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

learn more… | top users | synonyms

2
votes
1answer
669 views

compute minimum distance between point and great arc on sphere

Suppose I have a point $P$ on a unit sphere whose spherical coordinates are $(\theta, \varphi)$, and a great arc from point $Q$ to point $R$, also specified in spherical coordinates. I want to find ...
12
votes
1answer
276 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 ...
1
vote
0answers
133 views

to find volume of spherical tetrahedron and triangle area

I have two question though they are different in some way Could any one tell me how to find area of spherical triangle in a easiest way? How to find the volume of Spherical tetrahedron! which is in ...
1
vote
1answer
3k views

Spherical Geometry (SG) Vs Euclidean Geometry(EG)

Could any one tell me what are the fundamental contrasts with postulates of Euclidean Geometry and Spherical Geometry? I myself see these things, please tell me if there are more: Lines in EG are ...
3
votes
1answer
146 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 ...
0
votes
1answer
84 views

How to minimize the length of a curve on $S^2$

The length of a curve $\gamma$ starting from a point $p$ and ending at another point $q$ on $S^2$ is given by the formula $$l_{\gamma}(S^2)=\int_{0}^{1}\sqrt{(d\phi/dt)^2+ \sin^2\phi ...
0
votes
0answers
123 views

two points on a unit sphere

Consider the two vectors to the points on the unit sphere, $${\bf v}_i=(\sin\theta_i\cos\varphi_i,\sin\theta_i\sin\varphi_i,\cos\theta_i)$$ with $i=1,2$. Use the dot product to get the angle $\psi$ ...
2
votes
3answers
601 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) ...
2
votes
2answers
738 views

Discretize a circle on a sphere with a given center and radius

I would like to draw a discretized circle on the surface of a sphere (the Earth in that case). The input of the algorithm would be the center of the circle (expressed as longitude and latitude), the ...
3
votes
1answer
588 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 ...
0
votes
1answer
315 views

Given an arc length and an angle, how do I get a sphere coordinate?

Assuming I start at the top of a sphere and am given the radius of the sphere, an angle to turn, and a distance to walk along the sphere, how could I find my destination in the sphere coordinate ...
1
vote
1answer
147 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: ...
2
votes
1answer
1k views

how to calculate distance from a given latitude and longitude on the earth to a specific geostationary satellite

As the title suggests, I would like to know how to calculate the straight-line distance from a given latitude+longitude point on the earth to a given satellite in the geostationary belt. Perhaps a ...
1
vote
0answers
508 views

Generating evenly distributed points on a sphere

How could I write an algorithm to generate n points distributed 'evenly' on a sphere? I already wrote an algorithm to generate points distributed uniformly on the surface (here), but by 'evenly' ...
1
vote
1answer
136 views

Fréchet mean of the spherical shape space

The Fréchet mean of a general subspace is defined as $$F(x)=\int_M dist(x,y)^2d\mu(y),$$ where $\mu$ is the probability measure on a general metric space $(M,dist)$. I think the sample mean of ...
1
vote
1answer
283 views

Distance measurement between latitude/longiture pairs.

I need to calculate the distance between two lat/lng coordinate pairs. In addition, If given an initial lat/lng coordinate, angle of travel, and distance, I need to calculate the resulting lat/lng ...
1
vote
0answers
213 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 ...
2
votes
0answers
66 views

Average distance from point to spherical square?

I need to calculate the average distance from a point to a $4$ sided spherical polygon. Can someone point me to the right direction? I guess either the average point of a spherical square or centroid ...
8
votes
1answer
258 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
0answers
176 views

calculating the area on the surface os a sphere created by intersection of two spherical caps!

Consider a spherical object composed of two compartments (A and B, not necessarily hemispheres) sitting at the interface which is characterized by a plane separating 1 and 2. For this case, ...
2
votes
1answer
378 views

Finding the area of a spherical triangle

I am asked to calculate the area of a spherical triangle of points $(0,0,1),(\frac{1}{\sqrt2},0, \frac{1}{\sqrt2})$ and $(0,1,0)$. I know I will have to use Gauss Bonnet formula , after having found ...
1
vote
0answers
182 views

Is there a general equation expressing the area of a spherical quadrilateral using two polar angles, an azimuth, and a radius?

