Tagged Questions

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

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Simplest way to see that the affine isometries of a regulara $n$-gon are linear?

What is the simplest way to see that the set of affine isometries of the plane that fix a regular $n$-gon centered at the origin are in fact linear? One can see this by showing that the origin is the ...
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0answers
8 views

Is the standard embedding of the torus the tight embedding?

Definition of tight: A mapping of a surface into $\mathbb{R}^3$ is called tight if its image, equipped with the induced metric, has minimal total absolute curvature. (A definition of this kind is ...
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1answer
23 views

Volume of water in a cone

Let, slant height of a cone be 6cm, and radius be 3cm and the cone be uniform. Let a uniform & solid sphere of radius 1cm be put in the cone & fill the cone by water. What is the minimum ...
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1answer
29 views

Shortest path between two points that touches a line [on hold]

A man starts from from the point $P(-3,4)$ and will reach the point $Q(0,1)$ touching the line $2x+y=7$ at $R$. What are the coordinates of $R$ on the line if it is chosen so that he will travel in ...
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1answer
13 views

Given normal to plane and a point in it, find a unit vector in plane

I have a plane for which the unit normal vector and a point in the plane are known. I want to find a unit vector lying in the plane (any one). While there are dirty ways of doing this, I remember from ...
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2answers
15 views

Geometry homework help: Ratios

Alright so I have done part a, but I don't even understand where to start with part b. I keep ending up in circles where I basically just show that PQ/AB=PQ/AB in some complicated way. I would like ...
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1answer
19 views

A General GRE Quantitative Problem: triangle inside circle

Question: What I know about this question: I know nothing about the relationships among the sides of the triangle. If I were to draw a line down from $A$ to what looks like the center of $AO$, I ...
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1answer
21 views

Whitney's Embedding

The Whitney embedding theorem says that any smooth manifold of dimension $n$ may be embedded in $R^{2n}$. I am just beginning to study differential geometry for application to physics (general ...
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31 views

All polygons have a supporting triangle.

Show that every convex polygon $Q ⊆ \mathbb{R}^2$ has three vertices which form a triangle ∆, such that the line through either vertex of ∆ parallel to the opposite edge is supporting Q.
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1answer
58 views

Optimization of a Cylinder In a Sphere WITHOUT Using Calculus

I have a quick question. I'm curious as to how to do an optimization question WITHOUT using calculus. Question: Determine the dimensions of the cylinder of maximum volume that can be inscribed in a ...
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1answer
42 views

Finding the angle in a square

Hello, I don't know what to with this problem: I have to find angle ABS in given square (image). Where S is the last important unnamed point in the picture. Thanks a lot.
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1answer
80 views

Invent transformation mapping ellipsoid to unit sphere

Invent a transformation that maps the ellipsoid $ x^2+8y^2+6z^2+4xy-2xz+4yz=9$ onto the unit sphere. I don't even know where to begin with this question, any help would be appreciated.
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1answer
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A connected path between shapes

This is a follow-up to this question: A continuous path between shapes . Let $A$ and $B$ be two measureable, bounded, connected subsets of $\mathbb{R}^2$ such that $A\subseteq B$. Does there exist a ...
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2answers
37 views

Given the area and perimeter of a triangle, find its coordinates

How can we find the coordinates of a triangle, given its area and perimeter? (We can find any triangle that satisfies the given area and perimeter) I tried to find the lengths of the sides of the ...
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0answers
10 views

Angel function and continuity

I have the function $w:\mathbb{R}^2\backslash\{0\}\rightarrow\mathbb{R}$ given by $\cos(w)=\frac{x_1}{||x||_2}\text{ and }\sin(w)=\frac{x_2}{||x||_2}$ after some manipulation I got ...
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1answer
18 views

Find the sum of the lengths of line segments $BD$ and $CE$

sorry for the drawing. From a point $D$ on side $AB$, a line $DE$ is drawn through a point $E$ on side $AC$ such that angle $AED$ is equal to angle $ABC$. If the perimeter of the triangle $ADE$ is ...
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1answer
48 views

Prove the ratio between the following areas is 2 [on hold]

We have two non parallel lines. The distance between each points of each line is the following: $AB=CD=EF=HG=1$ and $BC=GF=2$. Prove that the area of $ADEH$ is twice the area of $BCFG$. Here is a ...
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0answers
9 views

parametric representations 3d object

I'm trying to model a 3 dimensional body that is sort of ellipsoidal and am looking for parametric representation of 3D objects similar to the quadratic surface representation of a sphere or ...
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2answers
30 views

How Do We Find Points On A Circle Equidistant from each other?

