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

6
votes
1answer
1k 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?
4
votes
3answers
5k views

Area of a spherical triangle

Consider a spherical triangle with vertices $A, B$ and $C$, respectively. How to determine its area? I know the formula: $A = E R^2$, where $R$ is radius of sphere, and $E$ is the excess ...
9
votes
1answer
499 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 ...
5
votes
4answers
2k 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 ...
4
votes
2answers
56 views

Finding the northernmost latitude in a great circle that passes through two points on the sphere

I'm trying to solve the following problem from Smart's Text-Book on Spherical Astronomy (exercise 5 on p.23 of the 6th ed.): $A$ and $B$ are two places on the earth's surface with the same ...
1
vote
2answers
2k views

How to find the intersection of three spheres (full solutions)?

The three equations of spheres are given $(x-x_{1})^2+(y-y_{1})^2+(z-z_{1})^2=a^2$ $(x-x_{2})^2+(y-y_{2})^2+(z-z_{2})^2=b^2$ $(x-x_{3})^2+(y-y_{3})^2+(z-z_{3})^2=c^2$ How do I find $(x,y,z)$ ...
2
votes
1answer
3k 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
2answers
441 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. ...
8
votes
4answers
287 views

How to calculate the area covered by any spherical rectangle?

Is there any analytic or generalized formula to calculate area covered by any rectangle having length $l$ & width $b$ each as a great circle arc on a sphere with a radius $R$? Note: Spherical ...
6
votes
2answers
314 views

Elementary arguments concerning the stereographic projection

How does one give a proof that is short; and strictly within the bounds of secondary-school geometry that the stereographic projection is conformal; and maps circles to circles?
3
votes
4answers
2k views

Intersection of two arcs on sphere

I have two arcs on a sphere that are defined as pair of points: (θ₀, φ₀), (θ₁, φ₁). I need to find a point where they intersect, or some indication if they don't. What is important is that they are ...
3
votes
1answer
144 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 ...
3
votes
5answers
460 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 ...
3
votes
2answers
251 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 ...
2
votes
2answers
766 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 ...
0
votes
0answers
75 views

Spherical coordinates and the Law of Cosines

I have one question on my project. I am assuming earth is a perfect sphere. How can I get from the Law of Cosines $$\cos(c)=\cos(\operatorname{lat} A)\cos(\operatorname{lat} ...
3
votes
2answers
280 views

Number of reflection symmetries of a basketball

Excerpt from John Horton Conway, The Symmetries of Things, pg. 12. Basketballs have two planes of reflective symmetry, as do tennis balls. I read this sentence and it immediately struck me as ...
2
votes
1answer
135 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 $A,B,C\in \mathbb{S}^2$, ...
2
votes
1answer
3k 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 ...
1
vote
1answer
177 views

Project point onto line in Latitude/Longitude

Given line AB made from two Latitude/Longitude co-ordinates, and point C, how can I calculate the position of D, which is C projected onto D. Diagram:
1
vote
1answer
269 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 ...
1
vote
2answers
214 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
1k 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' \ ...
1
vote
1answer
159 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 ...
0
votes
1answer
154 views

Determine if one point lies between two other points on a sphere

My question is rather simple. Can I use the dot product to determine if a coordinate lies between two others? With coordinates I mean a Point P(latitude, longitude) on the surface of the sphere. I ...
0
votes
0answers
127 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 $\psi$ ...
0
votes
1answer
651 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 ...
0
votes
2answers
204 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 ...