For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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2
votes
1answer
31 views

How many equilateral triangles can be inscribed in a triangle? [on hold]

Given any triangle ABC find points D, E and F not A, B or C, where D is on segment AB, E on segment BC and F on segment CA, such that triangle DEF is equilateral. How many such triangles exist? I ...
9
votes
1answer
368 views

What is the Sequence that Maximizes this Distance?

Suppose we are given $n$ segments $l_1,...,l_n$ in $\mathbb{R}^2$ such that $|l_i|=i,\ \forall\ i=1,...,n$, where $|l_i|$ is the length of $l_i$. Let $\alpha_1,...,\alpha_{n-1}$ be $n-1$ angles such ...
0
votes
0answers
4 views

How to properly clamp Beckmann Distribution

I am trying to implement the Cook-Torrance Microfacet BRDF shading model and I am having some trouble with the Beckmann Distribution: Beckmann Distribution with width parameter ...
2
votes
0answers
15 views

Parking Lot Optimization Problem — How To Find the Minimal Path In A Periodic Set?

When I was a commuter student, I would park in a very large parking that that had a set of stairs in a corner that I had to climb. In general, I had to park far away from this corner in an almost full ...
1
vote
0answers
22 views

Finding distance of point from 4D ray

I'm working on a programming project. In this project, a ray is fired from a point in 4-space. I need to find the distance from this ray to a number of other points in 4-space. I attempted to solve ...
0
votes
0answers
6 views

Zooming views relative to point

I'm making a viewer for a fractal generated by convergence to a root of some polynomial using a root finding algorithm. (Examble fractal) in the complex plane. I then made an interactive viewer, ...
0
votes
2answers
25 views

How many square based pyramids are in a bigger pyramids?

The biggest challenge to solve the problem is that I can't really picture a pyramid. And it is hard to make a model. The pyramids I am trying to find include those on all tiers.
0
votes
1answer
50 views

Algebra, Geometry and Algebraic Geometry

I want to know, what is the difference between Algebra, Geometry and Algebraic Geometry ? Your reply is highly appreciated.
1
vote
4answers
56 views

Find the value of $(a,b)$

The point $(4,1)$ is the midpoint of $(a,b)$ and $(-1,5)$. Find the values of $a$ and $b$ considering this statement. I know the midpoint formula is: $$ ...
0
votes
0answers
21 views

How to find the equation of a line which intersects these lines at 90 degrees?

How to find the equation of a line which intersects these lines at 90 degrees? $p\equiv \dfrac{x}{2}=\dfrac{y+1}{0}=\dfrac{z-2}{1}$ $q\equiv \dfrac{x-1}{1}=\dfrac{y-2}{1}=\dfrac{z+5}{0}$ Since the ...
0
votes
3answers
51 views

Find the value of $a$ if the distance between $(3,-2)$ and $(4,a)$ is $\sqrt{7}$

Find the value of $a$ if the distance between $(3,-2)$ and $(4,a)$ is $\sqrt{7}$. Do I use the distance formula with the variable or not?
0
votes
1answer
20 views

How to find how many rectangular prisms ( including cubes) are in a n by n by n cube?

I somehow got the answer to be [(n+1)!/2!(n+1-2)!]^2 *n Each part of the equation represents the height, length, and width of the possible rectangular prism in the big cube. You can multiply the ...
28
votes
3answers
1k views

Ambiguous Curve: can you follow the bicycle?

Let $\alpha:[0,1]\to \mathbb R^2$ be a smooth closed curve parameterized by the arc length. We will think of $\alpha$ like a back track of the wheel of a bicycle. If we suppose that the distance ...
0
votes
1answer
32 views

Volume of the pyramid…

I have such a problem from geometry: Five edges of a regular triangular pyramid have the length of $6$ $dm$, but the sixth- $4$ $dm$. Determine the volume of the pyramid. For me the problem is quiet ...
0
votes
1answer
18 views

Order of $A/2A$ for $A$ an Abelian variety

Let $A$ be an Abelian variety over $\mathbb R$ of dimension $g$. Then the size of $A(\mathbb R)/2A(\mathbb R)$ is $(\# A(\mathbb R)[2])/2^g$. I'm wondering how one might go about proving such a ...
0
votes
0answers
12 views

Question on Cobweb Diagram

Let f be a real map and assume that the points a and b form a limit cycle of order two of f. Derive a simple formula for the derivative of the second iterate of f at a.
1
vote
1answer
27 views

Intersection of an $n-$sphere and a plane (when non-empty and not a point)

Let the n-sphere of radius $r$ centered at $(0,0,...,0,y)\in\mathbb{R}^{n+1}$ be defined by $$ \mathcal{S} \iff {x_1}^2 + {x_2}^2 + ... + {x_n}^2 + (x_{n+1}-y)^2 = r^2 $$ and consider the function $d$ ...
0
votes
1answer
14 views

construction of a line.

