geometry assuming the parallel postulate of Euclid: in a plane, given a line and a point not on that line, there is exactly one line parallel to the given line through the given point.

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90 views

Determining the distance between 2 points based on their know distances to several other points.

I have 2 points in 3-dimensional space. I need to know the distance between them. The problem is, I do not know their coordinates. The only thing I have to go on are 6 other points in this space. The ...
5
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2answers
436 views

Are there any geometries/spaces where pi is a simple (or at least rational) constant?

I found this article on pi: http://blog.plover.com/math/pi.html and while I found it very interesting, it seemed unfinished. The basic point of the article is that pi is complex (for example e has a ...
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1answer
96 views

Need some help to understanding the formula

This is pinhole camera model (I don't get, is there [R t], or (R, t)) This formula is used to model the projection from a space point M to an image point m. Projection drawing Tilde over vector, ...
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0answers
78 views

Polynomial function from $S^3$ to $S^3$ and quaternions

I am searching the polynomial functions from $S^3$ to $S^3$. We say $g$ is a polynomial function from $S^3$ to $S^3$, if there exists $f_1,f_2,f_3,f_4 \in \mathbb{R}[X_1,X_2,X_3,X_4]$ and ...
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1answer
1k views

Construction of a regular pentagon

In Robert Dixon's Mathographics, a regular pentagon is constructed with straightedge and compass only. It is the pentagon $ABCDE$ pictured below. I am having trouble seeing why the central angles ...
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2answers
695 views

how can one calculate the minimum and maximum distance between two given circular arcs?

how can one calculate the minimum and maximum distance between two given circular arcs? I know everything of each arc: startangle, endangle, center, radius of arc. The only thing I don't know how to ...
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3answers
316 views

Closest Points in Euclidean Space

Let $C \subset R^n$ be the convex hull of the set of points $C' \subset R^n$ such that for every $c \in C'$, $c_i \in \{0,1\}$. Let $b \not \in C$. Is there an algorithm that will generate the ...
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1answer
2k views

Can a regular heptagon be constructed using a compass, straightedge, and angle trisector?

Euclid has a magical compass with which he can trisect any angle. Together with a regular compass and a straightedge, can he construct a regular heptagon?
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778 views

Sangaku: Show line segment is perpendicular to diameter of container circle

"From a 1803 Sangaku found in Gumma Prefecture. The base of an isosceles triangle sits on a diameter of the large circle. This diameter also bisects the circle on the left, which is inscribed so that ...
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3answers
390 views

Systems of equations finding right triangles

I need help setting up the equation for the question, "Find all right triangles for which the perimeter is $24$ units and the area is $24$ square units." I know that the area is $A = \frac12 b h$ ...
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1answer
760 views

Similar Triangle Theorem in the Incommensurable Case

The following is a geometry theorem whose proof is examinable in the Irish 'High School' Exam. Let $\Delta ABC$ be a triangle. If a line $L$ is parallel to $BC$ and cuts $[AB]$ in the ratio $s:t$, ...
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Finding angles in a parallelogram without trigonometry

I'm wondering whether it's possible to solve for $x^{\circ}$ in terms of $a^{\circ}$ and $b^{\circ}$ given that $ABCD$ is a parallelogram. In particular, I'm wondering if it's possible to solve it ...
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1answer
75 views

What is the range of longitude/latitude coordinates given a radius of area on Earth that I want to cover?

Given a pair of coordinates, I would like to have a range for both long/lat, that would cover, say 100 meters. How can I do this?
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2answers
209 views

Finding the position of a person on a grid, when you know the $(x,y)$ coordinates of transmitters and the signal strength at the person

I have a $100\times100$ grid. I have a transmitter on each corner, $4$ in total. $$\begin{array}{rl}\text{Transmitter (a) is at}&(0,0);\\ \text{(b) is at}&(100,0);\\ \text{(c) is ...
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3answers
1k views

Why Euclidean geometry cannot be proved incomplete by Gödel's incompleteness theorems?

Wikipedia mentioned the limitation of Gödel's theorems. According to it, Euclidean geometry doesn't satisfy the hypotheses of Gödel's incompleteness theorems, that is, it cannot define natural ...
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1answer
199 views

What is wrong with this proof that isometries must be surjective?

