geometry as on the surface of a sphere, where "lines" are great circles and any pair of lines must intersect

learn more… | top users | synonyms

0
votes
0answers
18 views

Describing Area of Spherical Cap as Sum of Spherical Triangles

I was wondering how one could express the area of a spherical cap in terms of a sum of triangles. The area of a triangle on a sphere is: $A = E R^2$ where E is the excess of the triangle ...
0
votes
0answers
17 views

Area of Spherical Zone

"Let $\mathcal S$={$\mathbf x \in \mathbb{R}^3 : ||\mathbf x||=1$} Prove that the area of the part of $\mathcal S$ that lies between the two parallel planes given, say, by $x_3=a$ and $x_3=b$, is the ...
0
votes
1answer
20 views

Proving that there are only five Platonic solids using spherical geometry

In 100 Great Problems of Elementary Mathematics by Dorrie, it is proved that there are only five possible tessellations of the sphere using congruent regular (spherical) polygons: $4$ regular ...
0
votes
1answer
24 views

Cartesian to Spherical coordinate conversion specific case when Φ is zero and θ is indeterminant

Following is the conversion for spherical to cartesian coordinate \begin{align} x &= r \cos\theta \sin\varphi \\ y &= r \sin\theta \sin\varphi \\ z &= r \cos\varphi \end{align} and we are ...
0
votes
1answer
20 views

Problem finding coordinates in a earth like coordination system

A picture with the problem Hey guys Given: two coordinates $A(a_1,a_2), M(m_1,m_2)$ , the distance between $B$ & $C$ is known as $w, d(B,C) = w$ d(B,M) = d(M,C) where d is the great-circle ...
1
vote
0answers
23 views

rotation of spherical surface in spherical coordinates

I need to plot a spherical surface in computer (like the surface of a lens). I know the normal vector (as an example, say $\ n=(1,2,3) $) of this surface and it originates from the centre of the ...
2
votes
0answers
20 views

congruency of triangles in hyperbolic and spherical geometry

In Euclidean geometry, we have the following congruencies of triangles: side-side-side, side-angle-side, angle-angle-side = angle-side-angle (because of the angle sum) and side-side-angle (only if the ...
0
votes
1answer
29 views

Rotations of sphere $\mathbb S^2$

In the picture bellow; How to prove that the result of rotation about $P$ through angle $\theta$, followed by rotation about $Q$ through angle $\varphi$ is rotation about $R$ through some angle? ــ ...
1
vote
0answers
27 views

Constructing a spherical triangle of a given surface area

Given three points $p_1$, $p_2$ and $p_3$ on the unit sphere $S^2$, we can construct a spherical triangle. The angles associated with the points are $A$, $B$ and $C$. Keeping the points $p_1$ and ...
0
votes
1answer
55 views

Spherical circle - Area

I am looking at the following exercise: The spherical circle of centre $p \in S^2$ and radius $R$ is the set of points of $S^2$ that are a spherical distance $R$ from $p$. If $0 \leq R \leq ...
0
votes
1answer
34 views

Similar spherical triangles are congruent

I look at the following exercise of A. Pressley: Show that similar spherical triangles are congruent. I have no clue how to show it. Can you give an idea how I can do that? In the book ...
0
votes
0answers
19 views

Proof equivalence of equal dihedral angles and vertices on a sphere for regular polyhedra.

I know that the following theorem is true: Theorem: Provided that all faces of a polyhedron are regular poygons, the statement ``all the dihedral angles are congruent'' is equivalent to saying ...
25
votes
6answers
2k views

How to generate random points on a sphere

How do I generate $1000$ points $(x, y, z)$ and make sure they land on a sphere whose center is $(0, 0, 0)$ and its diameter is $20$? Simply, how do I manipulate a point's coordinates so that the ...
0
votes
1answer
18 views

Project a line onto a sphere to calculate parameterized spherical coordinates

I have a line segment and I want to find the arc that it projects to on a sphere. I know there are two arcs; I'm interested in the one that's closest to the line (or intersects it). The easy way to ...
1
vote
1answer
28 views

Relationship between regular tessellations in general space

Using the notation for regular tessellations $\{p,q\}$ denoting the tessellation consisting of p-gons , q of which meet at each vertex [e.g. $\{3,6\}$ in $\mathbb{R}^{2}$ is the equilateral triangle ...
0
votes
0answers
18 views

Triangles in spherical/elliptical geometry

In Euclidean geometry the sum of internal angles of a triangle is always $180^\circ$. In non-Eudlidean geometries it may be either less than this (in hyperbolic geometry) or more (spherical/elliptical ...
1
vote
0answers
26 views

Non commutative flows on the 2-sphere.