Two non parallel lines l & m are given. For given two angles A & B we have to construct a line n such that it makes angles A & B with lines l & m respectively. Line n intersects l ...
0
votes
0answers
4 views

Finding angles in Barycentric system

How to find the angles of a triangle given the barycentric coordinates of its corners? Does it work if i take the first two components of every coordinate, and find the angles in the triangle (on the ...
0
votes
1answer
24 views

Bézier curve limits

Can be any curve of any shape (without sharp edges) described by Bézier curve with unlimited (but finite) number of control points? The answer to the question above would probably be no, because I ...
1
vote
1answer
46 views

How to find how many cubes are in a n by n by n cube?

I tried finding the answer using combinatoric by determining how many different length and width ans height are there for a cube, given the size of the bigger cube. But the formula I got turns out not ...
0
votes
2answers
34 views

A circle on the plane.

I have this problem: Let $C$ be a circle in the $xy$-plane with center on the $y$-axis and passing through $A=(0,a)$ and $B=(0,b)$ with $0<a<b$. Let $P$ be any other point on the circle, let ...
1
vote
3answers
86 views

What is the different between these two triangles? [duplicate]

What is the different between rigorous proof and proof based on intuition on this problem? It seems to me that these triangle are equivalent in area.
38
votes
6answers
4k views

How come $32.5 = 31.5$?

Below is a visual proof (!) that $32.5 = 31.5$. How could that be?
6
votes
1answer
67 views

isometries of the sphere

There is a theorem by Pogorelov that if a $C^2$ surface $M$ in $\mathbb{R}^3$ is isometric to the unit 2-sphere, then $M$ is itself (a rigid motion of) the sphere. What is known about isometric ...
1
vote
2answers
493 views

How to find the coordinates of the intersection of median

Given the triangle $ABC$ with its vertices $A(0,1)$, $B(-2,1)$, $C(8,-8)$. Determine the intersection point of the median $AM$ and the line $l$, if $l\parallel AB$ and $C$ is element of $l$.
1
vote
1answer
20 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.
1
vote
0answers
52 views

What if segments are not infinitely divisible?

I almost got myself mixed up I a philosophical discussion again. Somebody was talking about the Planck time and length which are, according to him, the minimal possible time and distance, and how ...
6
votes
0answers
112 views
+50

Set geometry and inclusion

I would like to prove that the set of the symmetric positive semi-definite matrices which is defined as $\Delta_2= \{S\in\mathbb{S}_{m,m} \quad \text{s.t.}\quad ...
0
votes
1answer
12 views

Find a vector in cartesian coordinates given its relative location to another vector in spherical coordinates

Here is my problem: -I have an arbitrary normalized vector N in cartesian coordinates -I am trying to find normalized vector M, also in cartesian coordinates -I am given the azimuth and polar ...
0
votes
2answers
30 views

Explanation for the uniformity of the distance between a Gaussian variable to its nearest integer?

earlier I asked the question Expected distance for a gaussian variable to its nearest integer. and got a good answer. The expected distance is highly close to $1/4$, which is very similar to the ...
1
vote
2answers
18 views

What is the result of these scalar products?

We know that : ABCD is a square. BGFE is a square. AEB and BCG are equilateral triangles. AB = 1. Here is the figure : I have already calculated the scalar products of BC.BE, DA.BE, EA.BE and ...
19
votes
7answers
3k views

How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$?

I would like to find the apothem of a regular pentagon. It follows from $$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$ But how can this be proved (geometrically or trigonometrically)?
2
votes
1answer
75 views

Curvature and the Arrow Pratt Absolute Risk Coefficient

So I'm in my first year of grad school, and I'm taking a decision analysis course. One of the topics we're covering is risk aversion, and with that comes discussion of the Arrow Pratt Absolute Risk ...
1
vote
2answers
51 views

How do I prove that this triangle is equilateral?

