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

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1answer
55 views

How to prove that there is a unique geodisic segment that is pependicular to two other geodesics? [duplicate]

Here is the question I'm not sure how to proceed with this question. An idea that I have is that I assume that geodisic from l to m is perpendicular to one of the geodisics, then I have to show that ...
0
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1answer
581 views

Minimum number sticks requirement

(Objective question) A box of sticks of equal lengths is provided. The minimum number sticks needed to build a frame to enclose a 3 dimensional volume is 1) 6 2)12 3)3 4)8 Solution:- 2)We can ...
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2answers
258 views

the osculating planes of a curve pass through a fixed point $\rightarrow$ the curve is a plane curve.

If the osculating planes of a curve pass through a fixed point, the curve is a plane curve. How to prove it?
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0answers
52 views

What is the problem? [duplicate]

Where has the one block gone in lower image,after we rearrange the triangles? (Found it on G+ as a post.)
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0answers
102 views

How to project points in 3-space into a 2D subspace while minimizing the maximum change in Euclidean distance?

We have a small set of points in $\mathbb{R}^3$ (around 4 to 10 points, say). I would like to project these points onto a 2D subspace such as to minimize the maximum change in Euclidean distances. ...
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1answer
73 views

Using Euler's formula

I have a question related to Euler's formula and whilst I understand the formula I'm not really sure about the question: Let $V_{k}$ be the number of vertices of P from which exactly k edges emanate, ...
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2answers
73 views

The zero set of $z_0^2+z_1^2-1$ in $\mathbb{C}^2$.

Recently, I read the notes "Vector bundles on Riemann surfaces" by Sabin Cautis (http://www-bcf.usc.edu/~cautis/classes/notes-bundles.pdf). On the sixth page of these notes, there is a statement ...
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1answer
256 views

Can one sample uniformly from the surface of an $n$-sphere of non-unit radius using normal r.v.'s?

One can sample coordinates of the surface of a unit radius $n$-dimensional sphere uniformly using the following method: independently generate a vector of $n$ standard normal random variables ...
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1answer
129 views

Generating vectors of the face-centered cubic lattice

I can't find a reference for a set of generating vectors for the Tetrahedral-octahedral honeycomb lattice. I would like to know the "canonical" set and if possible a more general set described by ...
2
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1answer
512 views

Finite Vertex-Transitive Planar game of Civilization?

If you have played games in the Civilization series, you will have noticed that the Earth is represented in a simplified and profoundly unsatisfying way. It is wrapped around the curve of a cylinder ...
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3answers
389 views

Cantor Set and Fractals

I have read that the Cantor set is considered a fractal. I am referring to the Cantor set in which the middle third of a real line is removed recursively. I see that this is recursively defined, but ...
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3answers
132 views

Semigroup generators and Cones

In $\mathbb{R}^2$, let $C$ be the cone (non-negative linear combinations) on the two vectors $$ v_1=\begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad v_2=\begin{pmatrix} 2 \\ 3 \end{pmatrix} $$ Consider ...
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2answers
49 views

Hyperbolic quadrilaterals with two adjacent right angles

For convenience we'll work in the hyperbolic upper half plane $H$. We are given a hyperbolic quadrilateral $Q$ with vertices $a,b,c,d$ and geodesic segment edges $[ a,b ]$ $[ b,c ]$ $[ c,d ]$ $[ d,a ...
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0answers
308 views

Calculating hyperbolic distance between two points

I was looking for a formula to calculate the hyperbolic distance between two planes and came across this formula $$\ln\left(\csc b-\cot b\over \csc a - \cot a\right)$$ where $a$ and $b$ are the ...
4
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1answer
151 views

Bounding the magnitude of the Fourier Transform of the indicator function of a Voronoi cell

Suppose I define the set $A \subset \mathbb{R}^3$ to be a single cell within a larger 3D Voronoi tessellation on a set of known points. $A$ is thus convex and simply-connected. Without loss of ...
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2answers
167 views

Find an orientation preserving isometry $f (z) = \frac{az+b}{cz+d}$ such that $f (i) = 17 + 3i$

This is probably a very simple questions but I am not clear on Möbius transformations and how to solve this problem. I'd appreciate if somebody can point me towards a method to do these sort of ...
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0answers
223 views

Facets of the convex hull as solution of an optimization problem?

