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

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-3
votes
0answers
27 views

Mathematics doubt. [on hold]

What is mathematics. I need a proper definition mathematics.
0
votes
0answers
9 views

Specific polygons in \R^{3}

Find all (convex) polyhedra P in $\mathbb{R}^{3}$ with following property: for every two vertices $v, u \in P$: $[u,v] \in \partial P$ . Here $[u, v] = \{w \; \vert \; w = tv + (1-t)u, \; t \in ...
2
votes
0answers
7 views

Projective line intersecting 3 projective subspaces

I am trying to solve the following problem: Let $\mathbb{P}(U),$ $\mathbb{P}(V)$ and $\mathbb{P}(W)$ be projective subspaces of dimension $k,$ $l$ and $m$ respectively in $\mathbb{P}_K^n$. Suppose ...
0
votes
2answers
22 views

AMC: Triangle area problem

In rectangle ABCD below, points F and G lie on segment AB such that AF = FG = GB and E is the midpoint of segment DC. Also, segment AC intersects segment EF at H and segment EG at J. The area of ...
0
votes
2answers
18 views

How find the length of the third of the chord for of a circle of radius r?

Three chord of a circle of radius $r$ are the sides of the triangle inscribed in the circle. Two of these chords have a length $\frac{r}{2}$ and $r\sqrt{3}$. How find the length of the third of the ...
1
vote
0answers
5 views

Identify the combination formed by first applying the glide reflection $γ_{A,B}$ and then applying $γ_{C,D}$

Question: Identify the combination formed by first applying the glide reflection $γ_{A,B}$ and then applying $γ_{C,D}$ where $A = (0, 0), B = (2, 0), C = (1, −2), D = (1, 0)$. Before, I jump in ...
1
vote
0answers
30 views

Volume of a section of the unit sphere

Let $v$ be a vector on the unit sphere in $\mathbb{R}^n$ and let $S(\epsilon)$ be the set of vectors $s$ on the same sphere such that $$ |s \cdot v| \leq \epsilon.$$ What is the surface area of ...
10
votes
5answers
551 views

Can someone explain 4th dimensional objects?

I'm not sure if I should ask this in mathematics or in physics. From what I can tell, there are only 3 dimensions: X, Y, and Z. However, I have seen a lot of things about fourth and even fifth ...
3
votes
1answer
22 views

Tangent lines of a smooth curve $C \subseteq \mathbb{P}^2$

Let $C$ be a smooth curve, given as the zero locus of a homogeneous polynomial $f \in \mathbb{K}[x_0,x_1,x_n]$. Consider the morphism $\varphi_C:C\rightarrow\mathbb{P}^2$ such that ...
0
votes
0answers
9 views

Finding closest rectangle to another using concept of closest edge [on hold]

I know coordinates and size of rectangles. My goal is to find 'the closest' rectangle to one special rectangle using the concept of closest edge and also to find distance between them??
0
votes
1answer
9 views

A question on the requirement of a quadrilateral being an adventitious quadrangle

There is a special type of problem called Langley’s Adventitious Angles. See http://en.wikipedia.org/wiki/Langley%E2%80%99s_Adventitious_Angles The problem was solved and has the following ...
1
vote
3answers
29 views

Calculating last two angles of a quadrilateral when two angles and (relative) side lengths are known.

I've recently run into a problem that I can't seem to get. Given a quadrilateral, $ABCD$, with angle $D$ equaling 54 deg, and with the length of $BC$ equaling $CD$ while being double the length of ...
-3
votes
0answers
19 views

Find out the vertices of a regular polygon, given one side and number of sides [on hold]

We are given: 1)one side of the regular polygon , $ (x_1,y_1) $ and $ (x_2,y_2) $ 2)number of sides of the polygon 3)construct the polygon in counterclockwise direction And are asked to find all ...
0
votes
0answers
28 views

How to calculate the volume between points? [on hold]

Say we have a set of finite points: $$ A=\{ (x_1,y_1,z_1), (x_2,y_2,z_2) , \dotsc , (x_n,y_n,z_n) \} $$ Assuming these are surface points of a 3D shape. I want to calculate the volume confined ...
-2
votes
2answers
32 views

In a $\Delta ABC$, G is the centroid. prove that [on hold]

In a $\Delta ABC$, G is the centroid.Prove that $AB^2+BC^2+CA^2=3(GA^2+GB^2+GC^2)$
4
votes
2answers
35 views

How to calculate line-line distance when cross product of directions is 0?

I have the lines $$\frac{x-1}{2} = 1-y = \frac{z-2}{3} \tag{1}$$ and $$\frac{x+1}{4} = \frac{4-y}{2} = \frac{z+1}{6} \tag{2}$$ I want to compute the distance between them. I started by putting ...
1
vote
1answer
29 views

Why is this a convex polygon?

