The tag has no wiki summary.

learn more… | top users | synonyms

15
votes
6answers
967 views

Tetrahedra with prescribed face angles

I am looking for an analogue for the following 2 dimensional fact: Given 3 angles $\alpha,\beta,\gamma\in (0;\pi)$ there is always a triangle with these prescribed angles. It is ...
2
votes
1answer
1k views

Finding a minimum bounding sphere for a frustum

I have a frustum (truncated pyramid defined by six planes) and I need to compute a bounding sphere for this frustum that's as small as possible. I can choose the centre of the sphere to be right in ...
5
votes
1answer
561 views

Malfatti Circles - Limiting point

"Three circles packed inside a triangle such that each is tangent to the other two and to two sides of the triangle are known as Malfatti circles" (for a brief historical account on this topic, see ...
6
votes
4answers
795 views

How to compute the average distance till intersection within a triangle in R^2?

Lots of simple questions because I am a noob. You are given 3 points in R^2; A, B, C forming a triangle with area > 0. You pick an arbitrary point inside ABC and an arbitrary direction. After some ...
8
votes
0answers
116 views

Generalized Hlawka inequality

Let $E$ be a vector space over the real (the complex case is interesting too). We consider functions $f:E\rightarrow\mathbb R$ which satisfy the homogeneity property $$f(\lambda x)=|\lambda|\,f(x).$$ ...
-4
votes
0answers
33 views

How many vectors exist satisfying the angle between any two vectors equals to a constant beta(0<beta<pi) in a n-dimension Euclid space? [closed]

At first, if beta=pi/2, we know that at most n such vectors exist, that is, orthogonal vectors. It's obvious that the number of vectors is influenced by the angle beta. Assume we've already found k ...
3
votes
1answer
124 views

Random non-intersecting circles in the plane

If I give a finite region of $\mathbb{R}^{2}$ and place $k$ circles of radius $r(k)$ uniformly at random inside, are there any known results for the probability that the circles do not overlap? ...
3
votes
0answers
58 views

Containing a “fuzzy” ellipsoid within an ordinary ellipsoid

Consider the ellipsoid described by the inequality $(x - x_c)^T P^{-1} (x - x_c) \leq 1$, where the vector $x_c \in \mathbb{R}^n$ denotes the center of the ellipsoid and the symmetric positive ...
2
votes
1answer
924 views

How to find the Fermat Point using the construction of the tangent to ellipse?

Be done the triangle ABC, it is known the method to finding the point Q that minimises the sum QA+QB+QC among all points Q in the plane (The Fermat point). I want a hint for solving this problem using ...
4
votes
7answers
2k views

Side-Angle-Side Congruence and the Parallel Postulate

Is there a link between the side-angle-side congruence of triangles and the parallel postulate? Specifically, does it follow from Euclid's first four axioms alone? In fact, does it even follow from ...
5
votes
1answer
197 views

Maximum number of general-position points with mutual rational distances?

Richard Guy has shown that there are six points in the plane—no three collinear, no four cocircular—such that all interpoint distances are rational. Guy, Richard. Unsolved Problems in ...
3
votes
0answers
89 views

Dubins car shortest paths: Decidable?

A Dubins car follows a Dubins path in $\mathbb{R}^2$, with constant wheel speed and limited turning radius. It is known that the shortest Dubins path in the absence of obstacles follows circular arcs ...
5
votes
2answers
424 views

Historical question re: ellipses obtained by certain geometrical constructions

I am a faculty member in the Forensic Science Program at PennState (UP). I am trying to obtain information of a historical nature concerning two closely related topics. I seek historical references ...
5
votes
2answers
262 views

What can be said of the structure of a metric space without isosceles triangles?

This is a question that has been bothering me in the back of my head for quite some time. Suppose we have a metric space $X$ with metric $\mathrm{d}$. By an isosceles triangle we mean a tuple of ...
2
votes
2answers
105 views

Maximum possible number of similar three-colored triangles

I want to maximize the number of similar triangles with vertices from three fixed sets, one vertex from each set. For example, if you fix two points $X$, $Y$ (i.e. two sets with only one member), then ...
17
votes
4answers
832 views

Metrics for lines in $\mathbb{R}^3$?

I seek a metric $d(\cdot,\cdot)$ between pairs of (infinite) lines in $\mathbb{R}^3$. Let $s$ be the minimum distance between a pair of lines $L_1$ and $L_2$. Ideally, I would like these properties: ...
53
votes
9answers
5k views

Geometric proof of the Vandermonde determinant?

