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

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550 views

Vertices of intersection between N spheres

Just wanted to know what is the best algorithm (in terms of speed and accuracy) to determine the intersection of N spheres (in 3D). With intersection I mean the following; in 2D and in the case of two ...
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1answer
351 views

Trying to pick a random point on sphere end up picking from a lune

I was inspired by this question to play around a little bit (its a weekend). I was pretty confident of my derivation and thought it might be nice to supplement it with a pretty picture. However, ...
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1answer
525 views

Arc length of a great circle which is the hypotenuse of an isoceles right triangle on the sphere

I am doing a problem which requires me to find the arclength of the hypotenuse of an isosceles right triangle. (The book calls it a 2 Dimensional Sphere but I hope that is a typo) I start at the ...
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1answer
393 views

quaternion representation of the rotation of a sphere into plane displacement

I do have a sphere of known radius which does have a coordinate frame rigidly attached to it. Let's call the coordinate frame attached to the sphere XYZs. The sphere can be rotated and displaced ...
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1answer
493 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 ...
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3answers
696 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 ...
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2answers
229 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 ...
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2answers
206 views

Calculating probabilities on a spherical map

A black and white colored sphere is given. We are looking at a random starting point on the sphere below us, which has a certain color. A random rotation can change the color of the spot below us. ...
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1answer
683 views

Formula for the coordinate of the midpoint in spherical coordinate system

Please let me know the formula for the coordinate of the midpoint of 2 points in spherical coordinate system . If possible , I want the answer includes the exact formula as , midpoint = point1 + ( ...
2
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0answers
366 views

Geographic coordinates to pitch+yaw+pitch

I'm creating a very basic simulation which involves air travel across the world and am trying to correctly position and orient my aircraft in a rendered 3D representation. I am representing the Earth ...
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1answer
577 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 ...
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2answers
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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 ...
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2answers
186 views

Simplest form for locus of latitudes/longitudes equidistant from two given latitudes/longitudes?

Given two latitudes/longitudes (th1,ph1 and th2,ph2), I want to find a simple formula for the locus of th3,ph3 that are equidistant from th1,ph1 and th2,ph2. Mathematica happily spits out an answer ...
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1answer
645 views

Using the spherical law of cosines

Compute angular length c of the great-circle route between these two cities: Daytona Beach (location A): $29^\circ12'\ N, 81^\circ1' \ W$. Sidi Ifni (location B): $29^\circ23' \ N. 10^\circ10' \ ...
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1answer
377 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 ...
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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 ...
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4answers
384 views

Distance between points on a face

Given a 2D picture of a face, how is it possible to measure the distance between two different points on the surface of the face? Thanks Joel
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1answer
374 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. ...
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1answer
242 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 ...
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1answer
137 views

How to test whether spherical caps intersect?

I have a unit sphere, on the surface of which are defined spherical caps. I typically characterize the caps by the unit vector $n$ from the center of the sphere to the top of the cap, and the angle ...
3
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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 ...
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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 ...
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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 ...
6
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2answers
839 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 ...
6
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1answer
856 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?