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

learn more… | top users | synonyms (1)

2
votes
1answer
49 views

Why is it against common sense?

The question is this. A man can walk at speeds of 6kmph uphill, 7.5kmph along level surface and 10kmph downhill. He travels from A to B in 3 hours and from B to A in 1 hour. What is distance AB? I ...
1
vote
1answer
30 views

Euler's Formula: $V-E+F=2$ by using spheric triangles

I just have a question to a proof found here: https://nrich.maths.org/1384 At one point it says: As eight copies of $\triangle$ will fill the sphere without overlapping. Why this? Why can I "...
1
vote
3answers
67 views

Area of the triangle formed by circumcenter, incenter and orthocenter

Lets say we have $\triangle$$ABC$ having $O,I,H$ as its circumcenter, incenter and orthocenter. How can I go on finding the area of the $\triangle$$HOI$. I thought of doing the question using the ...
0
votes
1answer
33 views

Prove the inequality of area of convex polygon X is A is less than or equal to $\frac{\pi d^2}{ 4}$ [closed]

I want to prove that convex polygon X in the plane has diameter d, its area is less than or equal to $\frac{\pi d^2}{ 4}$.
2
votes
4answers
63 views

Show that the unit sphere is connected [duplicate]

I need to show that $\{(x,y,z)\in\mathbb{R}^{3}:x^2+y^2+z^2 = 1\}$ is connected. Intuitively I understand that it is path connected and, therefore, connected. However, I don't understand how I would ...
4
votes
1answer
65 views

Splitting Line Segments and Finding Expected Value

Consider a line segment which has a length of $2n-3$. It is split into $n$ segments at random. It is guaranteed that $n\ge 3$ and $n\in \mathbb{Z}$. These smaller lines are then used as the sides of a ...
2
votes
0answers
23 views

I have a convex hull (generated from a library) in 3D. I only have the vertices. How do I compute the volume of the hull.

I have a library (quickhull in C++) that I am using to create a hull from a set of points. I am able to see the vertices of the hull but not the facets. I would like to compute the volume of the hull. ...
6
votes
1answer
52 views

Circle packing – How to get the minimum length?

In an a past admission paper from a local university, I came across a problem I couldn't solve. Given $n$ circles with their respective radii $r_1, r_2, \dotsc , r_n,$ we are to find the minimum ...
0
votes
0answers
15 views

What is the minimum number of sets of Euler angles to cover $SO(3)$?

This is a question I was asked to answer from a drone-robotics check assignment. What is the minimum number of sets of Euler angles to cover $SO(3)$? $$SO(3)=\{R\in\mathbb{R}^{3\times 3}|R^TR=RR^T=I\...
3
votes
0answers
48 views

What curves will satisfy this very intersting property?

Let $c_1,c_2\subset\mathbb R^2$ be differentiable curves. Given that for any rigid transformation $E$ (i.e. combination of reflections, translations, rotations), if $c_1,E(c_2)$ intersect ...
3
votes
0answers
76 views

cutting an equilateral triangle to $n$ equal pieces

We have an equilateral triangle and we want to cut it into $n$ equal pieces. For which $n$ is it possible? My Attempt: I found these possible numbers $2,3,4,6$ and also I proved every $n$ of the ...
0
votes
2answers
58 views

How many sphere on the boundary of a big sphere?

I don't know exactly how to ask this in a comprehensible way. I am trying to find a solution to my problem which is to find how many sphere of radius r are lying on the boundary (which means that in ...
0
votes
0answers
29 views

Properties of polyhedron solving constrained max problem

This is a question for people who don't have trouble to think in more than two dimensions. Don't hesitate to ask clarifying questions! Let us suppose we have $n$ random variables $X_i$ that are iid ...
0
votes
0answers
22 views

Smallest enclosing cylinder

I have a set of 3D points that approximately lie on a cylinder. This cylinder is straight and can be oriented in any direction. I would like to compute the minimal enclosing cylinder for the set; taht ...
0
votes
0answers
38 views

Given set of points in 3D, find group of points closest to each other

Given a set of any 8 points in 3D space. I want to find a subset of points that are closest to each other. Application: Assume in a 3D space, I have any 8 colors(represented in RGB). I know how to ...
0
votes
0answers
23 views

meromorphic function on torus

Consider the familly of meromorphic function on the square torus (endowed with the corresponding complex structure) with $p$ simple poles and $p$ simple zeros and $L^1$-norm equal to $1$ : $\mathcal ...
4
votes
1answer
64 views

In $\triangle ABC$, if $\tan A$, $\tan B$, $\tan C$ are in harmonic progression, then what is the minimum value of $\cot \frac{B}{2}$?