The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...
-3
votes
2answers
95 views

Hexagon Formed by connecting Trisections of triangle sides [closed]

Is there a theorem for the area of the hexagon formed by connecting the points formed when the sides of a triangle are trisected? It appears that the ratio of the area of the triangle to the area of ...
2
votes
1answer
224 views

Tools for Removing Radicals from Equations

I am currently doing some investigations on Sylvester's 4 Point Problem Probability of 4 Points being in Convex Configuration and repeatedly face the problem of solving equations between sums of ...
8
votes
3answers
640 views

On maximal regular polyhedra inscribed in a regular polyhedron

Let T, C, O, D, or I be regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron, respectively. Suppose that the outer polyhedron have edge-length 1. For example, it's easy to prove that ...
1
vote
0answers
95 views

A conjecture about cross sections of a pyramid [closed]

Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon. This is a conjecture I came across while trying to solve this problem. I was ...
17
votes
3answers
1k views

Angle of a regular simplex

I find the following question embarrassing, but I have not been able to either resolve it, or to find a reference. What is the vertex angle of a regular $n$-simplex? Background: For a vertex $v$ ...
1
vote
0answers
130 views

Continuity of minimizers to distance function from point to convex set

Suppose I am minimizing the Euclidean distance in $\mathbb{R}^{n}$ between a point $y$ and compact convex set $U$ (where $y\notin U$): $\min_{x\in U}\|x-y\|$. I believe the minimizer $x_{U}^{*}$ is ...
3
votes
1answer
226 views

Conditions for a parametric curve to avoid self-intersection?

Suppose a planar curve $C$ is defined by parametric equations in $t$: $x(t)$ and $y(t)$. Are there conditions on these two functions that guarantee that $C$ does not self-intersect? For example, the ...
3
votes
1answer
239 views

Generalising Euler's formula to ellipses and three dimensions

Let $D$ be the closed unit disk, $T$ a triangle and $E$ an ellipse with $E\subset T \subset D$. Without loss of generality say that $E$ is centred at Cartesian coordinates $(c, 0)$ with $0\leq c \leq ...
2
votes
0answers
142 views

Probability of 4 Points being in Convex Configuration

Background of my question is, that I would like to implement a parallel preprocessing for a constructing the convex hull of very huge number of points in the euclidean plane; the idea is to process ...
4
votes
1answer
416 views

Reference request: a differential equation in elementary geometry

15 hours and four up-votes but not a word from anybody. That's the result of this question to stackexchange. My question is where the following differential equation arises naturally and where it ...
14
votes
1answer
3k views

spiral of Theodorus

A long time ago when I was in college I read about making a spiral out of right triangles with sides 1 and $\sqrt{N}$. (A google search seems to indicate that this is called the Spiral of Theodorus.) ...
1
vote
5answers
1k views

Quadrilateral from 4 random points

Given 4 random points in 2D, how do I compute the area of the quadrilateral formed by the points? I'm aware of formulae giving the area when I know the sides a,b,c,d and the diagonals p & q. But ...
3
votes
2answers
295 views

Points contained in a disk [closed]

I have a question, but not sure how to prove this. We are given $n$ points in the Euclidean plane such that there exists no disk of radius $a$ which contains all of the points. Conjecture: There ...
9
votes
3answers
680 views

Neusis constructions

Is there some simple description of which complex numbers are "constructible" with straightedge and compass and neusis? See http://en.wikipedia.org/wiki/Constructible_number and ...
2
votes
1answer
146 views

Approximation of a convex body by a contained polytope

This question deals with approximating a convex body (a compact convex set of $\mathbb{R}^d$ with non-empty interior) by convex polytopes. For a given $\delta$, let $n_\delta$ be the number of faces ...
7
votes
2answers
198 views

How to find a tetrahedron that covers four points?

I’m looking for an explicit formula for the vertices of a regular tetrahedron that covers four given points. In particular: Given four distinct real numbers $a_1$, $a_2$, $a_3$, $a_4$, is there a ...
16
votes
9answers
2k views

Determine if circle is covered by some set of other circles

Suppose you have a set of circles $\mathcal{C} = \{ C_1, \ldots, C_n \}$ each with a fixed radius $r$ but varying centre coordinates. Next, you are given a new circle $C_{n+1}$ with the same radius ...
3
votes
1answer
360 views

What is the correct preposition? (And is there one?)