Let $\phi : \mathbb{R}^2\rightarrow\mathbb{R}^2$ be an isometry. Suppose $\phi$ is not surjective, that is there exists some $v \in \mathbb{R}^2$ whose fiber $\phi^{-1}(v)$ is empty. Then by the ...
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1answer
48 views

Angle $x\widehat{O}y $ and a point A inside it. Is it true that $d(A,Ox)*d(A,Oy)=c$

Let there be an angle $x\widehat{O}y $ and A a random point inside it(excluding the rays Ox and Oy). Is it true that the product $d(A,Ox)*d(A,Oy)$ is constant regarless of A? If so, provide the proof ...
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0answers
178 views

Which chapters of Euclid's elements would be helpful for drawing a grid?

I am drawing a $19 \times 19$ grid on my desk. For aesthetic purposes, I don't want to use a ruler. Rather, I want to use Euclidean theorems to 'prove' to myself that such and such line meets at a ...
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3answers
961 views

intersection between a line and a sphere

I have a plane sphere inscribed in a cube like in the image below. Both the sphere and the the cube are centered at the origin. The cube's edge has a unit length (so one edge of the cube is ...
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2answers
309 views

Elementary arguments concerning the stereographic projection

How does one give a proof that is short; and strictly within the bounds of secondary-school geometry that the stereographic projection is conformal; and maps circles to circles?
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1answer
107 views

metric on the Euclidean Group

I am not an expert in this so I hope this doesn't sound so stupid: what is the common metric used when studying the Euclidean Group $\mathrm{E}(3)$. One could probably ask the same thing for ...
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1answer
203 views

combinatorial geometry: covering a square

I'm stuck with this problem. can anyone help me? A finite collection of squares has total area 4. show that they can be arranged to cover a square of side 1.
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2answers
635 views

How to calculate a specific area inside a circle?

I want to calculate the area displayed in yellow in the following picture: The red square has an area of 1. For any given square, I'm looking for the simplest ...
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5answers
1k views

What is the modern axiomatization of (Euclidean) plane geometry?

I have heard anecdotally that Euclid's Elements was an unsatisfactory development of geometry, because it was not rigorous, and that this spurred other people (including Hilbert) to create their own ...
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2answers
312 views

What is a natural way to enumerate the symmetries of a cube?

I want to define a labeling, by small natural numbers, of the 48 symmetries of a cube — affine transformations which do not change the volume it occupies. What is a ...
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4answers
864 views

Finding closest rectangle to another

how I would find line lengths highlighted with other color (with question marks) on the picture below? I know coordinates and size of these rectangles. My goal is to find 'the closest' rectangle to ...
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3answers
16k views

Check if a point is within an ellipse

I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane. How do I determine if a point $(x,y)$ is within the area bounded by ...
3
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1answer
238 views

Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry?

In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, then is there a similar thing do happen in Spherical and hyperbolic spaces? In ...
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2answers
277 views

Is there a generalized method of rotation for curves?

I know that we can rotate a curve in $R^2$ about a linear axis, as is common for first year calculus problems involving solids of revolution. But has anyone come up with a general method to take a ...
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0answers
913 views

History of mathematical symbols, especially the symbol for right angle

Yesterday a child asked me, why (historically) a right angle is denoted by an arc and a dot like in this picture: I dont't know it, but I am interested in it too, so I post this question to this ...
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Does a maximal inscribed square in a regular polygon have a side parallel to a side of the polygon?

Suppose $P$ is a regular (i.e.,equiangular equilateral) polygon in the Euclidean plane, and the number sides of $P$ is not a multiple of $4$. Then $P$ contains an inscribed square. (Citation.) Of all ...
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1answer
117 views

Cube containing cubes

What's the size of smallest cube containing 5 unit cubes? Is there a way to generalize result for 5? I'm more interested in proving minimality, than actaully calculating the size, though.
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741 views

Is there a formula for the solid angle at each vertex of tetrahedron?

A tetrahedron has four vertices as much as a triangle has three vertices. A tetrahedron therefore can have four solid angles as much as a triangle can have three angles. I am wondering: Is there a ...
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1answer
289 views

Fractional versions of euclidean space?

