Tagged Questions
-1
votes
0answers
37 views
Distance on the surface of a sphere
Given a sphere which radius is r.
There are two red points on the sphere. Given the location of the two points in spherical coordinate system.
If the surface distance between a point and a red point ...
0
votes
2answers
31 views
Ray-Lens Intersection
So imagine that I have a ray parameterized as $\vec{R} = \vec{O} + t\vec{D}$, where $\vec{O}$ = origin, $t$ = parameter and $\vec{D}$ = direction vector.
I also have a spherical lens with aperture ...
0
votes
1answer
27 views
Find position on surface of a lens
If I have a lens with coordinates UV on the lens surface where U, V are [-1, 1] and I want to find the real-world (x,y,z) coordinates of the UV point, how would I do that if I have the following ...
2
votes
1answer
55 views
Spherical geometry - relating angles of lunes and segments of great circles
Consider the picture below. I have a sphere of radius $r$, centered at $C$. The angle $\varphi$ is the dihedral angle between the plane defined by the shaded area and a plane through the indicated ...
0
votes
0answers
61 views
Help on non euclidean geometry
Hi guys I have a exam on this course tommorow and I was going through the practice problems and I ran across these problems. I dont know how to go about this problem, the questions are:
1) What are ...
1
vote
1answer
31 views
Help to find spherical Line
What is the spherical line through the points $(0,-1,0)$ and $\left(0,\frac{1}{2},\frac{\sqrt{3}}{2}\right)$?
I solved:
$G = \{(x,y,z)\in S^2 \mid \exists\ a,b,c \in \mathbb{R}, ax+by+cz = 0\}$
...
1
vote
2answers
67 views
Find an atlas of charts on $S^2$ with certain properties
Find an atlas of charts on $S^2$ for which each chart preserves area, and the transition functions relating charts have derivatives with determinant 1.
I have been thinking that I should consider the ...
2
votes
1answer
37 views
Geometry Question - Trihedral angles, planar geometry, spherical geometry
Two rays, $OX$ and $OY$, are drawn in the horizontal plane $\pi$, and the third ray, $OZ$, is drawn in space so that the rays $OX$, $OY$, and $OZ$ form a trihedral angle $OXYZ$. The planar angles ...
0
votes
1answer
47 views
Area of a geopolygon or polygon defined by longitude/latitude points
How would you go about finding the real area of a polygon which is defined by latitude/longitude points? Remember that the real area is different as a map is distorted towards the poles.
0
votes
1answer
25 views
The law of cosines for a sphere
$\cos(c) = \cos(a)\cos(b) + \sin(a)\sin(b)\cos(C)$
Prove that if $a$, $b$, and $c$ is approximately $0$, then $c^2 = a^2 + b^2 - 2ab~\cos(C)$.
I wasn't sure how to prove this. One thought I had was ...
1
vote
0answers
26 views
Are there 3D tilings of a 3D projective hyperplane or 3-sphere?
I noticed that pentagons tile the projective plane (a spherical dodecahedron). Something they do not do on a flat euclidean plane. Is there analogous 3D tilings (honeycombs) of a 3D projective ...
2
votes
2answers
225 views
Geometry of spherical triangle
Using the formula for the area of a spherical triangle, find and prove a formula relating the angle sum of a spherical polygon to its area
Thought:
Area (spherical triangle) ...
1
vote
1answer
57 views
Can someone identify this algorithm for great-circle distance?
The below is an algorithm used by the jscoord library to calculate the distance between 2 coordinates:
...
0
votes
0answers
65 views
Centre of a spherical triangle
Suppose I have a triangle defined by 3 unit vectors {$v_1, v_2, v_3$} in a 3 dimensional complex inner product space.
What would be the centre of such a triangle? I guess it should be something like ...
8
votes
1answer
149 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 ...
2
votes
3answers
153 views
Spherical projection
An image of a square is projected onto a sphere (radius $R$) as above (the dot is the centre of the sphere, and the red projection is marked out where the line from the centre-dot to a point on the ...
