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

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12
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1answer
228 views

Relation between area of a triangle on a sphere and plane

We know area of a plane triangle $\Delta=\sqrt{s(s-a)(s-b)(s-c)}$ where $s=\frac{a+b+c}{2}$. I was just thinking: let we have a triangle with arc length $a,b,c$ on a sphere of radius $r$, do we have ...
11
votes
1answer
578 views

What's the name of a parabola mapped onto a sphere?

It seems that an 'arc' is a line-segment mapped onto the surface of a sphere (although I don't know if that name still holds if the segment wraps around the sphere more than once, i.e., if the angle ...
10
votes
5answers
327 views

How to construct three mutually orthogonal circles in stereographic projection?

I'm new to spherical geometry and I enjoy doing ruler-and-compass constructions, so I'm trying to teach myself to do them in stereographic projection. I'm finding it challenging, to put it mildly. ...
10
votes
1answer
161 views

Are there exotic symplectic structures on $ S^2 $?

Besides the obvoius symplectic structure on $ S^2$ given by the area element in the standard embedding $ S^2 \to \Bbb R^3$, are there any other closed 2-forms on $ S^2$ which produce nonisomorphic ...
9
votes
1answer
382 views

Navigating though the surface of a hypersphere in a computer game

People in StackOverflow seems not so into this theme, so I thought I could have better luck in here. I had the idea of an spaceship game where the world is confined in the surface of an 4-D ...
8
votes
1answer
231 views

Circle on sphere

Foreword This question was inspired by initial mistakes in this question. I wanted to explore the strange circle with $A>\pi r^2$ and got lost into geometrical jungle. A spherical cap is usually ...
7
votes
1answer
388 views

Spherical geometry: Arbitrary point between two points

If A and B are two points on the earth, how could I find any arbitrary point between them along the shortest distance side of their great circle path? Points are in radians longitude = 0 to 2pi ...
6
votes
2answers
2k views

Is an equilateral triangle the same as an equiangular triangle, in any geometry?

I have heard of both equilateral triangles and equiangular triangles. (For example, this sporcle quiz lists both.) Are these always equivalent, regardless of geometry? I know they are the same in ...
6
votes
2answers
4k views

Why does every direction at the north pole point south?

Why does every direction at the north pole point south? Why doesn't this happen at any other point on (face of the) earth? Is this due to convention used by humans or is there a geometrical ...
6
votes
2answers
191 views

Map Earth surface so straight line distance is great circle distance?

Is there a function $f:$ (latitude, longitude) $\longrightarrow \mathbb{R}^n$ (for any finite $n$) such that the linear distance between $f(x)$ and $f(y)$ is the great circle distance between $x$ ...
6
votes
1answer
864 views

How to calculate a heading on the earths surface?

Given an initial position and a subsequent position, each given by latitude and longitude in the WGS-84 system. How do you determine the heading in degrees clockwise from true north of movement?
6
votes
3answers
700 views

What is the area of the portion of 1/8 of an sphere cut off by two parallel planes?

So the problem that I'm trying to solve is as follows: Assume 1/8 of a sphere with radius $r$ whose center is at the origin (for example the 1/8 which is in $R^{+}$). Now two parallel planes are ...
6
votes
3answers
415 views

The unsolved mathematical light beam problem

I have the following problem: Imagine that you have a sphere sitting at the interface of two media(like water and oil). And the position(the heigth) of the interface to the center of the sphere is ...
6
votes
2answers
840 views

How do I map a spherical triangle to a plane triangle?

My goal here is to make my own custom "polyhedral map" of Earth. If you print out something from the "Map Fold-outs" page, you will have something almost exactly like what I'm trying to make. I have ...
5
votes
4answers
1k views

How to find the distance between a point and line joining two points on a sphere?