In a $\triangle ABC$, if $\tan A$, $\tan B$, $\tan C$ are in harmonic progression, then what is the minimum value of $\cot(B/2)$? $\bf{My\; Try::}$ Here $A+B+C=\pi\;,$ Then $\tan A+\tan B+\tan C=\...
1
vote
0answers
36 views

split a rectangle with triangles into polygons as uniformly as possible

Given a rectangle $A$ and $n$ triangles $\{B_1,B_2,...,B_n\}$, I put the triangles inside $A$, at least one vertex of each triangle is not outside $A$ (inside $A$ or on the edge of $A$). So that A is ...
10
votes
3answers
447 views

Why are there two versions of a polar equation for a circle from geometric form

In class today we learned that a rectangular/geometric equation for a circle such as $x^2+(y-5)^2 = 9$ can be converted into a polar equation by reducing it to the quadratic equation $r^2-10r\sin \...
4
votes
1answer
83 views

Mathematical description on the interface of two adjacent bodies.

I am recently studying about a problem related to shortest path. I can briefly describe my idea but I am not sure if there is some "professional" mathematical description about it. In the following ...
0
votes
0answers
17 views

Integrating across a transformed ellipsoid

I am attempting to generalize an algorithm which processes point masses. Given two sets of points (lets call them $\mathbf q$ and $\mathbf p$), the inputs to this algorithm are a 3x3 matrix and a ...
2
votes
1answer
65 views

Circle Puzzle Geometry

Two friends are playing a game. One friend stands in the middle of a circle radius 100m. His objective is to leave the circle. He may take one step at a time, distance 1m, in any direction. However, ...
0
votes
0answers
34 views

Discovering length of line

I'm attempting to work out length of BD from below diagram : The length of BD is -2 +- some value. But since I do not know the y co-ordinate of B can the length of BD be determined from ...
2
votes
0answers
31 views

Derive equation for shear modulus $G=E/(1+2v)$

shear modulus, G young's modulus, E and Poisson's ratio, $v$: $G=E/(1+2v)$ I have always wondered how this relation is derived, but have never found a derivation that I could follow online. I ...
2
votes
5answers
77 views

How to find the coordinates where the altitude of a triangle intersects the base in 3 dimensions?

Assuming I know three completely random coordinates in 3d space that correspond with vertices of a triangle, how can I then find the point at which the altitude intersects the base? I know how to ...
0
votes
1answer
33 views

Triangle Inequality Problem…

In triangle $ABC$, the medians $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ concur at the centroid $G$. (a) Prove that $AD < (AB + AC)/2$. (b) Let $P=AB+AC+BC$ be the perimeter of $\...
0
votes
1answer
27 views

Calculating relative position of points when zoomed in and enlarged by a rectangle

There is a rectangle, defined by the top left point $R1(0, 0)$ and the bottom right point $R2(200, 200)$ (the $y$ $axis$ is inverted). In that rectangle, there are some points $P1(100, 100)$, $P2(50, ...
0
votes
1answer
30 views

Length and diameter of a spiral of nanotube

I was reading from this popular article (in french). Talking about nanotubes of carbon the author says (my translation): The diameter of the nanotube is of the order of a millionth of a millimiter....
0
votes
1answer
38 views

A Question about Triangle Inequality [duplicate]

Two sides of an acute triangle are 8 and 15. How many possible lengths are there for the third side, if it is a positive integer? I'm not sure if the word "acute" affects the problem. If not, is the ...
2
votes
2answers
36 views

Find the area of $S=\{(x,y)|\rm{\exists ~}\theta,\beta,x=\sin^2{\theta}+\sin{\beta},y=\cos^2{\theta}+\cos{\beta}\}$

Let $S$ be the domain defined by $$S=\{(x,y)|\rm{\exists ~}\theta,\beta,x=\sin^2{\theta}+\sin{\beta},y=\cos^2{\theta}+\cos{\beta}\}$$ find the area of $S$ This is middle school problem,so I think it ...
0
votes
2answers
36 views

Ellipse equation from center and point on ellipse [closed]

Is there a way to get the equation of an ellipse iw we know the center and one point on the ellipse ?
0
votes
3answers
50 views

How many combinations are there for the interior angles of a triangle?

Suppose the interior angles of a triangle are all Natural numbers. How many combinations of angles are there without repeating similar triangles? So for instance, {1,1,178}, {1,2,177},...But without ...
0
votes
0answers
25 views

How to visualize this geometric setup?

I am working on a derivation (physics) and the first part requires one to imagine a geometric setup. Since I'm not a native speaker of the language, I am having trouble in doing so. The relevant ...
-1
votes
1answer
35 views

Why should sum of coefficients of collinear position vectors be 0?