1
vote
1answer
85 views
Converting between spherical coordinate systems
Say I have the spherical coordinates of some locations, specifically their longitude (0 to 360) and latitude (latitude = 0 at equator, 90 at north pole, -90 at south pole) on a sphere with a radius of ...
2
votes
2answers
52 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 ...
1
vote
1answer
108 views
Graph Isomorphisms, Delaunay Triangulation on a sphere, and Kulikowski's Theorem
Suppose I have a collection of $n$ non-collinear points on a sphere, $\left\lbrace P_i\right\rbrace_{i=1}^n $. And I construct a mapping from this collection of points to the Delaunay Triangulation ...
1
vote
1answer
167 views
Moving points along a curve on sphere.
I have two points on a unit sphere. I also have their coordinates.
...
2
votes
1answer
901 views
Angle between GPS coordinates
I realize GPS Coordinates are spherical coordinates. However I know the earth is more of an ellipsoid. I need to compute with a fairly high degree of accuracy the pitch and yaw between two objects ...
0
votes
1answer
257 views
Area of a Spherical Triangle from Side Lengths
I am currently working on a proof involving finding bounds for the f-vector of a simplicial $3$-complex given an $n$-element point set in $\mathbb{E}^3$, and (for a reason I won't explain) am needing ...
5
votes
2answers
1k 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 ...
2
votes
2answers
220 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 ...
1
vote
1answer
195 views
radius around point
I have a question, I am trying to calculate a radius between two latitude and longitude points on the Google map.
I understand the coding part, but I do not understand the mathematical side to it. I ...
2
votes
2answers
78 views
angles between $n$ points on an $n-2$ dimensional sphere
$n$ points are placed on an $n-2$-sphere so that the smallest angle from the centre between any pair of the points is maximised. What is this smallest angle?
$n=1 \ \ \ \cos^{-1}{1}\\
n=2 \ \ \ ...
4
votes
1answer
480 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 ...
1
vote
0answers
127 views
Isomorphic triangles?
From my previous post I have learnt that spherical triangles can have different interior angle sums. Is this enough to argue that the triangles are not isomorphic? I am not sure how isomorphism works ...
0
votes
2answers
134 views
The globe, spherical disks and spherical straight lines
Does a spherical triangle with 2 equal sides necessarily have 2 base angles of size $\pi/2$? The reason I think this is that if we have a triangle $ABC$ and $AB=AC$ (in spherical distance), we could ...
1
vote
1answer
166 views
How to scale a polyhedron contained a 3-sphere?
In the 3-sphere simulator I am building, the viewpoint is contained in the space of a 3-sphere (the surface of a 4-D hypersphere), and the user is able to navigate through it.
There are some ...
6
votes
2answers
641 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 ...
1
vote
1answer
609 views
Transforming from one spherical coordinate system to another
I have a set of points on the surface of a sphere specified in one coordinate system (specifically, the equatorial coordinate system), and for each point I need to work on all its neighbouring points ...
2
votes
1answer
103 views
Ratio of geodesic segments on the sphere
Let $\mathbb{S}^2$ be the unit sphere. Let $0<\lambda<1$ be fixed. What is the smallest number $0<\mu<1$ (depending on $\lambda$) such that for any three points $A,B,C\in \mathbb{S}^2$, ...
3
votes
1answer
93 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 ...
1
vote
1answer
769 views
Maximum sum of angles in triangle in sphere
Recently my differential geometry lecturer demonstrated that the sum of the interior angles of a triangle in a sphere is not necessarily never $180^\circ$. This is one way to prove that the earth is ...
3
votes
5answers
329 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 ...
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 ...
1
vote
1answer
255 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
1answer
352 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
202 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.
...
8
votes
1answer
280 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 ...
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
213 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 ...
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 ...
6
votes
1answer
609 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?