This is going to be a somewhat vague question, but I'll be happy if you indulge me. Euclidean space $\mathbb{R}^n$ is equipped with a lot of nice (algebraic, metric, topological,...) structure and ...
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1answer
432 views

transforming vector potential with a coordinate rotation

In electrodynamics, given the vector potential $\vec{A}$, the magnetic field is defined as: $\vec{B} = \nabla \times \vec{A}$ I'm having trouble figuring out how a coordinate transformation (a ...
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4answers
1k views

Proving two lines trisects a line

A question from my vector calculus assignment. Geometry, anything visual, is by far my weakest area. I've been literally staring at this question for hours in frustrations and I give up (and I do mean ...
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2answers
131 views

Question on Triangles

In a right triangle, the length of hypotenuse is $c$. The centers of three circles of radius $c/5$ are found at its vertices. Find the radius of the fourth circle which touches the three given ...
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2answers
397 views

High-dimensionality and intuition

Over the years I've come across (usually as a tangential remark in a lecture) examples of how our intuitions (derived as they are from the experience of living in 3-dimensional space) will lead us ...
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1answer
134 views

Most symmetric triangulation of $\mathbb{R}^n$

Consider the $n$-dimensional Euclidean space with its standart metric. It is well known that it admits a triangulation by regular simplices only if $n\le 2$. So let us consider less regular ...
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665 views

rotate vector around another vector

If I have vectors $a = (1,0,0)$, $b = (0,1,0)$, and $c=(0,0,1)$ and I want to rotate them counterclockwise at rate $r$ rad/sec around vector (1,1,1). What are the formulas for $a, b, c$?
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242 views

Generalized Jordan curve theorem (and a related MAIN QUESTION)

Preliminaries A Jordan map is a continuous map $f: [0,1] \rightarrow \mathbb{R}^2$ such that $f(0) = f(1)$ the restriction of $f\ $ to $[0,1)$ is injective A Jordan curve is a subset $\gamma$ of ...
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1answer
86 views

Prove that $\vec{EA}+\vec{EB}+\vec{EC}+\vec{ED}=2\vec{EO}$

Let there be a circle $(O,R)$ and $AB,CD$ two perpendicular chords of that circle that intersect on point $E$. Prove that $\vec{EA}+\vec{EB}+\vec{EC}+\vec{ED}=2\vec{EO}$
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2answers
217 views

Is $\mathbb{R}^2$ minus a countable number of points 'skew-Manhattan connected'

Let $A \subset \mathbb{R}^2$ be countable. Then it is not too hard to show that $\mathbb{R}^2 \setminus A$ is path-connected. However it is not always Manhattan connected since if $A = \mathbb{Q}^2 ...
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1answer
202 views

Minimum cardinality of a difference set in $\mathbb R^n$

Given a finite set $S$ of $m$ points in $\mathbb R^n$ that do not all lie in the same $(n-1)$-dimensional hyperplane, consider the set of difference vectors: $\{x-y \, | \, x,y \in S\}$ What is the ...
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0answers
197 views

Projection of the area of a bounded plane over other bounded plane

I have two bounded planes $\pi$ and $\rho$ in three dimensional space. Each plane is bounded by a coplanar rectangle. How can I find the orthogonal projection area of $\pi$ over $\rho$? Thanks in ...
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3answers
178 views

Finding graph node positions based on edge weights

Let's say I have a complete weighted graph with $n$ nodes and strictly positive weights. I'd like to find the smallest $d\in\mathbb{N}$ such that there exists a set of $n$ points in ...
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1answer
773 views

What regular polygons can be constructed on the points of a regular orthogonal grid?

Besides a square, what regular polygons can be constructed so that the points of that polygon lie on the points of a regular, planar, orthogonal grid? Besides a triangle and hexagon, what regular ...
5
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1answer
170 views

Given the area and height of a rectangle, what is the width of the base of a circular segment with the same height and area?

Given a rectangle of height $h$ and area $A$, what is the width $c$ of the chord at the base of a circular segment with the same height and area? I've made a diagram of the problem: My progress ...
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1answer
237 views

An intuitive proof for one of the fundamental property of a parallelogram

"The sum of the squares of the diagonals is equal to the sum of the squares of the four sides of a parallelogram." I find this property very useful while solving different problems on ...
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4answers
315 views

A Sphere Containing Points of Pairwise Equal Distance

Suppose one has $m$ points in $\mathbb{R}^n$ with the property that the distance between any two of them is some fixed constant $d$. Is it true that there is a sphere (living in $\mathbb{R}^n$) ...