Questions tagged [plane-geometry]
Plane Geometry is about flat shapes like lines, circles and triangles , shapes that can be drawn on a piece of paper
501 questions
1
vote
0
answers
96
views
Pseudo-Droz-Farny circles
I would like to present a construction of 2 circles. These 2 circles are somewhat similar in appearance to the well known Droz-Farny circles that can be drawn for every isogonal conjugate pairs of ...
1
vote
0
answers
124
views
A center of convex planar regions based on chords
This is based on Chapter 6 of 'Convex figures' by Yaglom and Boltyanskii. This post also continues On two centers of convex regions.
A point $P$ in the interior of a planar convex region $C$ divides ...
1
vote
0
answers
112
views
A generic question on circles associated with a triangle
This question is inspired by two posed by P.Terzić (both given elegant synthetic proofs by F. Petrov). The starting point is a triangle $ABC$ and a triangle centre $G_1$. There are two classical ...
1
vote
0
answers
106
views
Length of isoline $x(1-x)y(1-y)=c$
For the integral appearing in this answer, it may be beneficial to derive the length $L(c)$ of the isoline:
$$x(1-x)y(1-y) = c,$$
where $x,y$ are ranging in $[0,1]$, and constant $c\in [0,\frac1{16}]$....
1
vote
0
answers
84
views
How can construct the equilateral $A''B''C''$ such that area of $A''B''C''$ is biggest
Let $ABC$ be arbitrary triangle in a plane. Let $A'B'C'$ and $A''B''C''$ be two equilateral triangles such that $A \in B'C'$, $B \in C'A'$, $C \in A'B'$ and $A \in B''C''$, $B \in C''A''$, $C \in A''B'...
1
vote
1
answer
227
views
On comparing planar convex regions of equal perimeter and area
Definitions:
The Hausdorff distance between two point sets is the greatest of all the distances from a point in one set to the closest point in the other set.
Given two planar convex regions $C_1$ ...
1
vote
0
answers
27
views
Complexity of tour-expansion heuristic for the planar Euclidean TSP
This is a followup question to this one: Computational Geometric Aspects of Greedy Tour Expansion.
Assume that the candidate point, whose insertion into current incurs the least tour-length increase, ...
1
vote
0
answers
361
views
Geometric interpretation of metric [closed]
For metric in $\mathbb R^2$ (polar coordinates) Pythagorean triangle elements can be visualized in a differential form as seen at top yellow triangle. This happens for metric:
$$ ds^2= dr^2 +(r d \...
1
vote
0
answers
84
views
Trapping lightrays under nonstandard reflections and/or paths
Almost every version of trapping lightrays with mirrors is either resolved---usually negatively---or open:
"It is unknown whether one can construct a polygonal trap for a parallel beam of light": ...
1
vote
0
answers
139
views
On non-convex polygons that tile convex polygons
Polyominoes are rectilinear polygons - all angles are 90 or 270 degrees. It is well-known that among non-convex polyominoes are several whose copies can neatly tile rectangle. And for several such '...
1
vote
0
answers
202
views
Some Problems On Apollonian Gasket
Since 2013, I found Some problems on Apollonian Gasket as following. These problem also is higher level of Eppstein Point. I am looking for a proof of one of these problems:
Let three $(A)$, $(B)$, $(...
1
vote
0
answers
142
views
Comparing Different Notions of Unicoherence in the Plane
Unicoherence is a generalization of simple connectedness that has been useful in topology in one and two dimensions. It is also a fundamental concept in shape theory, and thus has relations to Cech ...
1
vote
0
answers
218
views
Patterns in local winding number sequences
This is something of a followup to an earlier question
Can any sequence of consecutive integers be realized as winding numbers?, whose answer is Yes.
Now I would like to define a local winding number ...
1
vote
0
answers
62
views
When do quasiperiodic functions on $\mathbb{R}^2$ have an average over the plane?
For example, consider the following
\begin{equation}
\lim_{T\rightarrow\infty}\frac{1}{(2T)^2}\int^T_{-T}\int^T_{-T}\big[\cos(v/a)-\cos(u/b)\big]\cos(\sqrt{u^2+v^2})\ du\ dv,
\end{equation}
where $...
1
vote
0
answers
149
views
Determine sub-polygon from line segments with known member connectivity
Test Polygon:
Consider the following polygon as attached. Let the known parameter be as follows:
•Member to member connectivity, i.e. it is known that A – B, X – E, F – B, etc. for all the members.
•...
1
vote
0
answers
122
views
Generalization of Ellipse via Fixed Sum of 3 Distances to "Foci"
It is a well known fact, that ellipses can be defined as $$\{x\in\mathbb{R}^2\ \ |\ \ \|x-A\|+\|x-B\|-\|B-A\|=e\in\mathbb{R}_0^+;\ A,B\in\mathbb{R}^2\}$$
Question:
has the generalization
$$\{x\...