The title says everything, really. I'm looking for some flows on $S^2$ such that They do not commute. They are of some interest, or they are peculiar in some ways. Thanks
2
votes
3answers
44 views

Trying to create a inverse quare algorithm for expanding sphere

So, in a piece of software I am writing (this isn't homework), I want to have a sphere expand relative to time. I want it to expand quickly from start with the expansion slowing over time. I.e, the ...
1
vote
2answers
69 views

Find a spherical triangle with angle sum $5π/3$

Find a spherical triangle with angle sum $5π/3$ I am unsure how to answer this question and would like to be shown how I go about answering?
3
votes
1answer
53 views

Determine a point lie in bisect area between 2 circles on sphere

Given 2 points A,B,O on sphere of radius $R$. Point O is in middle of AB. E and F are deviation from O by geodesic distance $d$ (angle between EF and AB is $90^o$). Consider 2 circles $C_1,C_2$ on ...
2
votes
1answer
50 views

How can I efficiently check a point lie in 4 circles on sphere?

Given coordinate of 2 points $A,B$ (Cartisian or longitude-latitude coordinate) on sphere of radius $R_1$. Point $O$ is middle of $AB$, 2 points $E$ and $F$ is derivation from $O$ by a distance ...
1
vote
0answers
26 views

Locate a point on sphere with equal distance

Given 3 points A (lat1, lon1), B(lat2, lon2), O(lat3,lon3) on earth with geometric location longitude and latitude and a distance d, where O is middle point of A and B. Let GCD denote the great ...
0
votes
0answers
13 views

Find a point on extended great circle with given distance

Given 2 points on earth with longitude and latitude coordinate A(lat1, lon1), B(lat2, lon2), and a distance d. Find coordinate (in longitude and latitude) of 2 points C, and D on extended of great ...
4
votes
1answer
56 views

Area of a plane triangle as limit of a spherical triangle

We know that the area of an spherical triangle (in a unit sphere) is given by $A(\triangle) = \alpha + \beta + \gamma - \pi$, where $\alpha$, $\beta$, and $\gamma$ are the interior angles of the ...
-1
votes
2answers
238 views

Colour of bear in Earth's surface [closed]

A bear stands in one point of the Earth's surface. Walking one kilometer south , then walking one kilometer east and immediately after one kilometer north and reaches the point from which started.Find ...
0
votes
2answers
135 views

Find all Equiangular Platonic triangles

A spherical triangle A is called equiangular if its 3 angles are equal. A is called Platonic if copies of A tile the unit sphere. I need to find all such triangles. Don't we have an infinite amount of ...
2
votes
0answers
19 views

Sample from distribution taking spherical statespace

I have a probability distribution over a 2-sphere, with density function $f(\phi)$, a function of polar angle only. Is there an efficient way to sample from this distribution?
0
votes
1answer
19 views

How to find height of two objects stacked at an angle

Consider the following situation: If there are two balls of diameter 50mm and 60mm stacked inside a tube with internal diameter of 60mm. If the smaller ball is stacked on the big ball, it is easy to ...
0
votes
0answers
28 views

Project locus of points on sphere using Mercator projection

Given a the locus of points on a sphere that are the same great-circle distance from some point, what is the shape described when that locus is projected onto a 2D plane using the Mercator projection? ...
7
votes
1answer
60 views

Area of the surface on a sphere surrounded by three externally tangent circles?

Let us draw three circles of radius $\dfrac23$ on a sphere of radius $1$, all of which are mutually tangent (externally). How do I calculate the area of the surface surrounded by the three circles? ...
0
votes
0answers
54 views

Find intersection points of two circles on sphere (earth)

Problem Assume that we have two devices that can measure distance to the target. Devices installed on earth at coordinats A and B. We measure distances Da, Db in meters. How to find points of ...
0
votes
2answers
54 views

Calculating the percent of area “covered” by a vector pointing on a sphere

The question is inspired by rotational dynamics and how much of the sky could a camera "cover" when it rotates in a specific way. Let's say that we have solved the equations for rotational motion of a ...
0
votes
0answers
11 views

Projection of great circles in flatprojected sphere

I am trying to construct a map using fractal fault algorithms. What I do is find a random great circle in a sphere and change the height of half of the sphere. Doing this ten thousands of time a map ...
1
vote
0answers
62 views

Placement of protons and neutrons in the nucleus

So, I'm creating a program that would represent a given atom (also different isotopes) in 3d view. I'd need some kind of formula to calculate the position of protons and neutrons to form a nucleus. ...
1
vote
3answers
44 views

How to know if an arc segment intercepts a circle in a sphere?