Let $\text{E}(2)$ be the group of isometries of the plane $\mathbb R^2$. Then $\text{E}(2)=\text{O}(2)\times\mathbb R^2$ as a topological space and is the semi-direct product as groups. Let $G$ be a ...
-1
votes
0answers
33 views

definition of largest circle containing in a convex body

For a convex body B(compact convex set with non-empty interior) what does each of these following values mean? 1- $$\ \sup_{x\in B}\inf_{y\in cB} d(x,y) $$ and 2- $$\ \sup_{x\in B}\inf_{y\in ...
2
votes
2answers
17 views

Calculating the currvature of a tractrix

I'm trying to find the curvature of a tractrix expressed in the form $r(t)=(\sin{t},\cos{t}+ln(\tan{(\frac{t}{2}))} $. From what I've found on the Internet it appears that people arrive at the ...
0
votes
1answer
20 views

will the ladder reach the height of a 3.5m window.

A ladder of length 4m leans against the wall of a house. The foot of the ladder is 2m from the wall. Will the ladder reach a window 3.5m high?
1
vote
0answers
27 views

Point in four dimensions

To describe a point in $3$D: Three parameters $ r,\phi, \theta $ are used in spherical coordinate system. Taking them pairwise,two as $(r, \theta)$ in azimuth plane and two $(r,\phi )$ in meridian ...
7
votes
2answers
378 views

Is a circle classified as an ellipse?

I read that an ellipse had $2$ focal points. So, I thought if a circle had $2$ points that were simply infinitesimally close together wouldn't it be classified as an ellipse? Help would be ...
3
votes
0answers
53 views

How prove that : $ \sin x+\sin y+\sin z \le \frac{3}{2} $ for triangle $ABC$

Let $ G $ be the centroid of $ \triangle ABC $ , such that $ \angle{GAB}=x,\angle{GBC}=y,\angle{GCA}=z $, How do I prove that : $$ \sin x +\sin y +\sin z\le \frac{3}{2} $$
0
votes
1answer
32 views

Draw the line segment joining the centers of two circles. Where does it meet the circles?

I'm trying to construct a line segment between two circles. Given each radius and $x$, $y$ center of each circle, how can I find the endpoints for the blue line segment?
0
votes
0answers
20 views

What are some techniques of specifying a molecules structure using the least amount of information?

For instance say I have a water molecule I can describe it's structure by two bond lengths and a bond angle. Are there any neat math tricks or representations of objects that I could use to describe ...
1
vote
1answer
35 views

Curvature of plane curves

What is the neatest way to derive the formula for the curvature $\kappa =\frac{||y'x''-y''x'||}{(x'^2+y'^2)^{\frac{3}{2}}} $?
0
votes
0answers
16 views

Signed unit normal

I'm trying to study for one of my exams and the past papers have no solutions. I had to define the signed unit normal and the signed curvature. The signed curvature, $\kappa_s$ being such that ...
1
vote
2answers
61 views

Probability of triangle to be acute?

Suppose that someone randomly picks $3$ points $A, B$ and $C$ on a fixed circle. What is the probability of triangle $ABC$ to be acute?
0
votes
0answers
15 views

How do I find all lines of $\mathbb{P^{3}}$ meeting $L, M$ and $N$?

Let $L, M$ and $N$ be $3$ lines of $\mathbb{P^{3}_{R}}$ such that $$L \cap N = \emptyset, \quad M \cap N = \emptyset \quad \text{and} \quad L \cap M = \{ G \} \ \ \text{(one point)}.$$ I'm asked to ...
3
votes
1answer
18 views

Differential length of a logarithmic spiral

I am working on a problem that asks to find the magnetic field at the origin of a logarithmic spiral $r = e^\theta$ from $\theta = 0$ to $\theta = 2\pi$, where the angle $\theta$ is measured ...
0
votes
2answers
50 views

finding angle value inside this triangle

I need a method to calculate the angle X in the image below, I know its value (30 degree) but how ?!! thank you.
1
vote
0answers
33 views

Assume that for any pair of vertices $P_i$ and $P_j$ , there exists a vertex $P_k$ of the polygon such that $∠P_i P_k P_j = \pi/3.$

Let $P_1 P_2 \dots P_n$ be a convex polygon in the plane. Assume that for any pair of vertices $P_i$ and $P_j$ , there exists a vertex $P_k$ of the polygon such that $∠P_i P_k P_j = \pi/3.$ Show ...
2
votes
3answers
33 views

Why do lines in the poincare model meet the infinite edge at right angles?