1
vote
0
answers
109
views
Intersection points of closed curves inscribed in a convex polygon
Suppose that I have two distinct simple closed curves, $C_1$ & $C_2$, and each is inscribed in a convex polygon, D. By inscribed, I mean tangent to each side of D. In particular, I am most ...
1
vote
0
answers
66
views
Non-Convex Polygons with "Antipodal Visibility"
by "antipodal visibility" of planar, simple polygons I mean the following property:
if two points $p$ and $q$ on the polygon's boundary divide the polygon's boundary into two polylines of equal length,...
1
vote
0
answers
618
views
Which plane convex arcs have the smallest maximum curvature?
Let $p$ and $q$ be positive real numbers with $p \leq q$. Suppose that $H(p,q)$ is the class of all convex arcs $c$ in the Cartesian $x-y$ plane which satisfy the following conditions:
(1)The $y$-...
0
votes
2
answers
129
views
Planar curves identical to their inverses
Is the right strophoid
the only planar curve $C$ whose inverse curve w.r.t. some circle (in this case: centered on the origin)
is identical to $C$?
&...
0
votes
1
answer
84
views
Hyperbolic version of Sylvester co-linear problem
Is the hyperbolic version of Sylvester co linear problem true?
0
votes
1
answer
238
views
Does anyone know of an academic reference for this proof that the tangent to a hyperbola at a point bisects the angle between each focus to the point?
I am writing a report and as part of it I need to prove the property that for a point $P$ on a hyperbola, the tangent to the hyperbola at $P$ bisects the angle $\angle F_1PF_2$ where $F_1$ and $F_2$ ...
0
votes
3
answers
115
views
Calculating radii allowing for circular placement of polygonal linkage's joints
Given a planar polygonal linkage defined by a sequence of $n$ hinge joints $(j_0,\,\cdots,\,j_{n-1},j_n = j_0)$ with links of fixed lengths $\lbrace\|j_{k+1}-j_k\|=d_k\ |\ 0\le k\lt n\rbrace$ between ...
0
votes
2
answers
3k
views
Determine the boundary points of a set of points [closed]
I have a set of points $S=\{(x_1,y_1),(x_2,y_2),\ldots,(x_n,y_n)\}$. Then how to find the boundary points (which is a subset of $S$) of $S$?
There are methods like convex hull, concave hull and $\...
0
votes
1
answer
332
views
Finding the minimum-area parallelogram containing all vertices of a cuboid projected to a plane
Description edited after comments:
Let's say I have a 3d cuboid that I projected onto a 2d plane. Now I want to find the minimum-area parallelogram that contains all those projected vertices. How do ...
0
votes
2
answers
130
views
Algebraic planar curve with precisely $n$ closed components? [closed]
For each integer $n$ I am looking for a real-valued polynomial in two variables, $A_n(x,y)$, such that $A_n(x,y) = 0$ defines a curve with precisely $n$ closed components in the plane $\mathbb{R}^2$. ...
0
votes
2
answers
836
views
Why does the area function of a parallelogram have a nonintuitive geometric solution?
I was reading a blog post on a simple derivation of the cross product. I learned how to determine the area of a parallelogram enclosed by two vectors $A$ and $B$.
First, here is the proof of the ...
0
votes
2
answers
177
views
Radical line of two ellipses
The equation of an ellipse with focal points $(a_1,b_1)$ and $(a_2,b_2)$ which passes through the point $(a_3,b_3)$ is given by the equation
$$\begin{gathered}
\sqrt{ (x-a_1)^2+(y-b_1)^2 } + \sqrt{ (x-...
0
votes
1
answer
126
views
On 'special' points on uniform planar convex regions defined in terms of moment of inertia
The following can be easily proved using perpendicular axes theorem and intermediate value theorem:
Lemma: Given any uniform planar convex region $C$ and any point on it, $P$, there will be at least ...
0
votes
1
answer
80
views
Extending functional inequality from rectangles to parallelograms
Let $f$ be a function defined on the unit square $R = [0,1]^2 \subseteq \mathbf{R}^2$ satisfying
$f \geq 0$, $f(0,0) = 0$,
$\frac{\partial{f}}{\partial{x}} \geq 0$, $\frac{\partial{f}}{\partial{y}} \...
0
votes
1
answer
1k
views
Fast way to generate random points in 2D according to a density function
I'm looking for a fast way to generate random points in 2D according to a given 2D density function.
For instance something like this:
Right now I'm using a modified version of "Poisson disc&...
0
votes
1
answer
176
views
Expected area of a pentagon formed from a randomly broken stick [closed]
Suppose we break a stick of length one at four randomly and independently chosen points and that the resulting pieces form a pentagon.
Such a pentagon can be formed with probability $1-(5/16) = {11\...
0
votes
0
answers
44
views
Lattice points in the boundary of a Minkowski sum of two convex lattice polygons
Let $P$ and $Q$ be two convex lattice polygons in $\mathbb{R}_+^2$ and let $P+Q$ be their Minkowski sum. Given a set $S \subset \mathbb{R}^2$, we let $L(S) =\#( S \bigcap \mathbb{Z}^2)$.
