geometry as on the surface of a sphere, where "lines" are great circles and any pair of lines must intersect
1
vote
0answers
74 views
Analytical calculation of the resulting surface between a sphere and a spherical cap from another sphere
Let's say I have one spherical cap, resulting from cutting a sphere centered at origin and with radius R1 with a plane, whose normal goes into the direction of the ...
3
votes
2answers
88 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 ...
2
votes
2answers
260 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 ...
2
votes
0answers
138 views
How to calculate rotated global location coordinates (Long, Lat)?
Given the current global location coordinate system:
-180 <-> 180 Longitude
-90 <-> 90 Latitude
A rotation of the globe 90 degrees counter-clockwise around the Y-axis would bring the north ...
0
votes
1answer
125 views
Numerical software to solve partial differential equations in spherical coordinates?
Which numerical libraries / math software can allow me to solve partial differential equations in spherical coordinates? (my system consists of N degrees of freedom, each degree lives in ...
1
vote
1answer
173 views
How to use a Rhumb Line?
I am new to working with coordinate data and figured out the equation I am looking for is the Rhumb Line. I went to go research it and found a lot of equations and I still have no idea where to start. ...
0
votes
1answer
48 views
Spheres and converting formulas
If the volume V of a sphere with radius r is V=(4/3)πr^3. If the surface area is s=4πr^2, how can I express the volume as a function of the surface area S? My first thought was to set them equal to ...
0
votes
1answer
203 views
I have equations for getting x,y,z given latitude, longitude, and altitude. How do I reverse them?
I am using equations that look like the following to get x, y, and z given latitude, longitude, and altitude.
...
1
vote
1answer
257 views
How to calculate the number of latitude/longitude coordinates in a specific area at a given precision?
I am looking into ways of grouping large sets of latitude/longitude coordinates and am wondering if there is an easy/standard way to roughly calculate the number of different coordinates in a given ...
4
votes
3answers
354 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 ...
0
votes
2answers
411 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 ...
1
vote
1answer
240 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, ...
2
votes
1answer
289 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 ...
2
votes
1answer
284 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 ...
4
votes
1answer
353 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 ...
6
votes
3answers
524 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 ...
3
votes
2answers
203 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 ...
1
vote
2answers
179 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.
...
1
vote
1answer
362 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
votes
0answers
287 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 ...
11
votes
1answer
441 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 ...
5
votes
2answers
2k 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 ...
2
votes
2answers
140 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 ...
0
votes
1answer
255 views
Using the spherical law of cosines
Compute angular length c of the great-circle route between these two cities:
Daytona Beach (location A) 29º12'N., 81º1' W.
Sidi Ifni (location B) 29º23' N. 10º10' W.
Ok so I converted the ...
8
votes
1answer
281 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 ...
4
votes
4answers
761 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 ...
1
vote
4answers
338 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
3
votes
1answer
295 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.
...
0
votes
1answer
215 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 ...
1
vote
1answer
122 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
votes
3answers
2k 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 ...
7
votes
1answer
335 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
...
3
votes
2answers
2k 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 ...
5
votes
2answers
605 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
votes
1answer
611 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?
