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

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What's the meaning of the length of glide reflection?

enter link description here Over here problem 4, it refers to a glide reflection with minimal length, but what's the definition of the length here, does that mean the length of the translation in the ...
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6 views

Approxiamte a surface which contains seveal curves in 3D

I have a set of several closed curves in 3d (like image below is showing my set of curves from 3 views). Each closed curve is consist of many points and all points in each curve have same height. ...
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1answer
10 views

Tangent property and rhombus

A circle is circumscribed by a parallelogram. prove by using tangent property that the parallelogram is a rhombus. I tried to prove that the adjacent sides of the parallelogram are equal but I lack ...
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1answer
28 views

Integration by parts (Differential Geometry)

I am studying the proof of a theorem and I am stuck. It says that by integration by parts we get that: For $g(t)$ a variation of Riemannian metrics wih $g'(0)=h,$ $$\int_{M} (-\Delta (tr h) + ...
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3answers
18 views

Find coordinates of point that intersects circle

I got a circle of 900 radius, knowing its center coordinates A(x1, y1) and got another point with also known coordinates B(x2, y2). I draw a line between point A and B. It intersects the circle in a ...
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25 views

Find probability that random triangle covers centre of circumscribed circle

We are given the equilateral triangle A. On each edge of the triangle we pick a point: randomly (probability distribution is Gaussian) independently of others We construct new triangle B ...
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0answers
11 views

How to find the area projected by a right triangle?

Give the area of the inclined face of the right triangle $A$ which makes an angle of $\theta_i$ with the ground plane, can someone show how we can find the area of vertical face and the horizontal ...
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1answer
28 views

How to find the number of right angled triangles with integer sides and inradius 2009 ..

Problem : How to find the number of right angled triangles with integer sides and inradius 2009 Please help on this as I am not getting any clue how to proceed this problem. I know that ...
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1answer
29 views

Locus of a point $P$ inside $\triangle ABC$

$P$ is a point inside $\triangle ABC$. $X$, $Y$ and $Z$ are feet of perpendiculars from $P$ on $BC$, $CA$ and $AB$ respectively. Find the locus of $P$ is $XY=XZ$ and $A \equiv (4,3)$, $B \equiv ...
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1answer
28 views

How to evenly space a number of points in a rectangle?

Say I have a rectangle, with variable width and height, for example lets use: width = 20 height = 30 I would like to put n ...
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1answer
22 views

Trapezoid problem.

Given Trapezoid ABCD with EF as median (Mid-segment)and with diagonals AC and BD with 2 points G and H as points of intersection of the median and the said diagonals ( G being the intersection of EF ...
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41 views

Prove that $a^2(p-q)(p-r)+ b^2(q-r)(q-p)+ c^2(r-p)(r-q) =4(\delta)^2$

If $p$,$q$,$r$ are the perpendiculars drawn from the vertices of a triangle ABC upon any straight line meeting the sides externally in D,E,F. where a,b,c are the sides opposite to angles A,B,C in ...
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0answers
11 views

Sylvester Gallai theorem in the complex space

An extension of Sylvester-Gallai theorem to the complex space $\mathbb{C}^d$ by Kelly states that if every line passes through at least 3 points in the given set, then these points have to be ...
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1answer
14 views

Find Latitude of point given Longitude and Great Circle orientation

Given the orientation of a great circle as the cartesian components of the normal vector $(a,b,c)$ to its plane, i.e. all points on the circle described in Earth-Centered Earth-Fixed (ECEF) ...
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0answers
28 views

Stuck in Preissmann's theorem

I am stuck on following the proof of Preissmann's theorem, whose statement is that Let $(M,g)$ be a closed connected Riemannian manifold of negative sectional curvature. Then every nontrivial ...
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1answer
18 views

How to represent two coordinate system transformations as one

I'm working on a system of relative euclidean coordinate systems. I'd like to define every coordinate system relative to a global coordinate system, which I'll refer to as [0]. Then, for example, ...
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0answers
19 views

Lovasz number and bipartite complement

Let $G=(V,E)$ be a graph on $n$ vertices. An ordered set of n unit vectors $U=(ui|i∈V)⊂R^N$ is called an orthonormal representation of G in $R^N$, if $u_i$ and $u_j$ are orthogonal whenever vertices i ...
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14 views

Double circle representation using ideal rubber bands

Let $V$ be a 3-connected planar graph, $V^*$ be its dual graph. Let $(C_i : i \in V)$ and $(D_p : p \in V^*)$ be a collection of circles so that if any edge $\{i,j\}$ borders any faces $a$ and $b$, ...
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2answers
14 views

Translate a rectangle but keep within bounds

In x,y space, given a coordinate P that is within bounds 0,0,bWidth,bHeight and a rectangle r where r has width rWidth and rHeight (and rWidth<=bWidth and rHeight<=bHeight), present an algorithm ...
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1answer
34 views

Is the intersection between two $n$-spheres an $(n-1)$-sphere?

