greatest number of sides in a face of a given polyhedron. Then the number of ...
2
votes
0answers
70 views

geomtry and induction Discrete Math

Claim: Suppose that we draw any number of straight lines in the plane, with the restriction that no two are parallel and no intersection point belongs to more than two lines. The lines divide up the ...
0
votes
1answer
278 views

Among these figures circle, square, rectangle, isosceles triangle which has the greatest perimeter had the same area?

Among these figures circle, square, rectangle, isosceles triangle which has the greatest perimeter had the same area geometrically ?
1
vote
1answer
27 views

Subdivision method of triangulation

Is there a standard name for the subdivision scheme for a 2d simplicial complex, in which each face is subdivided into 4 (similar triangles), and each edge in half?
2
votes
0answers
40 views

getting PDF from a given Moment Generating Function

if the moment generating function mgf of a random variable w is M(t)=(1-7t)-20 find the i)pdf ii)mean iii)variance of w
0
votes
1answer
110 views

Graph Help - Discrete Math

The vertices indicate where cashiers are located; the edges denote unblocked aisles between cashiers. The department store wants to set up a security system where (plainclothes) guards are placed at ...
1
vote
1answer
50 views

Proof convex polyhedron with line does not contain a corner if closed

The excercise I am struggling with is the following: Given a convex closed polyhedron that contains a line, the question is, whether this polyhedron can also contain a corner. My idea was to make a ...
1
vote
0answers
43 views

Edge in a convex polytope

I want to show that a convex polytope $A$ that is an intersection of half-spaces contains an edge if $ A=\{x \in \mathbb{R}^n|Ax=0 \wedge x \ge 0\}$, where x greater equal 0 means, that all components ...
8
votes
6answers
296 views

Can we always draw $n/3$ disjoint triangles from $n$ points in the plane in general position?

Suppose we are given $n$ points in the plane, where $n$ is a multiple of $3$ and no three of these points lie on a line. Is it possible to group all of these points into sets of three, so that if we ...
2
votes
0answers
41 views

Kmeans on “symmetric” data

A set is said to be fully-symmetric if for every $x$ in it, negating one of its components results in $y$ such that $y$ is in the set as well. A set is said to be semi-symmetric if for every $x$ in ...
1
vote
3answers
101 views

Discrete math question… Finding largest area

Can anyone help explain to me how to go about solving this question ?: Of all the rectangles with perimeter 4, prove that the square has the largest area. I know how to solve this algebraically, ...
2
votes
0answers
119 views

What exactly is written on this blackboard? [closed]

The other day I walked into an office in the university where they are working on some kind of a project which I don't really know a lot about, and I found the blackboard filled with the following ...
4
votes
2answers
260 views

Distribution of points on a rectangle

Let $R$ be a rectangular region with sides $3$ and $4$. It is easy to show that for any $7$ points on $R$, there exists at least $2$ of them, namely $\{A,B\}$, with $d(A,B)\leq \sqrt{5}$. Just divide ...
3
votes
2answers
814 views

Graham scan convex hull algorithm - include all points on boundary

I have am implementing the Graham scan algorithm to find the convex hull of a set of (two-dimensional) points. (My implementation is in Haskell in case anyone wants to know.) The problem is that not ...
1
vote
3answers
372 views

Proving by induction that an equilateral triangle will always be divided into (n+1)^2 small triangles?

I'm working on a proof that looks like this: Let $n$ be a positive integer. Given an equilateral triangle, place $n$ points on each side, dividing the side into $n+1$ equal segments. Use the ...
2
votes
0answers
96 views

The relation between perfect difference sets and finite projective planes

Given a (finite) perfect difference set, it is easy to create a finite projective plane. I'm wondering: Given a finite projective plane, does there necessarily exist a corresponding perfect ...
8
votes
1answer
446 views

Into how many parts do $n$ ellipsoids divide $\mathbb{R}^{3}$?

What is the maximum number of regions into which $\mathbb{R}^{3}$ can be divided by $n$ ellipsoids? (Each ellipsoid has the same size). Let´s denote this number by $r_{n}$. Clearly $r_{1}=2$. But ...
1
vote
0answers
78 views

Count Exclusive Partitionings of Points in Circle, Closing Double Recurrence?

I am studying a problem that I have worked out is equivalent to the following: Problem Description Given N distinct points on the border of a circle, there are $B_N$ ways to partition them - where ...
3
votes
1answer
104 views

probability involving matching of discrete shapes on a square grid

Figure F exists on a regular square grid. T transforms F by any combination of horizontal or vertical reflection as well as rotation by 90 or 180 degrees. A larger background grid of X by Y contains ...
1
vote
1answer
268 views

To find the number of points on a 2D grid?

Given N points on a 2D grid of the form (X,Y) we need to find to find all the points (R,S) such that the sum of the distances between the point (R,S) and each of the N points given is as small as ...
4
votes
0answers
157 views

Convex hull of balls

The convex hull is defined as the smallest convex set containing a set of points. Now we want to generalize it to a set of balls. If these balls have the same radius, then it can be shown that a ball ...
3
votes
3answers
227 views

Help me name or find the existing name for this geometric concept!

This may have a proper name, if so - let's discuss. If not, let's name it. This is for a web application in C#, so whatever we call it I will start naming as such in my code. I'm taking GPS data as a ...
6
votes
5answers
2k views

If a plane is divided by $n$ lines, then it is possible to color the regions formed with only two colors.

I am self-studying Discrete Mathematics, and there is the following exercise. (in Portuguese) A plane is divided by many lines. Show that it is possible to color the regions formed with only two ...
0
votes
1answer
77 views

minimal po2 surface for sprite arrangement

I've got 64 images of 96x192 pixels. I have to arrange them on a rectangle. I need the height and width of that rectangle to be powers of two. Given the dimensions of my images, they don't perfectly ...
3
votes
1answer
168 views

Proof whether or not 1/k by 1/(k+1) rectangles fit inside a unit square

I am reading Concrete Mathematics and came across an interesting problem, number 37 of chapter 2. The answers to exercises lists no known answer to this problem: Will all the 1/k by 1/(k+1) ...
0
votes
2answers
49 views

What is the number of sides of the hollow area generated by juxtaposition of four $2n$-sided regular polygons?

Here's an horrible drawing that tries to explain what I'm asking: Trying this with small numbers gives me $f: 4 \to 0, 6 \to 4, 8 \to 4, 10 \to 8, 12 \to 8.$ This suggests that $$f(2n) = 4 \times ...