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

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3
votes
0answers
255 views

Euclidean geometry and the Euclidean group

At Wikipedia's Erlangen program I read that "quite often, it appears there are two or more distinct geometries with isomorphic automorphism groups". Some examples are given. But what are examples ...
7
votes
3answers
4k views

Tetrahedron inside a sphere

What's the largest regular tetrahedron (having side length $x$) you can fit inside a sphere with a unit radius?
8
votes
2answers
3k views

Numbers of circles around a circle

"When you draw a circle in a plane of radius $1$ you can perfectly surround it with $6$ other circles of the same radius." BUT when you draw a circle in a plane of radius $1$ and try to perfectly ...
0
votes
1answer
279 views

From geometrical figures to function

There's one basic mathematical thing that keeps bugging me: the fact that a really simple 2D geometrical figure (like a circle) might not be a function. I know what the definition of a function is. A ...
13
votes
7answers
8k views

Drawing heart in mathematica

It's not really a typical math question. Today, while studying graphs, I suddenly got inquisitive about whether there exists a function that could possibly draw a heart-shaped graph. Out of sheer ...
2
votes
0answers
322 views

Can elliptic arc be represented by quadratic Bezier curve?

Can elliptic arc (defined as part of an ellipse, with extent not greater than $90˚$) be represented by quadratic Bezier curve?
11
votes
3answers
1k views

How to prove that a torus has the same volume as a cylinder (with the height equal to the torus' perimeter)

I want to find the volume of a torus with a given thickness and a given radius. Let r be the radius of a circle with its midpoint at $M(0|b)$ ($b \geq r$). Now I want to rotate this circle about the ...
3
votes
2answers
4k views

How elliptic arc can be represented by cubic Bézier curve?

If I have an arc (which comes as part of an ellipse), can I represent it (or at least closely approximate) by cubic Bézier curve? And if yes, how can I calculate control points for that Bézier curve?
0
votes
2answers
82 views

Lower curve from A to B: axiom or theorem?

The length of any curve that goes from A to B (other than the line segment AB) is greater than the length of that segment. This statement may be a theorem? Or, necessarily, is an axiom? It seems to me ...
1
vote
3answers
865 views

Is it possible to imitate a sphere with 1000 congruent polygons?

Edit: The answers I seem to be getting here are that this is not possible, but I have come across a formulae for subdividing an icosahedron (which I dont have to hand, but goes something like this) ...
4
votes
2answers
1k views

Create a trapping region for Lorenz Attractor

I would like to show that the quantity: $-2\sigma\left(rx^{2}+y^{2}+b\left(z-r\right)^{2}-br^{2}\right)$ is negative on the surface: $rx^{2}+\sigma y^{2}+\sigma\left(z-2r\right)^{2}=C$ for some ...
4
votes
2answers
356 views

Alien vs Alien Hunter Puzzle

I found this puzzle posted on a student website I frequent, but no one including the poster nor I could solve it. So I'm posting this puzzle here with hope that some of math whiz in here could ...
17
votes
1answer
417 views

Equilateral polygon in a plane

Let $n$ be a positive integer. Suppose we have an equilateral polygon in the Euclidean plane with the property that all angles except possibly two consecutive ones are an integral multiple of ...
0
votes
1answer
254 views

How to find the intersection between the great circle and a hyperplane?

Let $s = (\frac{1}{\sqrt{d}}, \ldots, \frac{1}{\sqrt{d}})$ and $u \in \mathbb{R}^d$ be two distinct unit norm vectors in the first orthant. Consider moving along the great circle defined by $s$ and ...
9
votes
4answers
5k views

How to compute the angle between two vectors expressed in the spherical coordinates?

Given two vectors $u, v \in \mathbb{R}^d$ represented the spherical coordinates is there a simple formula to compute the angle between the two vectors? Without loss of generality, we can assume that ...
3
votes
1answer
300 views

Curve of a fixed point of a conic compelled to pass through 2 points

Suppose that in the plane a given conic curve is compelled to pass through two fixed points of that plane. What are the curves covered by a fixed point of the conic, its center (for an ellipse), its ...
7
votes
1answer
246 views

How to compute the change in the angle between two unit norm vectors as the $\ell_1$ norm of one vector changes?