Suppose $a,b,c$ are collinear position vectors then we know that $xa+yb+zc=0$ where $x,y,z$ are scalars. But in my book something additional is also mentioned that $x+y+z=0$,why must it be so ? I ...
0
votes
1answer
46 views

How to find the area shared by 4 quadrants inside a square?

I was to find the blue area in this question : As described about how it's a square with 4 quadrants of same radius intertwined with each other, now to find the blue part area I thought about ...
1
vote
0answers
46 views

Cutting a pie into 2 unequal peices with a single cut, minimising its length. [closed]

Suppose we have a circle with an area of 1, which we are to cut into two pieces, of area (x) and (1-x) respectively. Let x<0.5. How should we make the cut, to minimise its length? What is the ...
2
votes
1answer
29 views

Does shearing a sphere generate an ellipsoid?

In my preferred 3D modeling software I see that however I shear a sphere, I seem to be able to make a nearly identical shape using some combination of non-uniform scaling followed by rotation. Are ...
0
votes
0answers
53 views

What is the mathematical vernacular for $\Delta x$

I want to use the proper terminology when I discuss length scales associated with $\Delta x$, where $\Delta$ is the difference operator. In other words $\Delta x = |x_1-x_2|$. It is a measure of the ...
2
votes
1answer
52 views

Geometry Triangle Question 3

In the triangle shown, $n$ is a positive integer, and $\angle A > \angle B > \angle C$. How many possible values of $n$ are there? Two sides of an acute triangle are 8 and 15. How many ...
0
votes
1answer
23 views

Change of coordinates in $R^{n}$ where the diagonal goes to $x=0$

Say I have a system of coordinates $\{y_{1},y_{2},...,y_{n}\}$. I'd like to get a new system of coordinates $\{x_{1},x_{2},...,x_{n}\}$ where the diagonal $y_{1}=y_{2}=...=y_{n}$ is $x_{1}$, and I'd ...
0
votes
2answers
51 views

Geometry Questions: Triangles [closed]

Can you guys please help me with these problems? In the triangle shown, $n$ is a positive integer, and $\angle A > \angle B > \angle C$. How many possible values of $n$ are there? The ...
1
vote
0answers
12 views

Looking for a particular parameterization of $S^n$

Say we have take vectors $(x_1,..,x_d) \in S^{d-1}$ and we look at vectors $(a_1,..,a_d) \in (\mathbb{Z^+ \cup \{0\}})^d$ such that $\sum_{i=1}^da_i =k$ for some positive integer $k$. Is there any ...
0
votes
0answers
35 views

Convex, closed plane curve is a Jordan curve

The claim I'd like to prove or disprove is in the title. Here, convexity means every tangent line at any point on the curve has the whole curve in one half of it. The curve being closed means that it'...
2
votes
0answers
44 views

Find the ratio of slope

Note : Elevation $46000$ and all dimention in $mm$ (milimeter) The pipe will be installed on a surface of module structure, that module structure has different surface. I want to know " ratio ...
1
vote
1answer
30 views

Question re: derivation of formula for volume of cone

The diagram above comes from a derivation of the formula for the volume of a cone; it's one of the preliminary steps, and sadly, I'm stuck on it. What we're doing here is inscribing an "infinite" ...
1
vote
1answer
33 views

Linear application

Let $f:\mathbb{R}^3\to\mathbb{R}^3$ be a linear application and let $\{e_1,e_2,e_3\}$ the canonical basis of $\mathbb{R}^3$. We know that $\operatorname{Im} f=\langle(1,1,3), (0,1,1)\rangle$ and that ...
3
votes
1answer
35 views

Area of circle segment intercepted by a line

The problem I want to solve is to calculate the filled area in the following diagram - so basically the area between the two circular arcs but with the red line cutting off one side. I think I have a ...
0
votes
1answer
60 views

Finding centers of ellipses with two points and their respective tangents

I hope you can help me with the following, probably rather complex dilemma: I generally want to find an ellipse given two points and their respective tangents in 2-D space (X and Y coordinates). Now ...
2
votes
3answers
60 views

Sphere and tetrahedron

If we have sphere inscribed in a tetrahedron, and if the distances from the center of the sphere to the edges of the tetrahedron are equal, is it true that this tetrahedron is always regular? I'm ...
4
votes
0answers
80 views

How did Archimedes calculate rational bounds of pi from a 96-gon?

Archimedes famously determined that $223/71 < \pi < 22/7$ using the 96-gons circumscribed by and circumscribing a circle of unit diameter. But I haven't found a reference that explains the final ...