This problem is bugging for quite some time now, and I actually need to program the solution in my game. I have one point on a radius 1 sphere, and a rotation matrix. When I rotate the point with the ...
1
vote
1answer
23 views

Projection from triangle to spherical triangle

Consider a triangle, $T$, in $\mathbb{R}^3$ with vertices $(0,0,1), (0,1,0)$, and $(1,0,0)$. Let $S$ denote the sphere centered at the origin with radius 1 and let $S_1$ denote the portion of the ...
0
votes
2answers
103 views

How to find a plane that is tangent to 3 spheres?

So there are spheres with radius of 1 centered at (1,2,0), (4,5,0) and (1,3,2). How can one find a plane that is tangent to all 3 spheres? Visually, it looks like as if the spheres are sitting on a ...
-2
votes
1answer
89 views

Determining North-South Line Via Watch Method: Theory & Reason

I recently read that if you're in the northern hemisphere and have an analog watch, then you can point the hour hand at the sun and know that a south line lies between (bisection) the hour hand and ...
1
vote
1answer
68 views

Frechet mean of $k$ elements in the n-dimentional sphere.

The Frechet mean of $k$ elements $x_1, \ldots, x_k \in S = \{ x \in \mathbb{R}^n,\, \| x \| = 1 \}$ is defined as the $\text{arg}\underset{\|x\|=1}{\text{min}} \sum_{i=1}^n d^2(x_i,x)$. Where ...
2
votes
0answers
22 views

Area of a sphere bounded by hyperplanes

Say we have a sphere in d-dimensional space, and k hyperplanes (d-1 dimensional) all passing through the origin. Is there a way to calculate (or approximate) the area of the surface of the sphere ...
1
vote
1answer
84 views

Plotting a skew ellipse

I tried to make a geometers sketchpad toolkit for spherical geometry that is more exact (and easier to understand) than the present available one. but then I soon realised my background is not good ...
1
vote
2answers
61 views

Computing a double integral over a surface S, where S is the unit sphere,

$$ \int \int_S (x^2+y^2)d\sigma$$ Where S is the unit sphere centered at (0,0,0), and $\sigma$ is surface area. I arrived at the correct answer of $\large \frac{8\pi}{3}$, but I took an (educated?) ...
0
votes
1answer
47 views

Is this algorithm for 3D spherical interpolation correct?

I am attempting to write a spherical interpolation algorithm for for the application of smooth 3D animation in a game. The scripting language that the game engine uses is Lua. It is often easier for ...
1
vote
2answers
347 views

Why is the polar triangle useful in spherical geometry?

We can solve many problems in spherical geometry by using the polar triangle. I am looking for an intuition why (and when) this is easier than working in the original triangle.
2
votes
1answer
65 views

mean square displacement on the 3-sphere

I would like to compute the mean square displacement (MSD) for a particle moving on the surface of a 3-sphere of radius R. I see that I could eventually use the polar coordinates and get a polar ...
0
votes
0answers
25 views

lat/lon spherical coordinates to equidistant spherical coordinates

How to transform spherical data expressed in latitude/longitude pairs (parallels/meridians) in a new set of pair expressed just in parallels pairs? In other words, I need to transform data expressed ...
1
vote
0answers
82 views

Orthogonal transformation of a triangle on $S^2$

Let $v_1, v_2, v_3$ and $w_1, w_2, w_3$ denote the vertices of two spherical triangles $\bigtriangleup_1, \bigtriangleup_2$ with the property, that $\|v_i - v_j \| = \|w_i - w_j \|$, e.g. their ...
0
votes
0answers
25 views

The congruence of Two Spherical Triangles [duplicate]

I found in Wikipedea following claim : Two Spherical triangles with corresponding angles equal are congruent (i.e., all similar triangles are congruent). I know that the Area of Two triangles are ...
1
vote
1answer
79 views

Azimuth angle limit in Spherical co-ordinate system

In spherical co-ordinate system (r, θ, φ), θ can range from 0 to 2pi, but φ only varies from 0 to pi. Why is that?
3
votes
1answer
56 views

Direction of rotation to transform from Point a to b on a unit sphere

If I have two points $a$, $b$, on a unit sphere, I believe I can determine the angle between them, expressed as vectors, as follows: $$\theta = \arccos\left(\frac{a\cdot b}{\|a\| \|b\|}\right)$$ ...