It is true that the intersection between two $n$-sphere in $\mathbb{R}^n$ is a $(n-1)$-sphere if is not empty or a single point? I have tried to prove it but my only idea is to work with equations and ...
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0answers
21 views

find a triangle on S2 all of whose angles are pi/2

I am stuck on this question. I am not sure how specific to be but I am thinking it is the triangle formed by any three great circles passing on S2 since they pass through the origin. Please offer ...
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1answer
12 views

Rotate a vector into a plane spanned by two other vectors

In an application test that I had to do for a job recently, I was asked the following question (I quote): “Given three vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$. Compute the rotation ...
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2answers
32 views

General form of a Möbius transformation sending two points to two points and a circle to another.

Suppose I am given a circle $C$ in $\Bbb C^*$ and two points $w_1,w_2$. Given another circle $C'$ and points $z_1,z_2$, what is the procedure to find a Möbius transformation that sends $C\to C'$, ...
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18 views

Uniform vs variable geometries

Euclidean, elliptic and hyperbolic geometry are all different. But they do share a common property: every part of space is "the same". There are no distinguished points that have different properties. ...
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19 views

Spherical geometry vs elliptic geometry

Wikipedia says that "spherical geometry" and "elliptic geometry" are both the geometry of the surface of a sphere. It also asserts that these two geometries are not the same — but neglects to ...
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16 views

Meaning of a Hypersurface resulting from Lagrange Multipliers

Suppose we have a function $f(x_1,\ldots,x_n)$ that we wish to maximize under the set of $n-1$ constrictions $g_i(x_1,\ldots,x_n) = c_i$ for $i \in \{1,\ldots,n-1\}$. We write the Lagrangian ...
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1answer
34 views

Where to place a bridge over the highway?

I've got a problem to solve. I had 2 different ideas which didn't actually work for all cases. On the both sides of highway there are 2 houses K and L (as in the attached picture). Line, that passes ...
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1answer
23 views

Find the radius given only a few variables

I'm writing a program that allows someone to generate a vertical road segment in 3D given a HEIGHT and an ANGLE. The road starts off flat, curves (to the ANGLE), has a brief straight segment (SEGLEN), ...
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0answers
23 views

$[f=q]$ is closed implies $f$ is continuous. [on hold]

Define $H=[f=q]$ to be the set of points such that $f(x)=q$. Let us assume that H is closed. Then the complement of H is open and nonempty (since $f$ does not vanish identically - ""i don't understand ...
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0answers
23 views

Intersection of two hyperplanes

$G$ and $H$ are hyperplanes in $\mathbb{P}_n$ with coordinates $g=(g_0, \ldots, g_n)$, $h=(h_0, \ldots, h_n)$. How can I find a symmetric matrix $A_Q$ of a quadric $Q$ with $ Q = G \cap H$, where ...
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1answer
20 views

How to find a point at a certain distance to other points on the same line

Assuming the points A(x1,y1) and B(x2,y2) and distances between AB (d1) and AC (d2) are known. How can I find the point C(xp,yp)? Actually it has a trivial solution, writing the distance equation 2 ...
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17 views

Distinguish between left and right polygon

When using land information data such as polygons presented as arrays of points, sometimes I need to know whether a polygon's points turn clockwise or counter-clockwise. Please tell me what kind of ...
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33 views

Are 1 dimensional connected closed smooth manifolds diffeomorphic to the $S^1$? [on hold]

Are 1 dimensional connected compact smooth manifolds without boundary diffeomorphic to the $S^1$ ?
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1answer
22 views

Does the perimeter of a 2-D object “count” toward its area?