Is there a general equation expressing the area of a spherical quadrilateral using two polar angles, $\theta_1$ and $\theta_2$, an azimuth, $\phi$, and a radius, $r$? I know this can be done by ...
3
votes
2answers
244 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 ...
2
votes
3answers
513 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
339 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 ...
3
votes
2answers
88 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
142 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 ...
1
vote
1answer
131 views

Angles spherical triangles

In the paper 'Examples of spherical tilings by congruent quadrangles' by Ueno and Agaoka, I came across the following claim (p.142): the sum of two angles in a spherical triangle is less than the sum ...
0
votes
1answer
417 views

Uniform distribution of points on the surface of a circle around a randomly chosen point

In a Monte Carlo simulation i have encountered the following problem: given a unity vector u defining a point A on the surface of a unity sphere, i must randomly determine a new vector forming an ...
0
votes
3answers
2k views

Surface integral on a sphere using spherical coordinates

I have a question involving surface integral on a unit sphere. Suppose $s_1$ and $s_2$ are two points on a unit sphere with spherical coordinates $(\theta_1, \psi_1)$ and $(\theta_2, \psi_2)$, ...
1
vote
0answers
139 views

Name and properties of spherical polygon with small-circle sides

Just as the title says: is there a formal name for a convex polygon on a sphere, of which the vertices are connected not by great circle but by small circle segments? My end goal is to intersect two ...
1
vote
1answer
245 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
0answers
137 views

Find third geographic coordinate in triangle using spherical earth model

I'm trying to solve triangulation problems using geographic coordinates from a GPS. all calulations must use the spherical earth model (great circle distance). Given the points and lengths: Point A: N ...
1
vote
0answers
234 views

Apollonius Pursuit Problem

Most references in a brief search under 'Apollonius' concern tangent circles. The problem I am interested in is the Apollonius pursuit problem. In the plane, the question concerns the point at which ...
2
votes
1answer
124 views

Real numbers mapped onto a sphere

We can compare real values if they were greater, lesser but we cannot do same for complex numbers. What if we map real values(within some small range) onto a sphere and declare each one of them as ...
5
votes
1answer
304 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 ...
2
votes
1answer
546 views

Moving points along a curve on sphere.

I have two points on a unit sphere. I also have their coordinates. ...
3
votes
1answer
365 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 ...
2
votes
1answer
3k 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 ...
1
vote
1answer
273 views

dotting gradient in spherical coordinates with displacement vector

The gradient in spherical coordinates is given by: $\nabla f = \left(\frac{\partial f}{\partial r}, \frac{1}{r} \frac{\partial f}{\partial \theta}, \frac{1}{r \sin \theta} \frac{\partial f}{\partial ...
0
votes
1answer
65 views

Cardinality of Mappings on a 2-Sphere

I am wondering about the number of mappings from a point on a sphere to a neighboring point, and a not so neighboring point. If I take a 2-sphere, and place it on some $x,y,z$-axis and fix those so ...
1
vote
1answer
669 views

Coordinates of interception point Y with XY being the shortest distance of X to AB on sphere

How would one calculate the interception point $Y$ with $\overleftrightarrow{XY}$ being the shortest distance of $X$ to $\overleftrightarrow{AB}$? This answer to the question How to find the ...
3
votes
0answers
46 views

Internal angle of a vertex of degree $d$ in $\mathbb{E}^2$ and $\mathbb{S}^2$

I am currently working on determining the maximum number of times the minimum spherical distance can occur among $n$ points in $\mathbb{S}^2$, and I have the following question. In $\mathbb{E}^2$, ...
0
votes
1answer
640 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 ...
8
votes
2answers
7k 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 ...
0
votes
1answer
428 views

Integrate linear function over spherical triangle

In the plane I can integrate a linear function over a triangle as the average of the corner values times the area of the triangle. Then if I subdivide the triangle into 4 triangles by introducing a ...
3
votes
2answers
1k 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
267 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 ...
1
vote
1answer
165 views

Travelling Salesman Problem on the unit sphere

Let vertices be defined on the unit sphere by its longitudes $\lambda_i$ and latitudes $\phi_i$. The distance between $i$ and $j$ is defined as follows: $\arccos(\sin(\lambda_i) \sin(\lambda_j) ...
4
votes
1answer
198 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 ...