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

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2answers
162 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
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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?
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1answer
26 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
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0answers
31 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
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1answer
69 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? ...
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0answers
71 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 ...
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2answers
64 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 ...
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0answers
12 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 ...
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0answers
73 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. ...
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3answers
66 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
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1answer
24 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
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2answers
133 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 ...
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1answer
104 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
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1answer
73 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
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0answers
27 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
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1answer
122 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
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2answers
82 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
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1answer
65 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
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2answers
575 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
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1answer
80 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
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0answers
32 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 ...
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0answers
87 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 ...
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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 ...
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1answer
102 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
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1answer
67 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: $$\theta = \arccos\left(\frac{a\cdot b}{\|a\| \|b\|}\right)$$ ...
3
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2answers
90 views

Stereographic projection and lengths

We know the Stereographic Projection doesn't preserve areas except for the points on the plane such that $x^2+y^2=1$, because that's where $dA=dxdy$. I was wondering what would happen with lengths: ...
4
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1answer
54 views

constructing a spherical triangle using only the laws of sines and cosines

I have a spherical triangle with corners $A,B,C$, angles $\alpha, \beta, \gamma$ and sides $a,b,c$ (which are opposite to the corresponding corners/angles). I am given $a,b$ (with $a>b$) and ...
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1answer
58 views

Spherical Geometry and Playfair's Axiom.

Recently I came across a variant of the Parallel Postulate known as Playfair's Axiom: In Euclidean (planar) geometry there is at most one line that can be drawn parallel to another given one ...
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0answers
28 views

comparing models of spherical geometry

There more than two models of spherical geometry? 1) one for the half sphere (taking the north pole as center, the boundary is the equator) 2) one for the whole sphere (taking the north pole as ...
8
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1answer
146 views

Instruct geometer moths so you can learn about their true geometry.

I had a space-ship wreck in an unknown world of some kind of moths. I could observe geometer moths working. Everything looked strange. The moths claimed that they were using only straight edges and ...
2
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1answer
165 views

Tangent points on circle that placed on Earth surface

I need help about spherical geometry problem that I need to use it for my project. I try to calculate $T_1 $ and $T_2$ coordinates on $B$ centered small circle on the sphere and $AT_1$ and $AT_2$ ...
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1answer
34 views

How to find point on sphere from pitch and heading

I have a sphere of radius R and I would like to draw some vector positions on it given pitch and heading. I have a heading between 0 and 360 (0 being +x direction), and a pitch between -90 and 90 (90 ...
4
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2answers
140 views

Finding the northernmost latitude in a great circle that passes through two points on the sphere

I'm trying to solve the following problem from Smart's Text-Book on Spherical Astronomy (exercise 5 on p.23 of the 6th ed.): $A$ and $B$ are two places on the earth's surface with the same ...
2
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2answers
145 views

Finding the shortest distance to the north of the sphere

I found these problems in Alan F Beardon's Algebra and Geometry: Verify that any point with latitude α is a spherical distance R(π/2−α) from the north pole. Suppose that an aircraft flies on the ...
4
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1answer
1k views

How to calculate a solid angle (in Steradians) given only Horizontal Beam angle and Vertical Beam angle data.

I would like to convert a rectangular beam shape given in Horizontal and Vertical beam angle, into solid angle representing the surface area in steradians of projected light. For example a light ...
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0answers
131 views

How can I find the center of gravity of a hollow spherical cap?

I am looking to find the center of gravity for a hollow spherical cap. Could I use that point as the point at which the entire mass of the spherical cap is for newtonian gravity problems?
0
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1answer
101 views

Can the intersection of two balls be described?

Suppose, two spheres intersect. Subtracting the equations of the speheres, a linear equation appears which indicates the plane conataining all points belonging to the intersection of the spheres. But ...
4
votes
1answer
315 views

Circle from three points on surface of sphere

I need to compute the circle on the surface of a sphere given three points on that very surface. It is very easy to do that in Euclidean geometry, but the sphere has no x and y, but just two angles ...
4
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3answers
135 views

Construct the great circle (geodesic) in spherical or Riemanian geometry

Given: a circle $C$ with centre $M$ two points $P_1$ and $P_2$ inside circle $C$, so that $M$ is not on the line $P_1P_2$. Cunstruct an other circle $O$ so that: $P_1$ and $P_2$ are on ...
4
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2answers
141 views

Four circles touching one another on a spherical surface

The diagram above shows four identical circles, each having a flat radius $r$ (i.e. flat area $\pi r^2$), touching one another at six different points (i.e. each of four identical circles touches ...
1
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1answer
56 views

How to parametrize circles on a sphere by the distortion of the equator?

I guess am having a very silly problem right now. Considering a unit sphere $S^2$ and, for example, a curve, in spherical coordinates, $c(t)=(1, \frac{\pi}{2},t)$ that goes around the equator how can ...
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1answer
432 views

Point in a spherical triangle test

Given three latitude/longitude coordinates on a sphere forming a triangle, how do I test if a point p is inside that triangle? I know latitude and longitude implies Earth and Earth is not perfectly ...
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0answers
39 views

What is the mathematics behind the two animations?

I found two animated GIFs from a designer's website, which looks very impressive: My questions are: what is the mathematics behind them? How to obtain the mathematical formulas and equations of ...
0
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1answer
54 views

Spherical and Hyperbolic lines in the Extended Complex Plane.

We work in the Extended Complex Plane: $ \mathbb{C} \cup (\infty)$. Basically, say we have two points, $z_1$ and $z_2$. It can be shown that, on stereographic projection of the Riemann Sphere onto ...
2
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1answer
324 views

Area of Spherical Polygon

It appears to me that after repeated applications of Girard's theorem on the area of spherical triangles that we can obtain the surface area of a spherical polygon with interior angles ...
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0answers
98 views

On the equidistant distribution of $n$ points on a sphere $S^2$ by algorithm and their “validity” measures by statistical methods

I have found an algorithm for distributing $n$ points $P_0, P_1, ..., P_n$ (approximately) equidstantly on a sphere where $$\varphi_i = \pi(\phi - 1)i \qquad \theta_i= \mathrm {asin} (2i/n - 1), ...
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1answer
63 views

Stereographic Projection from an Arbitrary Point

Let $p \in \mathbb{S}^{n}$, then the stereogaphic projection is a diffeomorpshim $h:\mathbb{S}^{n} \setminus \{p\} \to \mathbb{R}^{n-1}$. Suppose that $p$ is the 'north pole' ($p = (0,0,..,1)$), then ...
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2answers
224 views

find the equation of a sphere with endpoints A and B where B is the point of tangency of the sphere and the plane

Find the equation of a sphere with a diameter that has endpoints $A(1, 8, −2)$ and $B$, where $B$ is the point of tangency of the sphere with the plane $−9x +6y + 2z = 2$. Now i know that i can get ...
5
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2answers
289 views

Proof for spherical polar law of cosine

I'm reading my textbook and for some reason, it does not present the proof for the spherical polar law of cosine which is: $$ \cos(a)=\frac{\cos(A)+\cos(B)\cos(C)}{\sin(B) \sin(C)}$$ It does present ...
2
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1answer
99 views

The locus of points forming a right angle, in nonzero curvature

Given a line segment $AB$ in the Euclidean plane, the locus of points which form a right angle with $A$ and $B$ is known to be a circle, with $AB$ as a diameter. Is this also true for a geodesic ...