How do I calculate the distance between the line joining the two points on a spherical surface and another point on same surface? I have illustrated my problem in the image below. In the above ...
5
votes
1answer
1k views

Deriving the Surface Area of a Spherical Triangle

A triangle on a sphere is composed of points $A$, $B$ and $C$. The $\alpha$, $\beta$ and $\gamma$ denote the angles at the corresponding points of the triangle: The Girard's theorem states that the ...
5
votes
2answers
287 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?
5
votes
1answer
223 views

spherical triangle inequality for multiple edges

Consider a spherical triangle, with side lengths $0 \leq a,b,c \leq \pi$. In texts on spherical geometry we see proofs that $a \leq b + c$ but this inequality is unnecessarily weak, in that some sets ...
5
votes
2answers
4k views

Proof that the angle sum of a triangle is always greater than 180 degrees in elliptic geometry

I've scoured the internet and have found many proofs showing that in Euclidean geometry, the angle sum of a triangle is always 180 degrees. I've also found many proofs showing that in hyperbolic ...
5
votes
0answers
66 views

Volume of a spherical tetrahedron

In the paper Jun Murakami, The volume formulas for a spherical tetrahedron a formula for the volume of a spherical tetrahedron is given. I am trying to work through the details for the specific ...
5
votes
3answers
193 views

How to understand and create quaternions?

I have to multiply two quaternions to calculate a so called spherical linear interpolation between two $R^3$ coordinate systems within the interval $t = [0, 1]$. I understand how to do the ...
4
votes
1answer
176 views

Why do derivatives of certain equations relating to circles yield other similar equations? [duplicate]

Possible Duplicate: Why is the derivative of a circle's area its perimeter (and similarly for spheres)? We all know that the volume of a sphere is: $V = \frac{4}{3}\pi r^{3}$ and its ...
4
votes
1answer
205 views

Ratio of Circumference to Diameter on a sphere

I was listening to an audiobook of Einstein when they started discussing spherical geometry and how Pi was no longer the ratio of a circle's circumference to its diameter, so I set out to find the ...
4
votes
0answers
70 views

How I cut my orange - spherical volume integral

I cut my orange in six eatable pieces, following some rules. My orange is a perfect sphere, and there is a cylindrical volume down through my orange, that is not eatable. In the diagram, the orange ...
4
votes
2answers
198 views

Algorithm to generate an uniform distribution of points in the volume of an hypersphere/on the surface of an hypersphere.

I am searching two simple/efficient/generic algorithms to generate a uniform distribution of random points: in the volume of a n-dimensional hypersphere on the surface of a n-dimensional hypersphere ...
4
votes
1answer
497 views

approximating geodesic distances on the sphere by euclidean distances of a transformed sphere

Is there a way to find a function $F:\mathbb S^2 \rightarrow \mathbb R^3$ of class $C^1$, minimizing $$\int_{\mathbb S^2\times\mathbb S^2}(d(F(x),F(y))−\delta(x,y))^2 dx dy$$ , where $d$ stands for ...
3
votes
3answers
4k views

Area of a spherical triangle

A triangle on a sphere is comprised of points A, B and C. How to determine its area? I know the formula: A = E * R^2, where R is radius of sphere, and E is the excess angle of (a + b + c - pi), but ...
3
votes
5answers
412 views

How do you parameterize a sphere so that there are “6 faces”?

I'm trying to parameterize a sphere so it has 6 faces of equal area, like this: But this is the closest I can get (simply jumping $\frac{\pi}{2}$ in $\phi$ azimuth angle for each "slice"). I ...
3
votes
3answers
522 views

Article or book about the history of spherical geometry?

I teach a course on non-Euclidean geometry to high schoolers. I'm looking for an article or book that gives a thorough and interesting history of spherical geometry and trigonometry. I'm looking for ...
3
votes
2answers
99 views

IS ASA applicable on triangles on the sphere?

$ASA= \text{Angle-Side-Angle}$ I was wondering if $ASA$ still worked on triangles for the sphere. I have a pretty hard time visualizing triangles on the sphere because I know the sum of their ...
3
votes
2answers
133 views

How do I apply a digital filter to points on a sphere

Given a set of points on a sphere, how can I implement a higher order low pass filter on them? At the moment, I am just multiplying the vectors from the input and output set by their weights and ...
3
votes
2answers
3k views

Napier's Rules applied to spherical distance calculations

I was in the middle of writing the same old geographic distance calculation using the Haversine formula when it occurred to me: shouldn't there be simpler way to do this? Haversine is of course ...
3
votes
2answers
116 views

Given a latitude how many miles is the corresponding longitude?

OK so lines of longitude (the distance/circumference around the earth horizontally) differ based on what latitude you are at (0 at north and south poles up to ~25k at the equator.) So given a ...
3
votes
2answers
216 views

Volume of a sphere using geometry

How to derive the formula for the volume of a sphere using geometry? $V = (4/3) \pi r^3$ Edit: How did Archimedes calculate the volume of a sphere? Integration wouldn't have been there at his ...
3
votes
2answers
697 views

How to calculate the area of a circle ( given: origin, radius ) on a sphere ( Earth )?