I know the lines are generated by projecting geodesics on a hyperboloid to a plane and the boundary of the disk comes from the asymptotic cone around the hyperboloid, but I just don't see why the ...
0
votes
1answer
16 views

What are the images of a point in a multi-layer mirror systems?

We all know that there are infinite images of a point which is located between two parallel mirrors. Also, the locations of the images can be easily obtained. Generally, how to locate the images of a ...
0
votes
1answer
21 views

Let the diameter of a subset S of the plane be defined as the maximum of the distance between arbitrary of points of S

Let the diameter of a subset S of the plane be defined as the maximum of the distance between arbitrary of points of S. Let$$S=\{(x,y):(y-x)\le0,(x+y)\ge0,x^2+y^2\le2\}$$ Then the diameter of S is ...
0
votes
1answer
23 views

Given sides and a bisection, find angles in a triangle

Consider a triangle $ABC$ where the angle $A$ is $60^{\circ}$. Draw its bisection intersecting $BC$ at $D$. Let $AB = x$, $BD = y$ and $AC=x+y$, $\angle ABC = \alpha$ and $\angle ACB = \beta$. Find ...
1
vote
0answers
11 views

Boundedness condition of Minkowski's Theorem

Statement: "Let L be a lattice in $R^n$ and $S\subset R^n$ be a convex, bounded set symmetric about the origin. If $Volume(S) > 2^ndet(L)$, then S contains a nonzero lattice vector. Moreover, if ...
2
votes
1answer
32 views

“Reverse engineering” of a geometric illustration

The following enigmatic illustration can be found here, unfortunately without any accompanied comment or short description: Can you deduce its meaning? What was the way it was constructed?
3
votes
2answers
43 views

Complicated Planar Geometry [duplicate]

A regular polygon $\mathcal{P}$ is inscribed in a circle $\Gamma$. Let $A$, $B$, and $C$ be three consecutive vertices of the polygon $\mathcal{P}$, and let $M$ be a point on the arc $AC$ of $\Gamma$ ...
-1
votes
0answers
15 views

Find out rotation and angle given a square

Consider I have a camera and a square, the projection of the square on the camera would give me a "distorted" square. Can I possibly find out the angle of the camera relative to the square and the ...
0
votes
1answer
14 views

what is the difference between an elliptical and circular paraboloid? (3D)

My textbook uses the terms interchangably, and they look the same in graphs, so I was wondering if there a difference between the two? Thanks!
0
votes
2answers
34 views

How to reverse this mathematical formula

I am using the following function to compute a "Y" value from a Latitude ...
1
vote
1answer
30 views

Relative distance to rotated object

An object (with center $O_{2}$) has been rotated by an angle $\alpha$. There are two images of the object taken by a camera (centered at $O_{1}$), and two points $x_{1}$ and $x_{2}$ that are actually ...
-1
votes
0answers
26 views

How can one show that the vector $AK=\frac{1}{3}AI$?

$ABC$ is a triangle then $I$ is the medium of $[CB]$ and $J$ the medium of $[AI]$ and $K$ the intersection of $(BJ)$ and $(AI)$. Then how can one show that $AK=\frac{1}{3}AC$ Do we have to add ...
1
vote
0answers
14 views

Cover the entire sphere with a moving cone

Assume you have a cone with half-angle theta in a 3D cartesian space, with the vertex of the cone in the origin, and that you want to rotate the cone along a curve so that it covers the entire sphere. ...
0
votes
0answers
13 views

Maximize the intersection over union of oriented rectangles

I have an oriented rectangle in the form region=(x1,y1, ..., x4, y4) I want to know which is the axis-aligned rectangle with the same center that maximize the intersection over union of the areas of ...
0
votes
0answers
23 views

How to calculate the solid angle subtended by the torus?

The diagram above shows a torus, with circular cross section, having inner radius $r=19 cm\space$ & outer radius $R=33 cm\space$. How to evaluate the solid angle subtended by the torus at a ...
0
votes
1answer
33 views

Volume of paraboloid

A paraboloid is formed by revolving a parabola, $y=kx^2$, about its axis of symmetry. The paraboloid is bounded by a plane cutting the axis of symmetry perpendicularly at the point (0,20). The ...
0
votes
2answers
35 views

Finding the max area for a rectangle inside of a circle

I know how to find the max area, inside of a normal circle. This circle however is the inside of an aircraft and the floor must be there. How can this be done? Here's a sketch of the inside of the ...
0
votes
0answers
20 views

Conditions for point lying inside triangle formed by three complex numbers.

The question states $z_1,z_2,z_3$ are three non-collinear complex numbers such that $$z=\frac{lz_1+mz_2+nz_3}{l+m+n}$$ lies inside the triangle formed by $z_1,z_2,z_3$. If $l,m,n$ are the ...