Motivation Suppose that $u \in \mathbb{R}^d$ is a unit-norm vector, $\|u\| = 1$, $a, b, c$ are some positive constants and $\xi \in [0,1]$ is another constant (usually chosen close to 1). I am ...
12
votes
1answer
204 views

Almost identical map

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be bijective map with following properties: 1) $f|_{\mathbb{Q}^2}=id$; 2) Image of any line under map $f$ is again a line. Is it right that $f=id$?
0
votes
1answer
1k views

Calculating center of the ellipse

How to find center of ellipse from two points (these are just points on the ellipse, not related to foci), and two radii ($r_x$ and $r_y$, from standard definition of the ellipse $\frac{x^2}{r_x^2} + ...
6
votes
4answers
426 views

Visualising extra dimensions

What is the : most useful prettiest way to visualise extra dimensions in shapes and charts?
2
votes
1answer
405 views

A system of geometric equations

I have a question about solving a system of geometric equations. I really hope someone here can help me, it's been several months since I try to solve the problem but without success. As I am not an ...
8
votes
5answers
24k views

Height of a tetrahedron

How do I calculate the height of a regular tetrahedron having side length $1$ ? Just to be completely clear, by height I mean if you placed the shape on a table, how high up would the highest point ...
1
vote
4answers
1k views

Unit circle metric

Let $S^1$ the unit circle in $\mathbb{R}^2$ and $$d: S^1\times S^1\to\mathbb{R}$$ $$d(\theta_1,\theta_2) = \left\{ \begin{array}{ll} |\theta_1-\theta_2| & \mbox{if } ...
5
votes
1answer
183 views

Can I draw a net for a Steinmetz solid with a compass?

Can I draw a net for a Steinmetz solid from two cylinders with a compass? That is, can we flatten the net? I often make a model using paper and a compass-- it looks about right... is it really a ...
2
votes
1answer
1k views

Dissecting rectangle into two equal area parts to form noncongruent rectangle

A rectangle can be cut into two connected pieces of equal area so that they can be rearranged to get a new rectangle with different side lengths than the original one. Show that this can be done in ...
30
votes
7answers
2k views

Quotient geometries known in popular culture, such as “flat torus = Asteroids video game”

In answering a question I mentioned the Asteroids video game as an example -- at one time, the canonical example -- of a locally flat geometry that is globally different from the Euclidean plane. It ...
6
votes
3answers
490 views

What type of triangle satisfies: $8R^2 = a^2 + b^2 + c^2 $?

In a $\displaystyle\bigtriangleup$ ABC,R is circumradius and $\displaystyle 8R^2 = a^2 + b^2 + c^2 $ , then $\displaystyle\bigtriangleup$ ABC is of which type ?
60
votes
4answers
7k views

Why is a circle in a plane surrounded by 6 other circles?

When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other numbers? I'm ...
2
votes
5answers
261 views

What type of triangle satisfies: $\cot \biggl( \frac{A}{2} \biggr) = \frac{b+c}{a} $

If in a $\displaystyle\bigtriangleup$ ABC, $\displaystyle\cot \biggl( \frac{A}{2} \biggr) = \frac{b+c}{a} $, then $\displaystyle\bigtriangleup$ ABC is of which type ?
1
vote
1answer
406 views

How to explain why this method of slicing a solid to find volume fails?

The area of a pyramid with a square base with side length $L$ and height $h$ is calculated as follows: In $\mathbb{R}^3$ place the pyramid upright with one side flush with the z-axis so that the ...
1
vote
2answers
3k views

How to calculate area of triangle having its points 2D coordinates?

We have points A, B & C in 2D plane. How having point coordinates $(x, y)$ to calculate area of triangle formed by them?
6
votes
2answers
842 views

Tractrix-like curves

Is there a common name for curves, obtained from dragging a point along another curve, similar to how tractrix is obtained by dragging a point along a line? What is a parametric equation of such ...
2
votes
1answer
315 views

How to calculate the length of line in a trapezium, given some other lines?