I know that the Earth isn't a sphere, not even an ellipsoid, but for my measurements, its an acceptable approximation. Assuming I have a coordinate(lat,lon) and a distance( e.g.: 1000km ), what is the ...
3
votes
2answers
388 views

Coordinates and distance in higher dimensional spherical and hyperbolic space

For n-dimensional spherical space, it seems to me the representation of points is easiest and most manipulable as unit vectors, with distance being the vector dot product (which is the cosine of the ...
3
votes
2answers
234 views

What is the average rotation angle needed to change the color of a sphere?

A sphere is painted in black and white. We are looking in the direction of the center of the sphere and see, in the direction of our vision, a point with a given color. When the sphere is rotated, at ...
3
votes
1answer
116 views

A property of Hilbert sphere

Let $X$ be (Edit: a closed convex subset of ) the unit sphere $Y=\{x\in \ell^2: \|x\|=1\}$ in $\ell^2$ with the great circle (geodesic) metric. (Edit: Suppose the diameter of $X$ is less than ...
3
votes
1answer
54 views

Energy of particles on sphere and uniform rotation

I have a computer program containing some particles on a unit sphere, characterized by their positions $\{(\theta_i,\phi_i)\}$. They have a total energy given by the arc distance between particle ...
3
votes
1answer
111 views

problem or doubt regarding visualizing angles of spherical triangle

I must confess that I am not able to visualize or understand what is the angle of a spherical triangle say $ABC$ where $A,B,C$ are vertices of the triangle which is formed by intersection of three ...
3
votes
1answer
409 views

Geodesics on a 2-sphere

I've been doing some work where I need to find the geodesics in a given Riemannian Manifold. Let's take the example of the two sphere, for simplicity, with unitary radius. The distance between two ...
3
votes
1answer
133 views

Integration on a sphere

I have an integral at hand which has the form of $$I = \int_{u\in \mathbb{S}^2} f(\mathbf{u}\cdot \mathbf{s}_1) f(\mathbf{u}\cdot \mathbf{s}_2) d\mathbf{u}$$ where $\mathbb{S}^2$ is the unit sphere ...
3
votes
1answer
377 views

How to construct the midpoint in spherical geometry?

I am looking for the the method of constructing the midpoint of two points in spherical geometry. The only tools allowed for the construction are a pair of spherical compasses and a spherical ruler. ...
3
votes
2answers
37 views
+100

How to work out miles between Longitude values based on a Latitude value.

We know that when Latitude is 0, the distance between Longitude values is roughly 69 miles. When the Latitude is +/-90, Longitude values are 0 miles. At 0 Latitude, the earths circumference is ...
3
votes
1answer
54 views

What is the difference between the sphere and projective space?

I know about the antipodal mapping. What I want to know is what the most significant differences between the sphere and projective space are, and how to think of each of them and their relationship ...
3
votes
1answer
135 views

Laplacian on Sphere of Function Only Depending on Angle Between Points

Consider a function $f:S^2 \to \mathbb{R}$ , with $S^2$ the unit $2$-sphere in $\mathbb{R}^3$. Let's say that $f$ depends only on the polar angle $\theta$ from the north pole (e.g., $f(r,\theta,\phi) ...
3
votes
2answers
76 views

What's wrong with an irregular digon?

I recently found that there were some things that could be said about the digon, the polygon with 2 vertices and 2 edges; in particular, the Wikipedia article notes that “in spherical geometry a ...
3
votes
1answer
254 views

Formula for the fourth side of a spherical quadrilateral

Given two sides $a,b$ of a spherical triangle and the angle $C$ between them, the spherical law of cosines gives an elegant formula for the missing edge length $c$: $$\cos c = \cos a \cos b+ \sin a ...
3
votes
0answers
35 views

Ratio of the Volume of n-spherical cap to the volume of n-sphere

Assume an n-dimensional sphere with radius $R$ and volume $V^{(n)}_s$. Also assume a corresponding n-spherical cap with height $h$ and volume $V^{(n)}_c$. what is the ratio of two volumes? ...
3
votes
0answers
317 views

General Formula for Volume of Spherical Triangle

I'm working on a problem for which I'm trying to divide a sphere into layers defined by integer radius values ($r\in{1, 2, ...}$) such that the segments in each layer all have the same volume. Doing ...