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

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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 ...
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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.
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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 ...
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0answers
15 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 ...
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1answer
33 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 ...
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1answer
19 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 $d$ ...
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2answers
31 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 ...
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3answers
80 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.
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1answer
17 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.
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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 ...
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2answers
29 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 ...
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1answer
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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 ...
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1answer
26 views

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

I think there are at least 1 but not more than 3
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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 ...
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1answer
25 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 ...
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2answers
49 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 ?
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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
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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 ...
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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?
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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?
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0answers
27 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 ...
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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 ...
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1answer
43 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 = ...
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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 ...
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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 ...
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1answer
39 views

Calculate depth using triginometry

I was asked a question like this on an exam today and I'm wondering if I got it right or not. ...
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Generate X, Y, Z coordinates of 3D triangular prism with Edge Rounding

I'm trying to create an interactive 3D visualization with Python and Mayavi for inputs to an analysis program. The program accepts certain primitive shapes which it combines (constructive solid ...
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0answers
15 views

Solving for and x,y,z coordinate in a 3D plane

This is hard for me to explain, but basically I am making a game and I want a 3rd person like camera. I have a lot of information about how the camera should be but I can't seem to get the camera to ...
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1answer
12 views

Centroid of contiguous polygons

Say that I know which are the centroids of two polygons. These polygons share a number of edges (they belong to a planar subdivision). I want to compute the union of the two polygons and also to know ...
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2answers
33 views

Angle between two vectors, where am I wrong?

I am facing a problem, I want to find the angle between the vector u and the vector v, here is what I am doing to get this angle (I used this method) : So what I am finding is an angle that is about ...
2
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2answers
40 views

Geometric intuition: Seeing the regions in double integrals

Context: solving double integrals. I had the formula $$x^2+y^2=1-x-y$$ yet I could not see what shape it had. This is even more true with 3D pictures like $$2x^2+2y^2 \le 1+z^2.$$ Is there a summary ...
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1answer
27 views

Is there a way to find a point on a circle, given another point and an arc length without using trig functions?

Emphasis on not using the trig functions. For example, the problem would be something like find the point $\pi/3$ units counterclockwise from the point $(1,0)$ on the unit circle, without using trig ...
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1answer
48 views

Radius of the circle…

can you please give an idea of how can I solve the following problem. Given that $|AO|=\sqrt5$ and that $|OC|=\sqrt10$ find the length of the circle with the center in point $O$. Here's a picture ...
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4answers
37 views

How to determine the side on which a point lies?

Suppose we have a linear equation and a point in the plane, then how can one determine on which side of the line the point lies?
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action of symmetry group of cube on pairs of opposite faces.

I want to solve the following problem from Dummit & Foote's Abstract Algebra: Explain why the action of the group of rigid motions of a cube on the set of three pairs of opposite faces is not ...
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3answers
21 views

Standard equation of a line

I'm a bit confused. I read in many places that the standard equation of a line in $R^2$ is the following: $w_1 x_1 + w_2 x_2 = d$ but I found a resource that mentions it as: $w_1 x_1 + w_2 x_2 + d ...
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1answer
27 views

Common perpendicular

In the Poincare plane $\mathbb{H}$ show that two distinct type $1$ lines are parallel but do not have a common perpendicular. Let $\mathbb{H}= \{(x,y) \in \mathbb{R}^2 | y > 0 \}.$ A type ...
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0answers
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Oblique Pyramids

(1) Can we have a rectangle cross-section (doesn't need to be parallel to the base) in an oblique square pyramid? (2) Can we have a square cross-section (doesn't need to be parallel to the base) in ...
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1answer
36 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 ...
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1answer
12 views

How to identify any point inside or outside the given cone?

The equation of a double circular cone with a vertex $p=(a,b,c)$ with the generating angle $t$ is given by $(x-a)^2+(y-b)^2= \frac{(z-c)^2}{t^2}$ How do I identify the point ...
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Is there a name for this partial order between metrics?

Suppose we have a set $X$ and two metrics $d_1,d_2$ on it (which may or may not attain $\infty$). Assume furthermore that $d_1,d_2$ have the same metric components (where a metric comoponent is a ...
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1answer
33 views

Isosceles triangle proof

In a neutral geometry, given $\triangle{ABC}$ with $A-D-B,$ $\ A-E-C,$ $\angle{ABE}\cong \angle{ACD},$ $\angle{BDC}\cong \angle{BEC}$ and the line segment $\overline{BE}\cong \overline{CD}$, then ...
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max and min values on symmetric polytope

Let $-N\leq t \leq N$. Let $A$ be regular $(N-1)$-dimensional simplex with vertices $(t,0, \ldots, 0)\ldots (0, 0,\ldots, t)$ and $B$ be regular $(N-1)$-dimensional simplex with vertices $(t-N+1,1, ...
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0answers
31 views

Calculate rotation given Points of a Triangle, Rectangle etc

If I have the all the points of a rectangle or a triangle, how do I calculate it's rotation ? EDIT : I meant the rotation of the whole rectangle or a triangle
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1answer
30 views

terms/names for these *things* [on hold]

So I am having difficulty finding the correct terms to describe the following things. multiple planes make up a space multiple spaces make up a universe (maybe? if not, what is it?) multiple ...
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54 views

A book suggestion -Algebraic geometry. (Arf rings and Hilbert Function)

I am studying algebraic geoemtry. And I need to learn Arf Ring & Hilbert funciton. Please suggest me books / lecture notes...etc. that explains this topic in detail. Thank you.
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43 views

What is the nature of this surface?

What is the nature of the surface whose equation is (it depends on $m$) $$x^2+2y^2+(m+1)z^2+2xy-2yz-2x+2y-4z+m^2+4=0$$
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Scalar Curvature of a metric on the hemisphere, from a paper on the Min-Oo Conjecture

I'm reading a paper on the Min-Oo Conjecture (http://arxiv.org/abs/1004.3088), and I'm stuck on the following step in a proposition: Given a metric $g_0(t)$ on the upper hemisphere $\mathbb{S}^n_+$, ...
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Approximating shapes using a predefined set of geometric shapes

Suppose I have a predefined set of geometric shapes.Is there a certain shape that has an advantage in approximating "constructing" arbitrary shapes "models" from mathematical "best fit & ...
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1answer
47 views

Parabolas intersection

We can get parabolic equation by using only focus and diretrix of parabola: $y^2 = 2px$, where p is a shortest distance between focus and directrix. But this equation defines parabola in coordinate ...