The equality $...
0
votes
0
answers
78
views
Coordinates of the centers of the insphere and circumsphere
Suppose we are working in $\mathbb{R}^3$ space and we have four non-coplanar and non-collinear points, $(x_a, y_a, z_a)$, $(x_b, y_b, z_b)$, $(x_c,y_c, z_c)$, and $(x_d, y_d, z_d)$. How does one ...
0
votes
1
answer
231
views
Divide angles by coefficients relate to Fibonacci sequence
In the left Figure, consider a right triangle $OPA$ with $\angle {AOP} = 90^\circ$. Let $\ell$ be the reflection of $PO$ in $PA$ and $\ell$ meets $OA$ at $A_1$. Let $O_1$ be the center of the circle $(...
0
votes
0
answers
41
views
Air-transmissive mirror (illumination problem for a wall)
I propose an alternative version to the illumination problem (where the mirrored walls surrounding a room prevent light from reaching some region).
Here, our building area is an infinite straight band ...
0
votes
0
answers
77
views
In how many ways is it possible to order the sides and diagonals according to their length for all n-gons?
If we'd do it for example for 4-gons, for quadrilaterals, we could start with all the possible quadrilaterals.
We could say that the four vertices are a,b,c and d.
And then we'd have 6 lines, I mean,
...
0
votes
0
answers
108
views
A surprising result with the Riccati difference equation
I was looking at the Riccati difference equation with positive and negative indices
$$
R_n=\frac{aR_{n-1}+b}{cR_{n-1}+d}\quad n\in[0,N]\\
R_n=\frac{-dR_{n+1}+b}{cR_{n+1}-a}\quad n\in[-N,0]\\
$$
along ...
0
votes
0
answers
128
views
Concurrencies determined by intersections of angle trisectors (and isogonal lines) in a triangle
The famous Morley’s theorem, states that in a triangle the interior angle trisectors, proximal to sides respectively, meet at the vertices of an equilateral. However the six trisectors meet at 12 ...
0
votes
1
answer
55
views
On 'axiality' of planar convex regions
Axiality has been studied under a definition given here: https://en.wikipedia.org/wiki/Axiality_(geometry)
Consider an alternative definition of axiality as follows: For a convex region C, consider a ...
0
votes
0
answers
36
views
Vertex configuration to tile repeat unit
I am working with elongated triangular tiling which has a vertex configuration of 3.3.3.4.4 and noticed that the representative symmetry is not the repeat unit. Is there a general formula to convert ...
0
votes
0
answers
610
views
How to find a point on a line that minimizes sum of distances from three given points?
Let there be given three points $x_1, x_2, x_3$ and a line $l$ on a plane. Does there exist an explicit method of finding such a point $p$ on $l$, that sum of distances of $p$ from $x_1, x_2, x_3$ is ...
0
votes
0
answers
36
views
What is the locus defined by those equations?
I would like to know what is the locus of $x \in \Bbb R_+^n$ ($n=2$ would already be fine) defined by
$\sum a_i \cdot x_i$ s.t. $a_i+\epsilon \geq 0$, $\epsilon \in \Bbb R$.
I know that if $\...
0
votes
0
answers
167
views
Infinity new equilateral triangles in one configuration of triangle plane
An equilateral triangle constructed from a reference triangle is a topic which is intersested by plane geometry lovers. See Napoleon equilateral triangle, Morley equilateral triangle....In this topic ...
0
votes
0
answers
89
views
What is the minimal number of lines needed to partition a simplex into cells of diameter at most $\epsilon$?
I am studying a problem that requires me to partition the simplex into cells using a particular family of hyperplanes. For concreteness, consider the 2-simplex. I would like to construct lines ...
0
votes
0
answers
332
views
Are finite projective planes isomorphic to PG(2,q), where q=p^k with p a prime?
I can't find any reference to the following question:
Are finite projective planes isomorphic to $PG(2,q)$, where $q=p^k$ with $p$ a prime?
I think it's meaningless to continue studying without ...
-1
votes
2
answers
240
views
Locus of points for which the sum of the angles subtended there by two different line segments is a constant
Given a line segment AB, the locus of points P such that the angle APB has a constant value is a 'biconvex lens' formed by 2 circular arcs that passes through the points A and B. Special case: if APB ...
-1
votes
1
answer
824
views
Johnson-Lindenstrauss lemma preserves angles
Edit (April 1, 2020):
I formalized the original question (from the reference below) wrong. Please see @AndreasBlass correction.
Also, all the suggestions proposed so far doesn't seems to be related ...
-2
votes
1
answer
587
views
Is the conjecture true for n-sphere $(n>2)$? [closed]
This is higher dimension conjecture of Problem 3845 in Crux Mathematicorum and Theorem 2 in here:
PS: This figure is very nice, this is also generalization of Brianchon’s theorem, The Pascal theorem, ...
-3
votes
2
answers
546
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 ...