I'm a programmer and I saw this question on stackoverflow which exactly does my job: http://stackoverflow.com/questions/13608186/trying-to-plot-coordinates-around-the-edge-of-a-circle. In this, the ...
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Are compact complete geodesics closed?

Let $(M,g)$ be a compact Riemannian manifold. Is there an example of a geodesic $c:\mathbb{R}\to M$ s.t. $c(\mathbb{R})$ is compact, $c$ is NOT periodic (i.e. be NOT a closed geodesic) ?
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3answers
21 views

On inversive geometry

I am given the following problem set: Observe the circle $K$ with center $0$ and radius $r$ in the complex plane $ \mathbb{C} \simeq \mathbb{R}^2$. Show that the inversion on $K$ is given by the ...
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0answers
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Equivalence of definitions for a conic

I have to prove that these two definitions for the eccentricity of a conic $C$ are equivalent: Ratio between the distance of the points $x$ in $C$ to $f$ its foci and $l$ its directrix. Ratio ...
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0answers
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Veryfing the equation of the Limacon

Let $A$ be a fixed point in a circle of diameter $a$, such that $AC$ is a variable chord. Let $P$ and $Q$ be points in $\overleftrightarrow{AC}$ such that $|PC| = |QC| = b$. If A is the pole and its ...
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2answers
32 views

Ellipse with center in origin

The purpose is to fit data to a ellipse which center is the origin $(x_0=0,y_0=0)$. I found the general quadratic curve: $$ax^2+2bxy+cy^2+2dx+2fy+g=0$$ Reference: ...
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0answers
44 views

Roots of sides of triangle “An inequality”

$a, b, c $ are sides of triangle . Prove that : $$ ...
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1answer
23 views

Funny interconnection between a triangle and the ellipse inscribed

Le $p\in\Bbb R[X]$ be a 3rd degree polynomial. Suppose it has one real root and two complex conjugate roots: these three points forms a triangle in the complex plane. Consider the ellipse inscribed ...
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0answers
17 views

Use of Delaunay Triangulation and Voronoi Diagram to find alpha shape using Edelsbrunner's algorithm

I am learning how to find the shape of a set of points in 2-D. I understand that Alpha Shape method is a good way to find the shape of a set of points. Alpha Shape was originally introduced by H. ...
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3answers
47 views

5 points on a plane with rational distances

Can you find 5 points on a plane whose Euclidean distances between them are all rational numbers and no 3 points out of them are co-linear? If the answer is yes, can we find a construction for ...
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1answer
27 views

how many lines can be drawn from a point in space with n degrees of freedom?

A friend of mine asked a very interesting, simple looking but very hard to answer question, due to lack of good mathematics knowledge and limited vocabulary I can't find the answer on Google! How ...
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1answer
25 views

Find the length of each side of a square containing regular hexagons

I have to find the length of each side of a square such that all the regular hexagons of same length side and radius lying inside the square have centers either inside the square or on the boundaries ...
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0answers
19 views

Geometrical Explanation of Borsuk Theorem

Assume $K$, $L$ are $n$-pseudomanifold, and $K$ is compact. Let $f$ be a simplicial map between $K$ and $L$. We denote $n$-simplexes of $K$ and $L$ by $S_n(K)$, $S_n(L)$. Define ...
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Is the hesse normal not a popular description for a line,plane?

I looked up the "hesse normal form" for lines/planes on Wikipedia. It covers this topic in only 3 languages. Why is the hesse normal form not more prominent ? Or is it also known under a different ...
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3answers
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Obtaining normal form of a line from the general form

This is a question relating to SL Loney's coordinate geometry book (article 56). We have $Ax + By + C = 0$ as the general form of a line. Want to arrive at $xcos(\alpha)+ ysin(\alpha) - p = 0$ as ...
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1answer
38 views

Triangle inscribed in an ellipse [on hold]

What is the maximum area of a triangle that can be inscribed in an ellipse with semi-axes $a$ and $b$?
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0answers
25 views

Surface area with double integral - how to parameterize?