I'm writing a quick Monte-Carlo simulation for a class in Matlab in order to estimate the value of pi as demonstrated in this gif: http://en.wikipedia.org/wiki/File:Pi_30K.gif However, I'm not sure ...
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1answer
23 views

Shortest distance between two given lines (Hint)

There seems to be this question that I can't seem to be able to solve. I'm hoping someone could help me figure out how to solve it. Question: Find the shortest distance between the lines ...
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2answers
72 views

What is the angle that an Archimedean conical spiral makes with the floor?

I have a spiral in the form $$r = r_0(1-{\theta\over2\pi k }) \{r \ge 0\}$$ where $r_0$ is an initial radius, and $k$ is the number of turns. (It is a spiral that decays from $r_0$ to $0$ as $\theta$ ...
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1answer
36 views

What is the meaning of “girth” of a rectangular box?

Here's an optimization problem. A parcel delivery service will deliver a package only if the length plus the girth (distance around, taken perpendicular to the length) does not exceed 112 inches. ...
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2answers
37 views

Reflections on a sphere

There is a sphere located in a point s with radius r. The Sphere is a perfect mirror. If i'm sitting in the point c, I want to cast a ray to the sphere such that I hit the point p after bouncing in ...
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0answers
15 views

How do I find the angle measurement on a triangle that has one curved side?

I have tried taking this from a circle and measuring the angles as if the width and the height are the quarter of a circle however it is not measuring correctly. I have looked at it as if it is the ...
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0answers
14 views

Finding coordinates of a point on the unit circle

let $A,B,C,D,E$ be points in clockwise order on the unit circle.set $f(P)=\alpha x+\beta y$ where $P$ is a point having coordinates $(x,y)$.If $f(A)=10,f(B)=5,f(C)=4,f(D)=10$ what is $f(E)$? i am ...
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2answers
23 views

Midpoint of the shortest distance between 2 rays in 3D

I would like to come with an algorithm to find the midpoint of the shortest between 2 rays a+tb and c+sd, where t and s are scalars. I have a scenario which I try to depict like this. One of the ...
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1answer
16 views

Find the area of the trapezoid ABCD

Given: AB$\parallel$DC, AB=7cm, AD=BC=5cm and the distance between AB and DC is 4cm. Find the area of trapezoid ABCD. WHAT I TRIED: Since AB$\parallel$DC and AL=BM=4cm, the figure must be a ...
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1answer
25 views

A right triangle's incenter problem by pure geometry..

$ABC$ is a right triangle such that $\angle B= 90^{\circ}$ and $BD$ is the altitude to $AC$. Given that: $I$ is the incenter of $\triangle ABC$, $I_1$ is the incenter of $\triangle ABD$ and $I_2$ ...
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33 views

On the average length of the Steiner net for $n$ randomly chosen points in the unit square

$n$ points are randomly chosen in the unit square with respect to the uniform measure. What is the average length $L$ of the associated Steiner net (tree of minimum length through each of the $n$ ...
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1answer
13 views

Convex hulls intesection

There are 2 datasets: ${(0, 0),(−0.1, 0.1), (−0.3, −0.2), (0.2, 0.1)}$ and ${(0.2, −0.1),(−1.1, −1.0),(−1.3, −1.2),(−1, −1), (1, 1),(0.9, 1.2),(1.1, 1.0)}$ I want to show that this data isn't ...
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0answers
32 views

Calculating the length of a circular arc

In the post, How do the power-series definitions of sin and cos relate to their geometrical interpretations?, I am having trouble following the logic the blogger uses in the "Calculating the length of ...
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0answers
18 views

Point order in congruent triangles

Is the point order of the triangles really relevant when it comes to congruence? Let us assume that: $\triangle ABC \cong \triangle DEF$ Which means that there is a congruence correspondence $ABC ...
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1answer
50 views

Measuring The Star

In a regular pentagon the diagonals are joined to form a star. The star occupies what percent(%) of the pentagon's area?
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1answer
33 views

How is the area of a set of points in $\Bbb R^2$ defined?

Let $S$ be a subset of $\Bbb R^2$. If no vertical slice of $S$ contains gaps, we could define the area of $S$ through the following. $$A(S) = \int_{-\infty}^\infty\left(\sup\{y\in\Bbb R\mid (x,y)\in ...
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1answer
17 views

Are two lines parallel if there are two perpendiculars to the second line between them?

I came across a question like this: In a triangle ABC, there are 2 perpendiculars to any line 'l' passing through A from B and C namely BM and CN. D is the midpoint of BC. MD and ND are joined. We ...