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

# Questions tagged [spherical-geometry]

791 questions

**11**

votes

**7**answers

### The vertices of a tetrahedron lie on a sphere

I am struggling a bit with the following (elementary) question:
How to prove that every regular tetrahedron admits a circumsphere, i.e. there exist a sphere on which all four vertices lie.
I would like to find a slick elegant proof, which is…

Asaf Shachar

- 23,159
- 5
- 19
- 106

**11**

votes

**2**answers

### Intersection of circle and geodesic segment on sphere

I am trying to find an efficient way of computing the intersection point(s) of a circle and line segment on a spherical surface.
Say you have a sphere of radius R. On the surface of this sphere are
a circle with center ($\theta_c$,$\phi_c$) and…

qsfzy

- 121
- 6

**11**

votes

**1**answer

### Probability a random spherical triangle has area $> \pi$

From Michigan State University's Herzog contest:
Problem 6, 1981
Three points are taken at random on a unit sphere. What is the probability that the area of the spherical triangle exceeds the area of a great circle?
I assume we always take the…

aschepler

- 4,754
- 14
- 22

**11**

votes

**4**answers

### Packing regular tetrahedra of edge length 1 with a vertex at the origin in a unit sphere

I'm currently venturing through Paul Sally's Fundamentals of Mathematical Analysis. This is an unusual textbook in terms of the difficulty of exercises. I've already been stunned by the very first one:
How many regular tetrahedra of edge length 1…

YuiTo Cheng

- 4,337
- 18
- 23
- 57

**11**

votes

**4**answers

### 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 illustration, the points A, B and X lies on a spherical…

brainless

- 211
- 2
- 5

**11**

votes

**2**answers

### Distance between two points on a sphere.

Say there is a sphere on which there is an ant and the ant wants to go to another point. The ant can't definitely travel through the sphere. So it has to travel along a curve. My question is what is the least distance between the two points i.e.…

Abhijith S. Raj

- 323
- 2
- 3
- 15

**10**

votes

**3**answers

### 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 angle of $(a + b + c - \pi)$, but how to determine the…

Rick C. Hodgin

**10**

votes

**5**answers

### How to construct three mutually orthogonal circles in stereographic projection?

I'm new to spherical geometry and I enjoy doing ruler-and-compass constructions, so I'm trying to teach myself to do them in stereographic projection. I'm finding it challenging, to put it mildly.
The following picture shows three mutually…

helveticat

- 972
- 5
- 15

**10**

votes

**4**answers

### Intersection of two arcs on sphere

I have two arcs on a sphere that are defined as pair of points: $(\theta_0, \varphi_0)$, $(\theta_1, \varphi_1)$. I need to find a point where they intersect, or some indication if they don't. What is important is that they are arcs, not circles, so…

Andrey

- 245
- 2
- 8

**10**

votes

**2**answers

### 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 geometry, the angle sum of a triangle is always less…

Ryan

- 1,591
- 4
- 11
- 18

**10**

votes

**1**answer

### 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 hypersphere (also called a 3-sphere). Thus, in seeing it…

lvella

- 705
- 3
- 13

**10**

votes

**1**answer

### Equation of a great circle passing through two points

I've searched everywhere for something to help me with this problem, but I can't find anything. What I want to calculate is the midpoint between two locations (latitude and longitude) on a sphere. The midpoint must lie on the shortest path between…

Hex4869

- 485
- 1
- 4
- 14

**10**

votes

**7**answers

### 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 spherical surface with a radius $R$? i.e. How to find the area $A$ of rectangle in terms of length…

Harish Chandra Rajpoot

- 36,524
- 70
- 73
- 111

**9**

votes

**3**answers

### What length would the sides of a triangle over Earth's surface be for the sum of its angles to be 180.1°

For simplicity's sake, let the Earth be a perfect sphere. Imagine you are drawing an equilateral triangle over its surface. How long should its sides be, for the sum of its angles to be 180.1 degrees?

srgb

- 203
- 1
- 6

**9**

votes

**3**answers

### The area of a right spherical triangle

Is there a compact formula for the area (excess angle – assuming a unit sphere) of a right spherical triangle given its side lengths $a$ and $b$?
As explained in an answer to an earlier question about the area of a generic spherical triangle, the…

S.G.

- 347
- 2
- 7