We know that : ABCD is a square. BGFE is a square. AEB is an equilateral triangle. AB = 1. Here is the figure : How can I prove that BCG is equilateral ?
2
votes
1answer
26 views

Collinear points in 3dimension

Given three $3D$ points: $A,B$ and $C$, what is the procedure to check if they are collinear? In general, given $n$ points in $m$-dimension, how should one find out, if these $n$-points defines a ...
0
votes
1answer
20 views

How do I find the dot product of these vectors?

We know that : ABCD is a square. BGFE is a square. AEB is an equilateral triangle. AB = 1. Here is the figure : How can I find the scalar products of : • BC.BE • DA.BE • EA.EB
1
vote
2answers
44 views

Given a 2D integer grid, how to choose three points (x,y), (2x,2y) and (3x,3y) such that their distance to the integer grid is maximal?

Given an integer grid $\mathbb{Z}^2=\{...,(0,0), (1,0), (2,0),...,(1,1),(1,2),...\}$, choose $x,y \in \mathbb{R}$ such that the points $(x,y)$, $(2x,2y)$, $(3x, 3y)$ have maximal (Euclidean) distance ...
1
vote
0answers
25 views

Why isn't my Pappus chain lining up?

TL;DR Why isn't my Pappus chain lining up? Visualisation: http://jan.jarfalk.se/pappus-chain/ Proof: http://jan.jarfalk.se/pappus-chain/debug.html Code: https://github.com/janjarfalk/pappus-chain ...
1
vote
1answer
55 views

Two touching circles inscribed in an angle

There are two touching circles inscribed in a $60^\circ$ angle. The distance between the vertex of angle and the center of smaller circle is $5j$. What is the ratio of the surfaces of two circles?
0
votes
2answers
22 views

Some help needed with a geometry question

What is a formula for all integers n for which a regular polygon with n sides can be constructed using a ruler and compass construction?
10
votes
5answers
175 views

Intuitive/direct proof that a rectangle partitioned into rectangles each with at least one integer side must itself have an integer side

A challenge problem asked to show that if rectangle $R$ with axis-parallel sides is partitioned into finitely many subrectangles $R_1,R_2,\ldots,R_n$ (also with axis-parallel sides), such that each ...
0
votes
0answers
28 views

Rectangle with side length of integer value. [duplicate]

There is a rectangle $D=[a,b]\times [c,d]$. This rectangle has finite partition with smaller rectangles with parallel sides $\{D_i\}_{i=1}^n$ $(n\in\mathbb{N})$. Let's put these rectangles as ...
2
votes
1answer
37 views

Intersection coordinates of a parabola with a circle

One point$(x_{0},y_{0})$ on parabola is given, the distance between the given point with two other points is $r^{2}$, $r^{2}$ is given. So this problem can also be described as: The circle equation ...
14
votes
1answer
630 views

Maximum total distance between points on a sphere

What is the configuration (set of locations) of $n$ points on the surface of a sphere such that the sum of distances is maximum for $n=1,2,3,...$? The sum of distances is measured by summing the ...
0
votes
2answers
25 views

Find the value of EF and AC.

In the figure given below, BA, FE and CD are parallel lines. Given that AB = 15 cm, EG = 5 cm, GC = 10 cm and DC = 18 cm. Calculate EF and AC. I think the answer is EF= 8.66 and AC = 25.66 but I ...
1
vote
1answer
46 views

Classical geometry statement in modern terminology

Given two line segments $\overline{AB}$ and $\overline{CD}$, it's always possible to find a third line segment whose length divides evenly into the first two. In modern terminology, if we assign $x = ...
0
votes
2answers
33 views

Unusual 3D Packing Problem

I made up this interesting problem playing with wire sculptures: If I have a $10 \times 10 \times 10$ clear box and inside I can put wireframe unit cubes, what's the maximum number of unit edges (or ...
12
votes
1answer
212 views

cutting a cake without destroying the toppings

There is a square cake. It contains N toppings - N disjoint axis-aligned rectangles. The toppings may have different widths and heights, and they do not necessarily cover the entire cake. I want to ...
0
votes
1answer
23 views

Is there a smooth map from the square to the deltoid?

Is there a $C^\infty$ map between a unit square in $\mathbb R^2$ and a deltoid like this one The deltoid is obtained by varying the angles $\theta_1$, $\theta_2$ in the equations \begin{align} x_2 ...