Given $N$ points $x_1, x_2, ..., x_N \in \mathbb{R}^n$, consider their convex hull $$\mathcal{C} = \text{conv}( \{ x_1, ..., x_n \} ) = \bigcap_{j=1}^{J} \{ x \in \mathbb{R}^n : \ A_j x \leq b_j \} ...
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0answers
52 views

transforming a straight band into a logarithmic spiral

I want to plot the labels and the graduations of an historical timeline onto a logarithmic spiral. If this timeline is on the $x$-axis, $-\infty$ would project to the center of the spiral, $+\infty$ ...
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1answer
270 views

What's the logical flaw in Euclid's construction of the triangle?

NJ Wildberger says in this video that there's a logical flaw in Euclid's construction of the triangle, that you're not really able to know (apart from the picture) if the circles intersect. He also ...
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1answer
3k views

Centroid of a semicircle vs. a semicircular arc

Why is the $y$ centroid of a semicircle and that of a semicircular arc different? Using Pappus' second theorem on a semicircle of radius $r$, $\bar{y}=\frac{V}{2\pi A}=\frac{\frac{4}{3}\pi r^3}{2\pi ...
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1answer
59 views

How to show a basic integral inequality?

The following inequality is quite clear for $R^1$: $$\int_{B_1}1/|x-y|^\alpha dx\leq\int_{B_1}1/|x|^\alpha dx,\quad\forall y\in B_1,$$ where $B_1$ is the unit ball in $R^1$, i.e., $[-1,1]$ and ...
3
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1answer
176 views

How to simplify this trigonometric expression?

I was trying to solve a problem taken from an Physics Olympiad when I came across a curious and complex mathematical expression. I can not prove with what I know so far about mathematics, does could ...
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1answer
301 views

Maximal volume for given surface area of an $n$-hedron

Is there a term for a polyhedron with $n$ faces (or, similarly, $n$ vertices) that maximises the enclosed volume for a given surface area (equivalently, minimises the surface area for a given volume)? ...
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1answer
106 views

Calculating mean velocity of an orbiting body as it moves towards a point.

I'm making a game, in the game planets orbit a central point in circular orbits, they move directly towards their targets and the vector is simply added to their orbital path. Whilst not realistic it ...
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2answers
105 views

What is a good way to simplicize the integer lattice?

I have a function $f$ defined on the positive integer lattice in $\mathbb{R}^n$ (i.e. the vectors in $\mathbb{R}^n$ whose coordinates are all nonnegative integers). I want to extend the domain of $f$ ...
4
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1answer
225 views

Closest point to 3 (or more) circles

I've been scouring the Internet for enlightenment but so far I've found very little that has helped. To be fair, I'm not a math major and might just not be using the right search queries. I'm working ...
3
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1answer
73 views

straight lines on spherical surfaces

I'm not sure how to explain this so here goes. I want to place a horizontal line of text on the side of a curved brandy glass. Is there a formula that can be applied to ensure the text stays ...
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2answers
134 views

Triangle has congruent sides but not congruent angles

Is it possible to build triangle with all the same sides but angles that are not congruent?
1
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1answer
58 views

Sphere on a grid

So, this is a little tricky kind of a question and I'm not totally sure if it's a mathematic question or a more programming one, but I nevertheless hope to find answers. I want to find out the error ...
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3answers
510 views

Find coordinates of n points uniformly distributed in a rectangle

I have a rectangle R of width W and height H. I have N points inside this rectangle. I need to find an algorithm to position my points in the rectangle in the most uniform way possible (no overlaps, ...
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1answer
207 views

Ceva's Theorem in the space

I heard about Ceva's Theorem in three dimensions. Can you give me more details and link something about that? Thanks.
2
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1answer
1k views

Rotation through any angle θ

They say that rotation of any point $(x,y)$ through any angle $\theta$ is given by $(x \cos\theta, y \sin\theta)$. Can anybody tell how was this derived? Please post here or send me by email.
3
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1answer
537 views

Using Cavalieri's Principle to find the volume of an ellipsoid

I understand how to use the triple integral + change of variable method to find the volume of an ellipsoid, but given an ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} \le 1$$ whose area is$$A= \pi ab$$ ...
2
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2answers
461 views

Is there any regular geometrical structure with maximum surface area and minimum volume?