Of course, it's best explained with an image: $L1$, $L2$, $h_1$ & $h_2$ are all given, and I would like to calculate the length of $L3$. Is it possible, and how?
1
vote
2answers
180 views

Parallel straight lines

Which are the parallel lines? I prove that $a$ and $b$ are parallel but can't prove that $c$ and $d$ are parallel. The angles, which are $135^o$ and $45^o$ are alternate angles.They aren't equal , ...
0
votes
3answers
312 views

How to solve this triangle?

Please observe the triangle shown in the figure below (In red). I consider the angle between $5$ and $4$. The base should be $5$, hypotenuse $4$ and perpendicular $3$, but according to the solution ...
1
vote
0answers
125 views

Is it possible to express the area of the intersection of 2 circles as a closed-form expression? [duplicate]

Possible Duplicate: The cow in the field problem (intersecting circular areas) A farmer ties a cow with a rope to a pole at the edge of a round field of radius r. How long must the rope be ...
10
votes
3answers
7k views

Which tessellation of the sphere yields a constant density of vertices?

One way to tessellate a 3D sphere is by iterated subdivision of an icosahedron. I am wondering whether this method gives a homogeneous surface density of vertices. To the eye, it seems to do so, and ...
3
votes
0answers
155 views

The $n$-shortest lattice vectors problem in $\mathcal{R}^2$

I am looking for an algorithm to compute the $n$ shortest lattice vectors in $\mathcal{R}^2$. The problem statement is as follows: Given a lattice $L: \{ m \vec{u}+n\vec{v} \} \in \mathcal{R}^2$, a ...
1
vote
1answer
150 views

How to solve for x-coordinate of the top of an isosceles triangle?

I think the following picture helps explain it easier than words: I know the coordinates of point a which is at a vertex of the triangle, as well as the coordinates of point b which is somewhere ...
3
votes
3answers
2k views

A parallelogram and a line joining a vertex to the midpoint of opposite side

In a parallelogram ABCD. M is the midpoint of CD. Line BM intersects AC at L and it also intersects AD extended at E. Prove that EL=2BL PS: This is not a homework problem. I was solving geometry for ...
3
votes
2answers
478 views

shadow simulation from buildings

is it possible to calculate shadow areas of buildings or simulate shadows of buildings in a city, using the heights of these buildings and the sun angle and azimuth? the basic light tracing concept ...
3
votes
1answer
227 views

Triangle centers

From Wikipedia's triangle center article: "Thus every point is potentially a triangle center. However the vast majority of triangle centers are of little interest, just as most continuous functions ...
0
votes
1answer
259 views

Axiomatic affine n-space and the (classical) affine variety $A^n_k$ only agree for K an alg. closed field?

So, wikipedia has a page about "affine space", where such a space is an object (set, topological space, blah) with a free transitive (blah, continuous, blah) action by the additive group of a free ...
28
votes
2answers
807 views

Optimal yarn balls

Winding yarn into a ball suggests some mathematical questions: Under some natural model, what paths should the yarn follow to achieve the most dense ball? One model is that used by Henryk Gerlach ...
4
votes
3answers
6k views

Endpoint of a line knowing slope, start and distance

In a Cartesian system, I've got the slope, start point and distance of a line segment. What's the formula to find the endpoint?
8
votes
1answer
339 views

How to parameterize an orange peel

I'm trying to parameterize the space curve determined by the boundary of a standard orange peel: for example, the one on this photo: For example, the ideal curve would be inside the unit cube; have ...
1
vote
1answer
101 views

Problem interpretation - Distance Formula?

If you've got multiple arrays like this: (24,36,28,28,16,27) (38,38,45,57,35,50) every array being 6 integers, each integer in range [0,60] I would like to find the distance between those 2 ...
10
votes
1answer
604 views

Results related to The Happy Ending Problem

Im giving a small talk for a combinatorics class on the Erdos-Szekeres conjecture regarding the happy ending problem (the paper is focused on recent work regarding the conjecture). I always find that ...
16
votes
3answers
2k views

What is the difference between a variety and a manifold?

I hear people use these words relatively interchangeably. I'd believe that any differentiable manifold can also be made into a variety (which data, if I understand correctly, implicitly includes an ...
0
votes
1answer
111 views

Repeating Tiled Piecewise Function Help

I have a piece-wise 2-input function that I would like to repeatedly tile diagonally across a grid. See this image. So here, I know the equations of the pink, blue and green areas in the range 0 - ...