Problem: Find the surface area of the part of the cylinder $x^2+z^2 = a^2$ that is inside the cylinder $x^2+y^2 = 2ay\;$ and also in the positive octant $( x \ge 0, y\ge 0, z\ge 0$). Assume $a > ...
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The Area of Right Triangle and Integral

Consider that we are dealing with a right triangle with constant base ($B=B_1$ and $\frac{dB}{dt}=0$). The values (of $B=B_1$ and of $\frac{dB}{dt}=0$) are maintained by nature to be constant from ...
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1answer
46 views

Smallest possible triangle to contain a square

I was looking at this stack exchange question* and started thinking about the case of a polygon with 4 sides: a square. The question asks for a program that can take a polygon of N sides and return ...
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1answer
42 views

Help Needed in Solving this Problem. [on hold]

AQ=2, Qb=4, BP=3, PC=5, CR=6, RA=4 and P,Q,R. What are The Midpoints Of ABC? What is The Area Of PQR
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2answers
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Smooth homotopy

Let $M,N$ be manifolds. Suppose that $f_0, f_1:M\stackrel{C^\infty}\to N$ are homotopic, i.e. there exists a continuous mapping $f:M\times[0,1]\to N$ s.t. $f(x,0)=f_0(x)$, $f(x,1)=f_1(x)$. Then is ...
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3answers
27 views

Finding the variable

I am not sure how to solve number one. I tried figuring it out. I am not sure how you find x first.
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2answers
16 views

Find center of rotation after object rotated by known angle (2D)

I need to be able to calculate and find the true center of rotation (x,y) of an object after it has been rotated by a known angle. Previously I would simply find the center of the object, rotate 180 ...
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0answers
18 views

Inner product exterior algebra

I have to prove that if V is a real vectorial space (dimV=n) with inner product (.,.) then if we define $$ (v_{1}\wedge v_{2}\wedge...\wedge v_{k},w_{1}\wedge w_{2}\wedge...\wedge w_{k}) ...
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3answers
73 views

Geometrical problem

I have a problem to calculate At the given picture the line segments AB , AC , FE ,GH , GI are known I want to calculate the line segment ED Is this posible ? Thanks in advance Nikos
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0answers
11 views

Pairing Two Point Clouds

So I have two point clouds $X$ and $Y$ each with $N$ points in the familiar $\mathbb{R}^3$ euclidian 3D space. I then have an inter-point distance $d(\vec x_i,\vec y_j)$ which is zero if $\vec x_i$ is ...
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0answers
45 views

Volume of a split log

When solving the following problem, I could not understand why my reasoning came up with an answer that's different than the one on the solution's manual. Question: Consider $(x,y,z)$ such that ...
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0answers
70 views

Different solution for MOSP(Mathematical Olympiad Summer Program) 2001 Test 9 Problem

Let $ABCD$ be a convex quadrilateral and let $O$ be the point of intersection of its diagonals. Prove that if the perimeters of $\triangle ABO$,$\triangle BCO$,$\triangle CDO$ and $\triangle ...
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1answer
23 views

A triangular inequality including squares of sides

Show that for any triangle ABC, the following inequality is true $$a^2 + b^2 + c^2 > \sqrt{3} \max\{|a^2-b^2|,|b^2-c^2|,|c^2-a^2|\}$$ where $a,b,c$ are the sides of the triangle
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3answers
74 views

Volume of sphere

I know the formula of sphere and have seen number of derivations using with and without calculus. I was using a common sense approach of using a slice and rotating this slice 360deg. Number of such ...
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0answers
22 views

Find boundaries of a shape

I have a bunch of points separated one from each other, that form a random shape (as shown on the picture) For each of the points there is known x,y coordinates on the vector. What I'm trying to do ...
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0answers
25 views

Prove minimizing angle in construction

I am a computer science student currently working on his master thesis. I stumbled across a geometric problem that seems obvious but I couldn't prove it for weeks. I attached a picture with the ...