I am interested to know a geometrical structure with maximum surface area and minimum volume. According to me double napped cone may have such property as surface area of a cone is $\pi rl +\pi r^2$ ...
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4answers
476 views

Is every prime number the leg of exactly one right triangle with integer sides? What's wrong with my argument that this is impossible?

The problem is: "prove that every prime number is the leg of exactly one right triangle with integer sides." However, I seem to have proved that this is impossible. What did I do wrong here? Let ...
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1answer
70 views

Optimizing the area of a triangle in space.

A triangle has two corners, $(8,0,3)$ and $(0,8,3)$ and a third curve in space that consists of all points $(8,8,a^{2}+3)$, where $a$ is a real number. Calculate the area of the triangle as a function ...
2
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3answers
131 views

How do I test if a 3d point is to the “left” or “right” of a triangle in 3-space?

I'm attempting to determine if a point in 3-space is inside or outside of a convex polyhedron with triangular sides. One strategy, I suppose, is to determine which side of each triangle the point is ...
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1answer
63 views

Determining if a point in 3-space is inside a polytope knowing only the distances to the polytope's vertices

If I have a point in 3-space, as well as a convex 3-polytope, and an unordered set of distances to the vertices of the 3-polytope (but not the position of these vertices) is there any way for me to ...
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1answer
37 views

What is the simplest non-convex polytope with strictly triangular faces?

What is the simplest (in terms of the number of faces) non-convex polytope with strictly triangular faces? Please note that "non-convex" also means non-weakly-convex.
1
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1answer
48 views

Help to find spherical Line

What is the spherical line through the points $(0,-1,0)$ and $\left(0,\frac{1}{2},\frac{\sqrt{3}}{2}\right)$? I solved: $G = \{(x,y,z)\in S^2 \mid \exists\ a,b,c \in \mathbb{R}, ax+by+cz = 0\}$ ...
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1answer
54 views

Decomposition of a single 4D rotation

I have a $4\times 4$ matrix $M$ which represents a general 4-dimensional rotation. $$ M = \pmatrix{a_{11} &a_{12} &a_{13} &a_{14}\\a_{21} &a_{22} &a_{23} &a_{24}\\a_{31} ...
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3answers
68 views

Finding a position opposite to a rotated point

I have a point that's rotated to a certain angle. I want to get a point opposite to the direction it's facing. Also it needs to have a certain distance from it. I made a little illustration to ...
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1answer
37 views

Find a point in a polytope that always cuts off a constant fraction of the polytope.

I have $k$ linear inequalities in $\mathbb{R}^n$, which we can express as $Ax \ge c$ (where $A \in \mathbb{R}^{k \times n}$ and $c \in \mathbb{R}^k$). Assume that the set of $x \in \mathbb{R}^n$ that ...
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1answer
848 views

What is duality?

I have seen some examples of duality. Sometimes applied to theorems, as for example Desargues theorem and Pappus theorem. Sometimes applied to spaces, for example the dual space of a vector space. ...
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0answers
16 views

Given an N-Dimensional point P, is each component P(c) = 0 for c > N or is P(c) undefined?

A concrete example: Given a 2D point P with coordinates (x,y), if it were looked at from a 3D perspective would it have coordinates (x,y,0)? This seems to be a common assumption but I have yet to ...
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1answer
87 views

Why mathematical knots are defined as closed geometries unlike physical knots?

From wikipedia: A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a ...
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2answers
151 views

Showing the function $f(x,y)$ is one by one

Yesterday, while teaching geometry, I was faced to a problem saying that the function below is an distance function: $$d(P,Q)=\Big|\ln\frac{\frac{x_1-c+r}{y_1}}{\frac{x_2-c+r}{y_2}}\Big|$$ where in ...
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2answers
178 views

Two lines intersect at $O$ at an angle of $60^∘$

(Refer to diagram below) Two lines intersect at $O$ at an angle of $60^∘$. Two circles are drawn such that they are tangent to each other and the two lines. $A$ and $B$ are the centers of the smaller ...
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2answers
727 views

How to calculate angles required to lay out flat pieces in a circle

I want to construct a wheel made of flat pieces of wood, something like this picture: I am unsure how to calculate the difference in angle between each of the flat pieces of wood that make up the ...
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1answer
76 views

Hexagon BCEGHJ surrounded by the rectangle

What is $m+n$ if $sin^2 \theta = \frac{m}{n}$?