I just stumbled upon a linguistic problem I wasn't able to resolve via web search. Suppose we're given some geometric set $A$ and subset $B\subset A$. Isn't there a compact way of saying that there ...
1
vote
2answers
72 views

Successive Inner or Outer Approximation of Simple Polygons with Hierarchies of Implicit Functions

The problem I want to solve, is to quickly decide, whether a point $p=(x^*,y^*)$ is inside or outside of a polygon $P := (p_1, p_2,..., p_n=p_1), p_i := (x_i,y_i)$, with $n$ potentially very large. ...
3
votes
0answers
42 views

Number of not self-intersecting closed paths spanning $n$ iid uniform points

Let $X_1,X_2,\dots,X_n$ be independent uniform variables in the square. What is the number of piece-wise linear paths which vertices are all the $X_i$ and that do not self-intersect? In other words, ...
1
vote
3answers
493 views

Characterization of Angles Trisectable with Straightedge and Compass

Lindemann's prove of the transcendence of $\pi$ has settled the question, whether an arbitrary angle can be trisected, using straightedge and compass alone, to the negative. In the following, ...
3
votes
1answer
84 views

Number of small projections

Suppose $X$ is a finite subset of the plane and for $0\leq \theta<\pi$, let $l_\theta$ denote the line through the origin having angle $\theta$ with the positive $x$-axis. For how many values of ...
10
votes
3answers
339 views

The intersection of $n$ cylinders in $3$-dimensional space

A standard question in vector calculus is to calculate the volume of the shape carved out by the intersection of $2$ or $3$ perpendicular cylinders of radius $1$ in three dimensional space. Such ...
12
votes
2answers
577 views

Right triangle with edge lengths equal to regular unit polygon edge lengths

This question came up naturally recently from a blog post of John Baez. There is an observation of Euclid that edges of a pentagon, hexagon, and decagon inscribed in a unit circle form the edges of a ...
33
votes
1answer
2k views

Probability that a stick randomly broken in five places can form a tetrahedron

The following problem was brought to my attention by a doctoral dissertation on Mathematics Education, but - as far as I know - the solution remains unknown. I have already asked this question on ...
11
votes
1answer
388 views

Heronian triangle with two sides that are prime

Can any prime number form a Heronian triangle with a second prime as another side? I cannot find a second prime to form a Heronian triangle with either 23 or 167. I have checked up to the 10^7th prime ...
11
votes
3answers
747 views

Efficient visibility blockers in Polya's orchard problem

Polya's orchard problem asks for which radius $\rho$ of trees at each lattice point within a distance $R$ of the origin block all lines of sight to the exterior of the orchard. ...
3
votes
2answers
185 views

Equipartition of the circle [closed]

Browsing an old technical studies pupil's school book, I have found the description of a method to place at equal distance $N$ points on the circumference of a circle. I am looking for a proof of this ...
54
votes
10answers
10k views

Is Euclid dead? [closed]

Apparently Euclid died about 2,300 years ago (actually 2,288 to be more precise), but the title of the question refers to the rallying cry of Dieudonné, "A bas Euclide! Mort aux triangles!" ...
18
votes
0answers
330 views

The Eyeball Theorem generalized

I have not seen the 2D Eyeball Theorem—that tangents from the centers of two circles, each encompassing the other, intersect each circle in the same segment length—generalized to higher ...
25
votes
3answers
954 views

About the ratio of the areas of a convex pentagon and the inner pentagon made by the five diagonals

Question : Letting $S{^\prime}$ be the area of the inner pentagon made by the five diagonals of a convex pentagon whose area is $S$, then find the max of $\frac{S^\prime}{S}$. ...
7
votes
10answers
4k views

College (Euclidean) geometry textbook recommendations

I will be teaching a mid-level undergraduate course in Euclidean geometry this fall. Has anyone taught such a course, who can recommend a good textbook? My students will mostly be future high school ...
4
votes
1answer
243 views

Inequality involving the side lengths of a quadrilateral

If $a$, $b$, $c$ and $d$ are the four sides of a quadrilateral, the problem is to show that $ab^2(b-c)+bc^2(c-d)+cd^2(d-a)+da^2(a-b)\ge 0$. I've verified it to be true for quite a large number of ...