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
249 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 ...
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0answers
114 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
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1answer
2k 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
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1answer
125 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
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1answer
81 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
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0answers
113 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
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2answers
481 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
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2answers
556 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
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1answer
506 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
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1answer
277 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
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1answer
139 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
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1answer
848 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
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0answers
446 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' ...
0
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1answer
119 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
259 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
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0answers
184 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
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0answers
65 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
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1answer
251 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
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0answers
155 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
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1answer
316 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 ...
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0answers
171 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
230 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
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3answers
411 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
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1answer
282 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
82 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
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1answer
140 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 ...
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1answer
127 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
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1answer
389 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
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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
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0answers
123 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
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1answer
214 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
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0answers
123 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 ...
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0answers
215 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
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1answer
116 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
247 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
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1answer
455 views

Moving points along a curve on sphere.

I have two points on a unit sphere. I also have their coordinates. ...
3
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1answer
331 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
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1answer
2k 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
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1answer
266 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
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1answer
59 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
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1answer
569 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
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0answers
41 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
568 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 ...
7
votes
2answers
5k 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
387 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
901 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 ...
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1answer
253 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
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1answer
154 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
193 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 ...
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0answers
384 views

How to project a spherical map onto a sphere / cube

I have this panorama, an spherical map from google streetview, and want to map this on a sphere/cube. Below are some examples and illustrations, i am going to implement it in c++ and are not sure ...